![Chapter 1
Introduction
Yeliz Karaca1
and Dumitru Baleanu2,3
1
University of Massachusetts Medical School, Worcester, MA, United States; 2
Çankaya University, Ankara, Turkey; 3
Institute of Space Science,
Magurele, Bucharest, Romania
Complexity, having existed since antiquity, entails the
understanding of the complex components’ origin, with
meticulous computations and causal processes. Nonline-
arity, self-organization, adaptation, synchronization, noise,
a high number of descriptive variables or dimensions
involved in the description of differential equation systems,
and reaction to responses in the external environment are
some of the numerous characteristics of a complex system
in which multiple interactions emerge. Along these lines,
complexity thinking and theory, one of the basic premises
of which is the acknowledging of the existence of a hidden
order to the behavior and evolution of complex systems,
requires a horizon that takes the subtle and hidden prop-
erties of different domains into account, necessitating their
own means of optimized solutions and applicability.
Bearing in mind the quote by Stephen Hawking: I think the
next [21st] century will be the century of complexity is
critically significant not only for this era but also for on-
wards. Accordingly, the idea of complexity is stated to be
part of a new unifying framework for science and a revo-
lution in our understanding of systems the behavior of
which has proved to be difficult in terms of prediction,
management and control.
In a complex system, different and multiple ways need
to be contemplated for the provision of solutions and
sorting out the problems. The system is likely to change
depending on these selections, which shows us the complex
systems’ adaptiveness. And, the more insight is developed,
the answers to the problems keep changing which enables
more learning in the process. Given this, modern science
has embarked on the attempts for a thorough, holistic,
multifaceted and accurate interpretation of natural and
physical phenomena, which has proven to provide suc-
cessful models for the analysis of complex systems and
harnessing of control over the various related processes.
Computational complexity, in this regard, comes to the
foreground by providing the applicable sets of ideas and/or
integrative paradigms to recognize and understand the
intricate properties and dynamics of many different com-
plex systems. The lenses of such transformative thinking in
conjunction with mathematics-informed frameworks
encompass chaos, fractal and multi-fractional ways as well
as the indispensable incorporation of technology, with
Artificial Intelligence, as a far-reaching leg, which are all
essentially required to be capable of addressing and tack-
ling complexity manifesting chaotic, nonlinear, and dy-
namic characteristics.
Chaos refers to irregular and unpredictable behavior
characterized by sensitive reliance on initial conditions. The
tendency of nature toward pattern formation, iteration and
creation of order out of chaos all point to the generation of
expectations of predictability. Chaos and its study in con-
sort with the advances in scientific realm are important
roots of modern study of complex systems that display
dynamic, nonlinear, open qualities and interconnection
with the environment constituting many interacting com-
ponents, with new unanticipated patterns emerging. Chaos,
in this context, is said to have somehow strict definitions
portraying a nonlinear world, addressing deterministic
systems with trajectories diverging exponentially in time,
which is also among the properties of behaviors in complex
systems. In mathematics and physics, chaos theory is
concerned with the nonlinear dynamical systems’ behavior,
which under certain circumstances exhibits a phenomenon
referred to as chaos marked by sensitivity to initial
conditions.
Fractals are also components of dynamic systems, being
the images thereof, driven by recursion, which is to say the
image of chaos. Accordingly, fractals are used for modeling
structures where patterns recur repeatedly and describe
random or chaotic phenomena. For the handling of
complex systems, the concept of progressive smoothness
on finer scales may not always prove to be useful as a
starting point from mathematical point of view. This
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![Chapter 2
Theory of complexity, origin and
complex systems
Yeliz Karaca
University of Massachusetts Medical School, Worcester, MA, United States
1. Introduction
Complexity, as an idea and scientific concept that has
existed since antiquity, requires the understanding of the
origin of complex components, entails lengthy and metic-
ulous computations as well as causal processes. Multiple
interactions emerge among the components in a complex
system whose research address characteristics like adapta-
tion, self-organization, noise, synchronization, high number
of descriptive variables or dimensions, nonlinearity in
description of differential equation systems and reaction to
responses in the external environment. Given that, the
characterization, definition, analysis and understanding of
complex systems encompass powerful relation between
variables and discrimination by noisy nondeterministic
phenomena, sensitivity to initial control as well as strange,
nonperiodic and unpredictable time evolution. As physicist
Nigel Goldenfeld put very aptly, “Complexity starts when
causality breaks down.” Having structure with variations
and far-reaching conditions like spontaneous order,
nonlinearity, feedback, robustness, lack of central control,
numerosity, hierarchical organization and emergence are
considered, complexity reveals many deep layers involved
in complexity. The large number of independent interacting
components and multiple pathways by which the complex
system can evolve further indicate some of the reasons why
causality breaks down as complexity starts. Along these
lines, causality is relative, being prone to fundamental
variations depending on perception, external factors, the
environment, space, time and so on. Time is a flying and
flowing phenomenon; and if we, as the human agents, are
the pilots of the time which is on an infinitive continuum,
the decision-making processes need to be prompt aside
from being efficient and robust. It would not be possible to
proceed along one way on this continuum since constancy
causes nonstationary and steady state; for this reason,
multifarious and integrative way of thinking and methods
unifying the elements of different disciplines would address
the different parameters of complex systems and their
related problems with optimized solutions. Multifarious,
meaning the possession of many varied parts and aspects as
well as happening in a great variety, in fact, can help
identify the solutions of the complex problems taking into
account nearly all possible parameters of complex systems.
To this end, it is necessary to identify the optimal model
and by providing cross-validations, multifarious methods
consider the fact that each complex problem has a different
nature and the solution to each problem needs custom-
ization. Complexity along with all the variations in net-
works demonstrates that our decisions are not based on
only one single parameter, but on multiple numbers of
parameters with hidden information being at stake as well.
Multifarious adaptive methods within mathematics-
informed frameworks come into prominence for the solu-
tion of complex problems, which enable that the solution is
not superficial or pretentious but reliable, robust and
smooth ensuring the maintenance of quality, sustainability
and meritocracy. Time is not only comprised of de-
notations, it is beyond the traditional thinking of science,
with representations and reflections on life, which apply to
all domains of life. Evolutionary processes, nonlinearity,
and all the other dimensions of complexity rest on time,
reveal time and occur within time. In the ever-changing
current landscape and variations, with causality breaking
down, Stephen Hawking’s quote, “I think the next [21st]
century will be the century of complexity” is critically
significant. The idea of complexity is sometimes stated to
be part of a new unifying framework for science and a
revolution in our understanding of systems the behavior of
which has proved difficult to predict and control thus far.
The goal of complexity science is to achieve a global un-
derstanding by considering the multiple interacting factors
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![of systems, many branches of possible states and high-
dimensional manifolds while keeping abreast of actuality
along the historical and evolutionary path, which itself has
also been through different critical points on the manifold.
Changing or removing the causes will not necessarily
change or remove the outcomes, and thus, modern scientific
way of thinking is geared toward the development of
models that benefit from local computations, interpolate
task-related manifolds in spaces with high dimensions
instead of just learning the rules or representations of the
world. Similar to evolutionary processes, models that are
overparameterized can be thrifty in terms of providing
applicable, versatile and robust solutions so that multifar-
ious set of functions can be learned and optimized out-
comes can be achieved. These applicable models and way
of thinking from which the models are derived pose broad
challenges to different areas of science if they rely on just
theoretical assumptions.
Evolution, order and complexity reveal the relationship
between natural and social worlds, which reflects a modern
way of thinking that definitely challenges the dichotomy
between the natural and the social. This study, accordingly,
provides a conceptual outline and historical account of
complexity and complex systems along with complex order
processes toward modern scientific path from Darwin and
onwards with a focus directed toward natural, applied and
social sciences.
The capacity of managing the modern societies ulti-
mately rests on a communication network that runs effec-
tively. Just like the neural nets of biological brains, those
networks determine the learning capability, which will
eventually help with survival. The dynamics of information
technologies in the framework of complex systems need to
be modeled within their natural, social, economic and
cultural environments, so as to say informational and
computational ecologies [1]. Under such complexity,
fundamental principles of complex thinking, such as being
holistic, human-oriented, and transdisciplinary, are all
dependent on the concepts of modern theory of evolution
and self-organization of complex systems. Along these
lines, nonlinearity of evolution, chaos, space-time elements
and complexity emerge as important aspects, which is very
well reflected by the quote of Ilya Prigogine: “Our vision of
nature is undergoing a radical change toward the multiple,
the temporal, and the complex” [2]. Aside from these no-
tions mentioned, the following concepts also come along
with complexity: openness, nonlinearity, chaos, self-
organization and synergetics. When something complex is
handled and defined, then all the other aspects including its
nature, structure, evolution as well as its principles should
be considered, which all point toward the definition of
modern science of complexity. In essence, the science of
complex systems has provided us with the conceptual and
methodological tools so that issues of evolution, self-
organization, emergence, and transformation can be
tackled and mechanisms of micro and macro levels can be
explained in terms of their behaviors over time [3].
Mechanistic thinking posits that the universe is under-
standable, analytical method is the only way of research,
and causality explains everything in the world. In contrast,
complexity way of thinking is totally different, stating that
different matters are interconnected and interaction shows
nonlinearity and noncausal determinism. Besides having
the self-generation and self-organization properties, this
strand of thinking includes uncertainty, noncontinuity,
inseparability and unpredictability [4]. Consequently, the-
ory of complexity enables us to gain powerful evidence and
elucidation to challenge traditional and mechanistic
thinking so that humanity can adopt a new way of thinking.
Complexity thinking requires a horizon that takes into
account the subtle properties of different domains, which
require their own means of solutions and applicability. In
neurological system complexity, “evolvability” is concerned
with the species owing their existence to the capability of
their ancestors with regard to evolving and adapting. Another
important point has to do with the correlation between the
complexity of brain design and optimality. The progress
made in the neurosciences has shown the complexity of even
very simple nervous systems, and complexity is manifested
in their structure, function, coding schemes used to represent
information as well as in their evolutionary history. These
viewpoints are of critical importance in the future science of
brain complexity as well [5].
The present and future science of complexity relies on
application aspects for optimal development and strategy
generation to solve the complex problems. To cite relevant
works, [6] introduced a measure, named as neural
complexity, to capture the relationship of two aspects of
brain organization, which are anatomy and physiology. The
complexity measure is applied in computer simulations
regarding the cortical parts of the brain so that the basic
principles of neuroanatomical organization limiting brain
dynamics can be examined. That approach is said to be
useful for the analysis of complexity in other areas related
to biology like embryogenesis and gene regulation as well.
Another work [7] is on the resting-state functional magnetic
resonance imaging (rs-fMRI) to record and analyze the
brain’s neural activity. Noting that different regions of the
brain show different levels of complexity, the study ex-
amines individuals, correlated entropy/complexity changes,
concluding that there are important differences at the level
of subject when there is a memory task which is linked to
performance opposed to being at rest. Finally, the study by
[8] employs functional magnetic resonance imaging to
explore the functional networks that underlie cognitive
reasoning in humans with an anatomical abnormality,
called corpus callosum dysgenesis. The findings of the
study demonstrate that resting-state functional brain
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![network that supports cognitive control is preserved
partially in people with corpus callosum dysgenesis.
Medicine and healthcare systems also display highly
complex properties, particularly during uncertain situations
where promptness of response, accuracy of diagnosis, drug
regimens, interactions between the related parties and so
forth are of vital importance. Furthermore, humans them-
selves are complex, considering their mind, spirit, body and
social contexts. Healthcare settings’ complexity adds
further elements to the situation like the organizational
infrastructure and multi-faceted dynamics there.
Complexity thinking and science can assist all persons
involved in health issues so that simplistic linear thinking
can be avoided and multi-dimensional processes related to
patient care and healthcare service organizations can be
considered within a nonlinear and complex framework.
Consequently, in the study [9], it is stated that complex
systems approaches use a collection of advanced analytical
and methodological tools to be involved in innovative
theory testing and development. The exploration of
dynamically complex systems is done in virtual labora-
tories, which would not be possible to realize by traditional
methodologies. With regard to neuroimaging, the study
[10] illustrates the basic properties characterizing complex
systems while evaluating the way they relate to brain
structure and function based on the experiments of neuro-
imaging. Neural systems are reviewed from the complexity
science perspective by matching the complex systems’
properties with evidence obtained from contemporary
neuroscience. On the topic of law, complexity and medi-
cine, the study [11] discusses the strategic decision-making
in healthcare system by systems methodology based on the
fact that society is a complex system. The models devel-
oped in the frame of systems dynamics as well as the other
means are presented in the related study with a focus on the
importance of feedback loops. As can be seen, complexity
has its implications with other theoretical assumptions,
applications and methods in the field of neurology and
medicine [12e23].
Being staggeringly complex, biological systems
including cells, organisms and ecosystems at large cannot
be analyzed by abstracting them from the whole. Extending
across space, biological networks go across the dimension
ranging from the microscopic scale of intracellular orga-
nization to the global scale of planet ecology. It is
acknowledged that integrated system of complexity along
the evolutionary path is a significant feature of living sys-
tems. Thus, it can be stated that everything biological needs
to be viewed in an evolutionary framework. In addition,
one example of the reason why reductionist approach
would fail is related to the function of one cell in a multi-
cellular organism. Even though one understands the func-
tion of the cell, that would not necessarily mean that the
organism’s physiology can be understood completely [24].
As each cell’s activity would be affected by that of other
cells in the organs, tissues and organ systems within the
organism itself abstraction and isolation would not be of
help. Rather complexity thinking would show that inte-
gration into the system enables us to understand that system
has emergent properties and should be interpreted from that
vision. If we are to note some of the works where
complexity and biology are addressed, it can be said that
the study by [25] is on complexity measures for the com-
parison of genomic characteristics. The authors demon-
strate that the fluxes, block entropy content, conditional
probability and exit distance distributions can be utilized as
markers, which can help the discrimination between
eukaryotic and prokaryotic DNA so that in many cases
details about finer classes can be discerned. The complex
measures handled in the study highlight the discerning
power of genomic observables for the prediction of the
evolutionary position of DNA sequence even if its origin is
not known. Another study [26] handles degeneracy, which
is a ubiquitous biological characteristic related to genetic
code and immune systems. Degeneracy is stated to be
required for and result of natural selection. Deeper under-
standing of the way degenerate systems become associated
and synchronized across levels is also said to be an
important challenge to deal with in modern evolutionary
biology. [27] is a study which extends self-organizing
approach for a bacterial genome for the purpose of
analyzing the raw sequencing of human data. The authors
indicate that metagenomics allows for the genomic study of
uncultured microorganisms, and more economical and
faster technologies can help the sequencing of uncultured
microbes, which are sampled directly from their habitats
can provide transformation related to the view for the mi-
crobial world. Finally, the authors of the paper [28]
developed a notion for a model to understand stage tran-
sition systems through hierarchical coordinate systems.
Through this way, an algebraic definition for biological
systems’ complexity is handled along with the comparison
with genome size and number of cell types. The complexity
measure of the authors is said to be unique for maximal
complexity to fulfill a natural set of axioms, which also
shows a strong connection between hierarchical complexity
in biological systems and global semigroup theory, which
is one area of algebra. Biological complexity has its
extensions in other studies with different theoretical
assumptions, models and applications [29e33].
Engineering systems’ uncertainty increases with the
exponentially growing complexity of the system, which
makes tolerance to the uncertainty an essential factor. En-
gineering processes involve many facets and stages, so
complexity is indispensable for engineering as well. [34] is
a study concerned with risk assessment of complex systems
in a supervised dynamic probability manner. The method
developed with this vision is said to improve execution
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![time of dynamic probabilistic risk assessment models, and
the optimization model is employed for the generation of
failure scenarios apart from the comparison of appropriate
optimization solution algorithms. The aim of the method
used in the study is that operators can monitor the risk level
of all probable failure scenarios in real time while helping
with the better decisions in situations that require emer-
gency, which are all properties of complex systems.
Regarding health and engineering, the work [35] proposes a
nonlinear data fusion method for composite health indicator
and derives the reliabililty measures in a computationally
efficient way. Another study [36] is on the integration
environment for engineering and science, which requires
multidisciplinary efforts from a multitude of specializations
working in harmony. For accuracy and efficient use of time,
the authors present a remote component environment to
allow the users to integrate disciplinary tools. The study is
important in terms of displaying the complex factors at play
during engineering processes like design and analysis. With
regard to extreme events, which require interdisciplinary
interaction, the study [37] reviews the existing approaches
for the definitions of extreme events along with the case
study that emphasizes the intricate properties in the defi-
nition of extreme events. Since addressed in a broad variety
of disciplines like climatology, mathematics, meteorology
as well as social sciences, extreme events due to their im-
pacts also require a complex systems understanding.
Complexity and engineering have provided studies with
different perspectives [38e41].
Being adaptive and responsive to changing circum-
stances, science is an evolutionary process itself, which also
applies the dynamics within the complex social systems.
The call for the adoption of complexity theory has also
been the case for social sciences so that it would be possible
to get away from reductionist frameworks. In addition,
more recent connectionist methods are sought to address
complexity and open social systems in a better way. The
study [42] provides the differentiation between general
systems theory (GST) and complexity theory along with the
benefits of them for social sciences. Complexity theory is
identified as a theory that can provide a new perspective
and also a novel method of theorizing as well as addressing
complexity by linking it with advances in technology,
markets, globalization, cultural changes and all the other
related future challenges as well as opportunities consid-
ering today’s complex problems. The paper [43] provides a
new definition of social complexity based on the number of
differentiated relationships individuals have. This definition
is argued to be an objective and cognitive one, stemming
from the view that social complexity is used widely but
measured poorly. The authors, who review previous defi-
nitions, posit that the number o differentiated relationships
is a flexible measure as well. The paper [44] handles social
complexity around the study of animal and human
societies. This author of this work also states that the
concept is defined and understood in a poor way. The
definitions for vertebrate and invertebrate societies are
reviewed with a critical outlook; terms like social structure,
social organization, care system and mating system are
defined and characterized along with the provision of an
outline of the different aspects related to evolution of social
complexity. The work [45] is on complexity and archeol-
ogy, which has complex elements like social roles, eco-
nomic roles, big permanent settlements, large populations
of people and other marker criteria. It is also maintained by
the author that computational and systems dynamics
modeling can provide the systematic study of complex
adaptive systems. A small-scale society is given as a
computational model to exemplify the complexity of even
simple societies, which shows that complexity is regardless
of size or organizational structure. New modeling methods
are also said to be of assistance to archeologists in their
studies. A much older work [46], on the examination of
complexity concept in sociology, shows that complexity
has been discussed for quite a long time period The author
discusses and shows that the complexity of social data are
no more complex than other sorts of natural phenomena.
The work [47] is on the complexity applications in lan-
guage and communication sciences, and the authors point
out the interdisciplinary approaches for human sciences.
For instance, the paradigm of complexity in sociology, the
reason and the way to model the complexity of thought
systems, the impact of social reputation in language
evolution, to name some aspects in the relevant fields. The
diversity of social science lends itself to the use of many
varied theoretical stances and methods within the
framework of complexity in different studies [48e52].
One of the basic premises of complexity theory is that
there exists a hidden order to the behavior and evolution of
complex systems. And that system can be an ecosystem, an
organization or economy of a country. Along this line of
thinking, complex adaptive systems are stated to be
revising and rearranging their building blocks in a constant
way based on their experience. Examples include pro-
motions for some employees in businesses and concluding
new trading agreements for countries. Complex adaptive
systems, including ecosystems, brains and economies are
governed by anticipation and feedback from the environ-
ment based on which models are improved. Their shared
properties are nonlinearity, aggregation, diversity and
flowing [53]. Moreover, there is a coherent system behavior
generated by cooperation and competition between actors
in complexity theory, which is another important governing
concept. Informed by natural and mathematics sciences,
complex systems approach is also stated to be a useful tool
for political economists to focus on the dynamic adapta-
tions and their nonlinear implications at different levels of a
social organization [54]. The survey [55] provides a
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![discussion of behavioral and experimental macroeconomics
through the emphasis on complex systems perspective.
Heterogenous expectations and heuristics switching models
also match the micro and macro behaviors in economics as
well. The work [56] is on the generalization of model of
economic growth with constant pace considering the im-
plications of memory. Fractional differential equations and
their solutions are obtained for the description of the output
dynamics brought about by the changes of net investments
and effects of power-law fading memory. The application
of fractional calculus for mathematical economics is
addressed in [57] together with the recent mathematical and
conceptual developments in the field focusing on memory
and nonlocality. Economic complexity has also relationship
with regard to phenomena related to climate; accordingly,
the study [58] posits that economic complexity leads to
intensity of greenhouse emission, which highlights that the
knowledge situated in the productive structure of complex
economies includes the knowledge needed for cleaner
technologies of production. Aside from business, including
finance and other fields that strive for the solution of the
complex problems, related studies have been carried out
[59e63].
Taken together, complexity, as an idea and scientific
concept, has been in existence since antiquity and discussed
along different origin-related, historical, philosophical and
epistemological viewpoints as addressed in this study
[53,64e69]. Complexity requires the understanding of
interacting components in the complex systems, along with
lengthy and meticulous computations as well as causal
processes for the solution of different complex problems in
different systems [18,21,23e53,63e69]. Different from the
previous works, this study attempts to encompass all
possible dimensions of complex systems in different fields,
not disregarding varying origin-related, historical and
epistemological viewpoints. Accordingly, the goal of pre-
senting the brief history of complexity in this study is
intended toward a global understanding of complexity,
taking into account various multiple interacting factors of
systems, many branches of possible states and high-
dimensional manifolds while keeping up with reality and
actuality along the historical and evolutionary path, which
itself, has also been through different critical points on the
path. Subsequently, the presentation of complex order
processes toward modern scientific path aims at under-
standing the conditions and requirements to handle com-
plex problems of the 21st century and onwards. The reason
for handling these details, in the parlance of complex sys-
tems, is for the sake of modern way of thinking depending
on the transition from evolutionary dimension as a revo-
lutionary stage, as a new paradigm for natural sciences and
social sciences so that the foundation for the complex
systems’ interpretations can be explored by related areas.
Succinctly, accounts of past, present and future in different
complex systems help a deeper understanding and guidance
along the way.
The rest of this study is structured as follows. Section 2
provides the A Brief History of Complexity and the Related
Areas of Different Complex Systems with the Section 2.1.,
which is entitled “Theories Pertaining to Complexity and
Their Historical Account”. Section 3 addresses Complex
Order Processes Toward Modern Scientific Path: From
Darwin and Onwards by providing Section 3.1 entitled “A
Conceptual Outline: Complexity and Complex Systems.”
Last but not least, Section 4 of the study provides the
concluding remarks and future directions.
2. Theory of complexity, origin and
complex systems
Complexity theory shows that units in large populations
can be involved in self-organization process, producing
patterns, storing information and engaging in collective
behaviors. In natural landscapes, natural patterns are
derived from nonlinear processes with properties that are
modified; and nonlinearity, in this regard, is usually viewed
to be one of the essential elements related to complexity
[70]. In social systems, complexity refers to the amount of
data or information required to be able to completely
describe the system which exhibits complex features.
Nonphysical systems like social structures are said to have
similarity in terms of behavior, character, or rules to be
conformed to; and within such settings, social hierarchies
can combine into systems getting ever larger via the
mechanism of representation [70]. Most of the things
related to the behavior of social systems refer to the inter-
action of its members instead of the individuality of those
members; and each social system exhibits specific charac-
teristics that may remain although all of its individual
members are replaced [71,72].
2.1 A brief history of complexity and the
related areas of different complex systems
Having existed as a term since antiquity, complexity as an
idea and scientific concept gained its explicit definition in
the late 1980s although its presence in mathematical sys-
tems had already been noted by Henri Poincaré in regard to
three-body problem. The 1920s saw the quantification of
complexity of simple mathematical formulae for statistical
models, whereas biological, social systems as well as other
systems were characterized by high complexity during the
1940s. A huge number of components that had different
types and behaviors with interconnections and in-
terdependencies used to be the common definition terms. A
decade later, complexity started to be handled in pure
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![mathematics, followed by information theory and the DNA
structure discovery, which underlined the information
content of complexity. The content of algorithmic infor-
mation came into existence in the 1960s and this pointed
toward the shaping of the definition of complexity.
Different areas used several other definitions in the 1960
and 1970s when computational complexity theory started to
head for a varied direction, with complexity defined in
terms of resources required to carry out the computational
tasks. Afterward, the 1980s saw the growth of research in
complex systems, with a focus placed on numerical mea-
sure of complexity [69]. Another related development is the
creation of the Santa Fe Institute whose founders wanted to
react to the specialization and reductionism in science, and
thus, to enhance the development of the science of complex
systems, in other words, complexity science [53,64e68].
The notion of complexity in different areas has many
dimensions. To start with, mathematics is concerned with
the study of arbitrarily general abstract systems and a very
high level of complexity in the behavior of many systems
which have rules that are actually simpler than the rules of
most systems in traditional sense. It can be said that the
traditional mathematical approach to science has contrib-
uted to physics, as another area, and it is nearly acknowl-
edged universally that physical theory is to be based on
mathematical equations. In theoretical physics, existing
methods are around continuous numbers and calculus,
probability as well at some times. Nevertheless, a greater
simplicity in that structure yields the identification of new
phenomena. In computer science, computational systems
established to carry out specific tasks have been the focal
point, and within this purpose, even the simplest con-
struction is capable of yielding a behavior that is
immensely complex. Computational ideas, in this sense,
can include all kinds of core questions regarding mathe-
matics and nature. As another field, biology encompasses
vast and profound details about living organisms and bio-
logical elements, with evolution by natural selection being
one of the most classical realms thereof since general
observations on living systems are customarily analyzed
based on evolutionary history instead of abstract theories.
Social sciences, varying from psychology to economics,
philosophy and sociology, also offer complexity with the
ever changing, adapting, and evolving features over time as
a function of people’s preferences and attitudes. Although
physical sciences require the formulation of solid theories
in terms of equations and numbers, social complexity
reflects behaviors of humans as ongoing and broader as a
result of complicated conditions of individual and group
existence through many different arrangements, patterns
and movements. For philosophy, on the other hand, issues
regarding the universe and the role of human beings
therein, besides the uniqueness of humans’ conditions, limit
to knowledge and the inevitable position of mathematics
are positioned at the core [69]. Engineering, as another
discipline, has its obvious association with complexity,
which also shows that even simple underlying rules can be
put into practice to carry out a sophisticated task. This
means construction of a system with complicated basic
rules is not always required in engineering. This is because
for the design and operation of engineering systems, the
aim is to reduce complexity so that the system can be
rendered robust, which assures long-term stability, system
reliability and cost minimization [39]. Rather than the
classical approach of “dividing and conquering,”
complexity engineering tackles adaptive, self-managing,
self-organizing and emergent features [73]. The important
changes in the foundations of complex and sophisticated
technology yield the application for purposes related to
humans, which enables the imagination of a whole new sort
of technology that can attain the same sophistication as
nature whose essential mechanism can be captured by rules
even in simple programs.
2.2 Theories pertaining to complexity and
their historical account
The investigation of studying complexity theory on its own
as a separate phenomenon dates back to the early years of
the 1980s upon the suggestion of Wolfram [69]. Afterward,
its popularity grew in time and theory has enabled the
development of the fundamental understanding of the
complexity as a general phenomenon as well as its origins.
Further, computational complexity theory, developed in the
1970s, intends to characterize the extent of difficulty
regarding certain computation tasks. Starting about a cen-
tury ago as a branch of mathematics, dynamical systems
theory concerns itself with the investigation of systems that
evolve over time in line with specific sorts of equations,
and mathematical and geometrical methods for the char-
acterization of the probable forms of behavior that can be
produced by related complex systems. Accordingly, the
field of fractal geometry, unlike conventional science and
mathematics, which deal with regular and smooth kinds of
shapes, starting in the late 1970s, underlined the signifi-
cance of embedded and nested shapes, which include
arbitrarily intricate pieces common in nature, since many
systems produce shapes that are inexplicably complex.
Dynamic system theory, on the other hand, stems intel-
lectually from mathematics, physics, meteorology, astron-
omy and biology explains the developments as the
probabilistic consequence of the processes’ interactions at
different multiple levels, with interactions of multiple
factors as well as systems on different levels and timescales
[74,75]. Relatedly, chaos theory is reliant on the observa-
tion of specific mathematical systems, which display
behaviors depending on the details of initial conditions.
This was spotted at the end of the 19th century and became
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![prominent following the computer simulation works of the
1960 and 1970s afterward.
One of the most contentious theories is evolution the-
ory, the Darwinian part of which with natural selection is
usually assumed to explain the complexity observed in
biological systems. The theory, which has no clear-cut lines
about why it should imply that complexity has generated,
has had applications outside biology as well in recent years
[69]. Darwin saw the path to the evolution of complexity;
individuals’ varied traits were also seen in each generation;
and new variations’ emergence and spread came to mean
the production of complex structures. While some of the
variations contributed to the increase in their survival, some
others enabled them to produce more offspring. On the
other hand, some scholars have recently suggested that
complexity can emerge through other routes. McShea and
Brandon claim there is a more basic biological law than
natural selection, which is diversity and complexity
increasing in evolutionary systems. The authors, including
Gould, Kauffman, and Lineweaver et al. also put forth that
complexity, increasing even without the action of natural
selection, refers to the number of parts of the amount of
differentiation between parts in an individual and it is
important that we do not consider only the number of parts
that constitute living things, but also look at the types of
those parts. Correspondingly, Kauffman [65] explains that
complexity emerges without natural selection to help it
along [76,77]. This argument, also famously put by Richard
Dawkins as “the blind watchmaker” postulates that
complexity is not the outcome of fine-tuning through nat-
ural selection which took millions of years. In other words,
it just happens [78]. The schematic depiction of evolution
process with complexity over time can be interpreted as the
shape of the tree of life. This suggests that there is a cu-
mulative type of increase in complexity, which provides
evolution with a direction and an arrow of time, as a
concept of asymmetry of time developed by the British
astrophysicist and mathematician Arthur Eddington in 1927
[79,80]. Entropy is another related concept in this context,
which is one of the few quantities in physical sciences that
require arrow of time. The entropy of an isolated system is
known to increase, not decrease, and the increase of dis-
order, or entropy, is what enables one to distinguish the past
from the future [66] (for further views on complexity and
how it differs in terms of its increasing quality, see [77]).
3. Complex order processes toward
modern scientific path: from Darwin
and onwards
The interaction of selection and self-organization is an
important theme in developmental and evolutionary
biology, which should include Darwin in a broader context.
Both simple and complex systems can manifest powerful
self-organization and spontaneous order is at stake for
natural selection, which is very aptly incorporated by
evolutionary theory. As for the adaptation processes, while
some systems can adapt readily, others may be exposed to
disruptions by minor modifications, which make the
adaptive improvement by selection and random mutation
occur hardly. Darwin was of the assumption that that kind
of an improvement was viable and the complex systems’
adaptation capability, as shown one century later, made us
be aware of the construction requirements to permit adap-
tation [65].
The first captivating view of Darwin is that natural se-
lection and branching tree of life spread from major phyla
to minor genera as well as species, reaching out to humans.
Although being enchanting, Darwin’s answer to the sources
of order appeals only to one force, that is to say natural
selection. Hence the reason why the view is considered to
be inadequate, since it is thought that the view does not
integrate the possibility that simple and complex systems
show spontaneous emergence of order. That’s why, the
mingling of self-ordering with Darwin’s evolution-natural
selection mechanism produces much unfolding and makes
the order we see sensible albeit the complexity of organ-
isms. The current stage of science has become so extraor-
dinary that while molecular biology leads us to the inmost
ultimate mechanisms, complexity and evolution capacity of
cells, studies carried out in mathematics, biology, chemis-
try, physics and so forth expose the powers of self-
organization [65]. These far-reaching effects help us
explore the sources of order and understand the order
inherent in complexity, which means that complexity is
associated with self-organization and spontaneity; and
natural selection is not the only reason thereof. An inte-
gration of this knowledge with Darwin’s basic insight is
recommended so that self-organization and selection
themes can be combined and evolutionary theory can be
expanded to stand on a broader structure, which is stated to
have three tiers according to Kauffman (1993): first of all, a
delineation is necessary for the spontaneous sources of
order as well as the simple and complex systems’ self-
organized properties. Secondly, it is noted that self-
ordered properties both enable and limit the efficacy of
natural selection. For this reason, organisms need to be
viewed in a different and new perspective with a striking of
balance and collaboration. Natural selection molds the or-
der which already exists, so selection is not the only source
of order in organisms, yet it needs to be acknowledged
somehow. Finally, adaptation capacity of the complex
living systems should be understood. In Darwin’s view,
accumulation of mutations was probable; however, the
capacity of doing that is not very clear, since some systems
cannot adapt at all. Thus, the fact that selection enables the
organisms adapting successfully needs to be considered
Theory of complexity, origin and complex systems Chapter | 2 15](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-38-320.jpg)
![carefully, which leads us to the concept of adaptation
capacity in a co-evolutionary process with organisms
whose selection operates on complex co-evolving systems.
As a universal process and dynamics, evolution leads to
diverse phenomenology of life and its theory brings about
rich phenomenology of life on Earth for modern biology
having been subject to modifications in terms of its nature
over the years. Although it is recognized that process of
evolutionary change does not necessarily cause more
complex organisms, evolution, in fact, is a process which
can give rise to more complex organisms. For this reason, it
is important to understand the theory of evolution as related
to the phenomena of life and the way complex systems
generally arise. Extremely improbable combinations of
nature’s building blocks are possible due to the current
complexity of living organisms. Therefore, the explanation
of their existence and scope of evolution, with the parts,
former being the formation of simple self-replicating or-
ganisms from molecules and the latter being the formation
of complex organisms from simple organisms, are still
issues under consideration and analysis [68]. Considering
the fact that evolution can produce more complex organ-
isms with selection, it is important to note that complexity
is seen in the co-evolution of hosts and pathogens, each one
developing more sophisticated adaptations. When we have
a glance at other fields, like economy, evolutionary theory,
in terms of individuals and institutions, and complexity
science, viewing economies as complex adaptive systems,
is seen to have their integration to enhance the under-
standing toward economics. In that regard, multi-level
selection, causation, human psychology and cultural
changes as evolutionary processes are products of gene-
culture co-evolution [81].
Evolution, order and complexity reveal the relationship
between social worlds and natural worlds, which also
reflects a modern way of thinking that challenges the
dichotomy of natural/social, presenting how the ideas from
biology can be put into practice [82]. It is well noted that
life requires structural complexity; yet, there is the chaotic
mixture of organic compounds, which are highly complex.
This points to the fact that life also requires a specific de-
gree of structural order. At this point, the dilemma shows
that neither complexity itself nor order on its own is able to
characterize a living organism. Thus, combination of these
two requirements, namely complexity and structural order,
can characterize the difference between living things and
nonliving ones, which also marks the course and results of
Darwinian way of evolution. Evolution as a robust mech-
anism drives order and complexity in several natural pro-
cesses, and this cycle leads to an increase of the structural
order in a system. This line of thinking makes it necessary
to define complexity and order as the integral characteris-
tics of life, which also point toward their use as parameters
to evaluate the related processes along the course [83]. This
perspective, with the combination of high order and high
complexity forming a functional unit and indicating a
universal type of biosignature, verifies the importance of
balance and integration, as has been discussed above. From
the Darwinian view, natural selection, in terms of balance,
is the idea of nature producing more individuals that have
variety than what is necessary. Those that are fit or well-
adapted get selected by nature and get to procreate,
which reveals highly dynamic and complex features of such
a process. In other words, nature itself contributes to this
complexity. Speaking once again of nature, in this context,
from the smallest atomic particle to the biggest galaxies, the
past, present and future of every living or nonliving thing in
the universe is marked with its connection to everything.
Thus, nature is characterized by its connection to anything
else in this sense where complexity, as the real part of
nature, comes to the fore as a significant notion with the
paradoxical combination of order and randomness, which
characterizes the multiple processes in nature in various
areas. To put it differently, complexity is not just an
abstract or philosophical concept but in fact a real part of
nature as well as the societies of all beings [84].
Order, as one of the most fundamental concepts in
science, is defined in mathematics in terms of the total and
partial orderings of a set, the order of a differential equation
or a group and so on, while order or disorder is generally
associated with entropy in physics. The minimal descrip-
tion of a pattern’s random aspects refers to the measure of
its primary disorder and the average primary disorder of an
ensemble of patterns is tied to the entropy of that ensemble;
with secondary order being beyond the entropy concept.
Additionally, the primary order has the tendency to
decrease in evolution, while the secondary one tends to
increase [85]. Regarding the temporal scale, when a
randomness-finding complexity is at stake as the measure
of complexity, the first-order complexity is proposed as a
measure of randomness of original time series, whereas the
second-order complexity is a measure and indicator of its
nonstationarity degree. To elaborate, it may be expected
that the second-order or macroscopic complexity of time-
series is higher, and its value could be reliant on the de-
gree of its nonstationarity; and so, a stationary random
time-series could possess a high value of the first-order
complexity measure, yet a low value of the second-order
complexity measure [86]. Order, according to Kruger
(1979), has two aspects, which are homogeneity and
symmetry [87]. Complexity, on the other hand, as a unique
way of estimating the information content of a pattern, is
viewed as means to measure disorder or randomness,
defined as the length of the shortest probable description of
a pattern or the shortest probable computer program to
generate a pattern. When selection, in this regard, is taken
as the only source of order, it becomes necessary to assume
that there would be only chaos without selection in
16 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-39-320.jpg)
![accordance with this basic Darwinian approach. Therefore,
selective aspect is required to attain and sustain order [65]
(for further details on these issues see [88e90].
3.1 A conceptual outline: complexity and
complex systems
The study of complex systems struggles to enhance our
capability of understanding universality that emerges when
systems are extremely complex with interconnected parts.
Across this line, if one attempts to understand the behavior
of a complex system, it is required to understand how the
behavior of the parts act together to form the behavior of
the whole. This means, it would be insufficient to try to
understand only the behavior of the parts. This consider-
ation underpins the significance of “whole” in complex
systems, which makes it evident that each part needs to be
described in regard to other parts and relation thereof; just
like the case of being unable to describe the whole without
describing each part. All these intricate elements and subtle
details make the understanding and analysis of “complex” a
difficult one [68]. Going on with evolutionary theory,
“wholeness” stands as another contentious aspect in view
of the evolutionary origin of complex “wholes.” The
consideration of self-organized collective properties of both
simple and complex systems provides an accurate account
of analysis. Apart from selection, another idea happens to
be self-organization. The balance of the self-organized
properties which is typical in selection and ensemble is
reliant on the level selection can move the population to
ensemble’s parts where typical order is not exhibited any
longer. In systems which are adequately complex, selection
cannot avoid the order which is displayed by the majority
of the members; and thus, such an order is at stake not due
to selection, rather despite selection, which suggests that
sorts of collective self-organization correspond to some of
the order which are displayed by organisms. As stated
above, consideration of self-ordering proves to be benefi-
cial owing to the concerted action of a huge number of
constituents in systems and when evolution is the question,
it is possible to face the question of how difficult it might
have been to capture a particular property or structure [65].
The interrelated or interwoven elements and self-
organization are two of the important qualities of com-
plex systems, as mentioned briefly above. In this section,
let us have a deeper glance at the other important properties
for a thorough understanding. Although not easy to define,
as stated previously, complexity characterizes many
evolving parts interacting with one another in varied forms
and displaying nonlinear patterns as aggregate. This set of
characteristics is referred to as a complex system, which
displays certain properties. One of the properties is cardi-
nality, which refers to many parts varying from particles to
agents making up the system. For example, a particular
pattern in the sand might be viewed as complex because of
having many regularities, not being symmetric perfectly.
The reason why it is considered complex is not due to its
imperfection, but rather those kinds of regular patterns form
regardless of the way wind blows, which shows that an
interplay exists between regularities and a sort of robust-
ness [70]. Diversity is another important characteristic
which shows that parts are different from one another, and
dimensionality shows that parts differ from one another in
many ways across different dimensions. A further quality is
the acting, interacting, and adapting of the parts through
networks, which is called connectivity. Adaptive interac-
tion is a characteristic which refers to the interacting agents’
modification of their strategies in diverse manners along the
accumulation of experience. Chaotic behavior, where small
changes in initial conditions produce substantial changes
afterward can also be noted along with fat-tailed behavior
where rare events happen more frequently than what would
be predicted by a normal, or bell-curve, distribution
[19e21,23e53,63e69]. Another noteworthy attribute of
complex systems is nonlinearity, which represents the
relationship between variables in a nonlinear fashion and
the aggregate of parts is not actually equal to the sum of
their actions or characteristics. Other attributes can be listed
as irreducibility, feedback, emergence, adaptiveness, oper-
ating between order and chaos and self-organization, the
last two of which have been briefly mentioned above [42].
Considering all these elements of complex systems, the
core of the problem that needs to be analyzed is how
complex systems self-organize their structures and/or how
they self-regulate their dynamics.
4. Concluding remarks and future
directions
Complexity is both an old and new scientific concept and
idea, which should entail the understanding of origin of
complex components. It is so profound that the inherent
complexity of the related phenomena and elements in the
related fields should exceed the reductionist outlook of
traditional science and mechanistic way of thinking. For
this reason, as has been discussed and pointed out,
complexity obliges us to adopt an understanding that ex-
tends across a class of complex problems with many subtle
and intricate attributes with working through more inno-
vative and novel ways of thinking as well as applicable
laws showing critical importance. In this setting, evolution,
order and complexity help the revealing of the relationship
between natural and social worlds, portraying a modern
way of thinking that challenges the dichotomy between
natural and social. Accordingly, this study has provided a
conceptual outline and historical account of complexity and
complex systems along with complex order processes to-
ward modern scientific path from Darwin and onwards
Theory of complexity, origin and complex systems Chapter | 2 17](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-40-320.jpg)
![concerned with natural, applied and social sciences. When
compared with the previous works done up until now
[18,21,23e53,63e69], this study has tried to include all the
possible dimensions of complex systems in varied fields,
taking into account the details of origin, history and episte-
mology. The reason why those aspects have been handled,
within the parlance of complex systems, is to emphasize the
transition from evolutionary dimension as a revolutionary
stage, and as a new paradigm for natural sciences and social
sciences, so that the foundation for the interpretation of
complex systems can be explored meticulously by related
areas. Concisely, the accounts of past, present and future in
different complex systems will facilitate toward a deeper
understanding and guidance along the way.
Based on these considerations, some of the following
future directions can be presented as such:
- Complexity thinking entails a broad horizon that con-
siders the subtle properties of the domains in question
and also other domains’ properties, which all require
their own means of solutions and applicability. These
will play an important role in the future science of all
sorts of complexity,
- The properties of evolution and adaptation can shed
light on the understanding of past so that the present
can be interpreted in a holistic way and future plans
and schemes can be designed appropriately and timely,
- When stuck in between two extremes of order and
chaos under uncertain conditions, complexity thinking
and theory can make the systems be adaptive, react to
the world and act spontaneously. This line of thinking
can also be of help to organizations, nations, scientific
research and all other related parties for systematic
and adaptive way,
- Being cognizant of complex systems enables to analyze
the core of the problem by understanding how systems
self-organize their structures and how they self-regulate
their dynamics,
- Theory of complexity ensures powerful evidence and
provides elucidation to challenge traditional and mech-
anistic thinking to steer humanity toward adopting a
new way of thinking.
In conclusion, complex systems, complexity thinking
and theory actually broaden the horizon and scope of
modern way of thinking that relies on transition from
evolutionary dimension as a revolutionary stage and as
a new paradigm for natural sciences and social sciences.
Multiple interactions emerge among the components in a
complex system for which research needs to address
characteristics like self-organization, adaptation, reaction
to responses in the external environment, nonlinearity
in description of differential equation systems
synchronization, noise and high number of descriptive
variables or dimensions. Therefore, the characterization,
definition, analysis and understanding of complex systems
all include powerful relation between variables and
discrimination by noisy nondeterministic phenomena,
sensitivity to initial control as well as strange, nonperiodic
and unpredictable time evolution. Taking all these into
consideration, this detailed strand of thinking ensures that
the foundation for the complex systems’ interpretations can
be explored by related areas along with the accounts of
past, present and future in different complex systems
helping us with a deeper understanding and guidance along
the way.
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![Chapter 3
Multi-chaos, fractal and multi-fractional
AI in different complex systems
Yeliz Karaca
University of Massachusetts Medical School, Worcester, MA, United States
Reality is more complicated beyond what is thought at the
initial condition.
1. Introduction
Complex and nonlinear dynamical systems are thriving as
models of natural phenomena, usually characterized by
unpredictable behavior whose analysis is hard to do as it
takes place, like the incidents in chaotic systems. The
essence of the problem lies in comprehending which sort of
information, particularly concerning their long-term evo-
lution, can be expected to be extracted from those systems.
Correspondingly, complexity, order and evolution all
manifest in the relationship between natural and social
worlds, representing a modern way of thinking that
certainly poses challenges against the dichotomy between
the natural and the social. Computational complexity, thus,
focuses on the amount of computing resources required for
certain sorts of tasks, and its theory enables the assessment
of resources that will be needed for the class of task at hand
for a classification to be made into different levels of
complexity. All these details and intricate elements pose
challenges, which are also the components of dimensions
of modern science and computational complexity, which
require innovative methods and unusual ways of thinking.
Given these, the idea of complexity is stated to be a
component of a new unifying framework for science and a
revolutionary way of thinking so that we can understand the
complex systems whose behaviors are difficult to predict
and control; and also, be able to come up with solutions to
complex problems. Through these developments, the use
of mathematical methods has become an indispensable
method and tool for the improvement of diverse disciplines.
The application of correct mathematical analysis methods
can enable the accurate extraction of important information
and prediction of future inclinations. Even though classical
calculus can be powerful as a tool to tackle many dynamic
processes in applied sciences, the existence of varied and
huge number of complex systems in nature cannot be all
characterized by classical integer-order calculus models,
particularly with regard to information processing and its
analysis. It is acknowledged that fractional-order system
model can be the key to describing the system performance
in a better way with predictive credibility and viability.
Computational technologies in different complex sys-
tems, which are based on mathematical-driven informed
frameworks, can also generate realistic and applicable
adaptive models under dynamic and evolving conditions.
Based on such transformative understanding in conjunction
with mathematics-informed frameworks embracing chaos,
fractal, and multi-fractional ways, the incorporation of
technology, with Artificial Intelligence, as the most prac-
ticable component, has become an essential necessity in the
current era. Through these means, we can tackle
complexity which shows nonlinear, dynamic, and chaotic
characteristics in addition to conceiving and implementing
optimized solutions in an efficient and facilitating way with
some required degrees of flexibility at the same time.
Chaos theory is concerned with the behavior of
nonlinear dynamical systems which under circumstances
exhibit a phenomenon referred to as chaos that is marked
by sensitivity to initial conditions. Three significant prop-
erties of chaotic systems are ergodicity, initial value
sensitivity, and unpredictability [1]. Related to chaos and
complexity, the study [2] handles finance and economics as
complex nonlinear systems affected by many external
factors, ranging from human action to conflicts, policy to
bilateral relations. By taking the time delays in a financial
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![system into account, it is noted that fractional-order cal-
culus alleviates the shortcomings which integer-order cal-
culus cannot express sufficiently and precisely. The authors
examine the dynamics and complexity in a fractional-order
financial system with time delays while observing capti-
vating transitions to deterministic chaos that incudes
cascading period doubling and high-level complexity. The
paper [3] explores the theoretical issues on drive-response
synchronization of a class of fractional order uncertain
complex valued neural networks (FOUCNNs) along with
time varying delay as well as impulses. Banach contraction
mapping principles, robust analysis techniques, and
Riemann-Liouville derivatives are used in the study so that
a new set of sufficient conditions regarding the uniqueness
and existence of equilibrium point of such neural network
system can be derived. In addition, Lyapunov functional
approach is applied to obtain the global stability of the
equilibrium solutions. The study, thus, provides an example
of a multifarious approach. Another study [4] aims at
improving the randomicity behaviors of some chaotic
maps, which represent chaotic systems, and proposes two
integrated chaotic systems for the generation of different
chaotic maps, by developing a new image encryption al-
gorithm using integrated chaotic systems whose theoretical
aspects are also provided. Concerning the unpredictability
of chaotic systems, the authors of [5] handle COVID-19
pandemic, which has had massive impact on tourism
sector through six illustrative examples about how a
research agenda should resemble, pointing out a paradigm
shift in tourism because of pandemic conditions. The
chaotic conditions in such a complex system show that a
new understanding is needed in a “new-normal” tourism
world by taking into account the greater global economic
and political context, by going beyond the purely descrip-
tive means. The research paper [6] deals with chaos, brain,
and divided consciousness as modern trends of psychology
and cognitive neuroscience stating that applications of self-
organization, nonlinear dynamics and chaos are important
for problems related to mind and brain relationship. Chaotic
self-organization is noted to provide a unique tool both in
terms of theory and experimental aspects for a more pro-
found understanding of dissociative phenomena. The
nonlinear methods employed in the study are for the anal-
ysis of marked changes in electroencephalography and
bilateral electrodermal activity during the experiencing of
dissociated traumatic and stressful memories as well as
psychopathological states. The analysis of the study cor-
roborates the possible role of chaotic transitions, and the
author suggests that self-organizing theory of dreaming is
significant related to memory formation and processing as
cognitive processes. The study is important in terms of
showing the dynamic ordering factors and self-organization
which lie under the brain physiology and psychological
processes, which are different examples of complex
systems. The study of [7] is on the key features of
complexity from different perspectives with relevance to
strategies proposed to combat the COVID-19 pandemic.
The author argues that critical systems thinking is the
approach to complexity that offers the most fitting under-
standing of the phenomenon and suggests that multi-
perspectival and multi-methodological approach of critical
systems should be adopted by decision makers to get ready
for and respond to similar crises during the turbulent times.
The author concludes that a variety of systems methodol-
ogies and different perspectives with alternative explana-
tions can be used to instigate informed ways. Again related
to the pandemic, chaotic dynamics is focused on [8]
through the mathematical examination of COVID-19 data
in different countries, which shows similarity to the chaotic
behavior of many dynamics systems just like logistic maps.
Apart from providing direction for public policy makers,
the study remarks based on the use of an interactive data
map that the pandemic’s scale and behavior are unpre-
dictable because of the chaotic systems’ properties. Chaos,
referring to irregular and unpredictable behavior charac-
terized by sensitive reliance on initial conditions, can be
shown by some of the nonlinear differential equations. The
tendency of nature toward pattern formation, iteration, and
creation of order out of chaos generates expectations of
predictability. Due to varying degrees of interaction be-
tween chance and choice as well as the nonlinearity of
systems, nature escapes the boredom of predictability [9].
The studies handling chaos and multi-chaos in various
disciplines [10,11] provide important insights into its
operation, dynamics, and properties when we consider that
characteristics of living systems play a prominent role for
surviving and thriving through chaos.
Fundamental change in the outlook of traditional ge-
ometry is essential to handle complex systems. Conse-
quently, fractional dimensions of objects, known as fractals
as never-ending patterns, are integrated into an integral
dimension space, and geometric fractals are described by a
procedure or an algorithm that generates them explicitly
self-similar. Fractals are, in fact, images of dynamic sys-
tems driven by recursion, which is to say the image of
chaos; and fractals are used in order to model structures
where patterns recur repeatedly and describe random or
chaotic phenomena. One of the related studies is on fractals
in the nervous system regarding neurodynamics [12] doc-
umenting the prevalence and showing the credence of
fractals at all levels of the nervous system along with their
functionality as well as paying attention to the relationships
between power-law scaling, self-similarity and self-
organized criticality. The overview by [13] provides the
philosophical, historical and basic concepts related to
fractal geometry discussing the ways neurosciences can
make use of computational fractal-based analyses. The
comparison of fractal with Euclidean approaches to analyze
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![and quantify the brain across its whole physiopathological
spectrum is also provided. The study of [14] is on the
predictive optimization of the classification of stroke sub-
types based on fractal and multi-fractional methods. Box-
counting and wavelet transform are the other ones used
and Feed Forward Back Propagation (FFBP) algorithm is
employed for stroke subtype classification. The study aims
at providing a new direction concerning complex dynamic
systems and structures that display multi-fractional prop-
erties. [15] is a study on Rescaled Range (R/S) fractal
analysis with wavelet entropy characterization for fore-
casting purposes based on self-similar time series
modeling. The proposed novel method with its multifarious
methodology can be applied in different applied sciences to
reach optical solutions in complex, dynamic and nonlinear
settings for critical decision-making processes. Concerned
with financial environments, the study [16] handles char-
acterization of complexity and self-similarity using fractal
and entropy analyses for stock market forecast modeling.
The study shows the critical importance of Hurst exponent
(HE) as computed by R/S fractal analysis employed as an
indicator along with Shannon entropy (SE) and Renyi
entropy (RE) related to future forecasting capability with
regard to the stock indices. Finally, [17] is a study on self-
similarity and multi-fractality in human brain activity
through a wavelet-based analysis regarding scale-free brain
dynamics. The authors propose a novel method to enrich
the characterization of scale-free brain activity via a robust
wavelet-based assessment; and for this they used magne-
toencephalography (MEG) to analyze human brain activity.
The study of [18] is on fractal stochastic processes on thin
Cantor-like sets, reviewing the basics of fractal calculus,
defining fractal Fourier transformation on thin Cantor-like
sets, introducing fractal generalizations of Brownian mo-
tion as well as of fractional Brownian motion. The fractal
derivative is said to be the progenitor of the fractional de-
rivative, which emerges if a certain fractal distribution of
events on the time axis is used. While fractals refer to
objects or quantities that manifest self-similarity at every
scale, dynamical systems describe the way a complex
system changes in time. Thus, research into these aspects
provides significant input and contributions for different
fields [19,21] to understand complexity deeper.
Manifesting evolution in time cumulatively, complex
processes are based on the occurrence of both huge and
number of phenomena with the mean time interval between
two subsequent critical events being infinite. Nonlocality is
one of the reasons for interest in applications of fractional
calculus and many interesting physical phenomena with
memory effects also suggest that their state is not only
dependent on time and position but also on the prior states.
Consequently, fractional differential equations (FDEs) are
seen as alternative models to nonlinear differential equa-
tions. In view of that, the study of [22] investigates the
behavior of dynamical process of complex systems within
two types of ideal ways. FDE is constructed to describe the
time evolution of the complex systems. The dynamical
behaviors are conveyed universally into a power law dis-
tribution with exponents determined by parameters that
characterize the system. The authors also demonstrate how
slow relation and super slow relaxation processes take place
in hybrid systems, with time intervals having been
measured at the logarithmic and double logarithmic scale.
The incorporation of fractional calculus models for the
description of diffusion in heterogeneous and complex
materials is steady, and [23] handles biological tissues
whose features can be encoded in the attenuation of the
MRI signal by the fractional order of time and space de-
rivatives. The authors describe different exponential and
fractional order models applied in MRI and investigate the
connection between model parameters and underlying tis-
sue structure. Fractional calculus is stated to provide new
functions like Mittag-Leffler and Kilbas-Saigo, which
characterize tissues concisely, pointing toward success in
treatment. Fractional differential equations and fractal
analysis are accepted to be useful means to describe the
dynamics of complex phenomena characterized by spatial
heterogeneity and long memory. To construct the mathe-
matical modeling of many nonlinear phenomena, fractional
differential equations (FDEs), being viewed as alternative
models to nonlinear differential equations have varieties of
them, serve as important tools in various fields including
but not limited to mathematics, physics, fluid flow, biology,
control theory, signal processing, fractional dynamics and
systems identification [24,29].
Computational technologies, with machine learning as
the core component of AI, enjoy broad use and trans-
formative impacts, which enable us to train complex data to
automate or augment some human skills. Thus, the cross-
cutting nature of AI provides motivational power to
formulize research in a systematic way. Artificial neural
networks (ANNs), which are networks of computer systems
inspired by the human brain and biological neural networks
have the ability of learning and modeling complex and
nonlinear relationships. In the simplification, abstraction,
and simulation of the human brain, ANNs reflect the related
basic characteristics of this complex organ [30]. Among the
related studies, the paper by [31] is on the deep fractional
max pooling neural network (DFMPNN) with 12 layers for
the recognition and better diagnosis of COVID-19. The
model proposed by the authors is demonstrated to be su-
perior to 10 state-of-the-art models, and three more im-
provements are also proposed accordingly. Another study
[32] is on the adaptive fractional-order backpropagation
neural network for handwritten digit recognition problems
through the combination of population extremal optimiza-
tion algorithm, named as evolutionary algorithm, and a
learning mechanism based on fractional-order gradient
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![descent. The study emphasizes the significance of the
optimization of the initial connection weight parameters.
[33] is another work where a fractional-order deep back-
propagation neural network model is proposed with L2
regularization. The fractional gradient descent method with
Caputo derivative is used to optimize the network proposed
whose necessary conditions for convergence are also
illustrated, with the conclusion that the proposed network is
deterministically convergent and capable of avoiding
overfitting in an effective way. Moreover, [34] is a study
concerned with swarm intelligence technique employed to
solve FDEs. Feedforward ANNs are used to define the
unsupervised error, and particle swarm optimization is
performed for the learning of the errors’ weights. The
scheme proposed by the authors point to conceptual
simplicity, easy implementation, and extensive scope of
applications. Last but not least, the study by [35] shows the
way to use data-driven similar measures in an effective way
in standard learning algorithms in the context of life science
as well as other related complex biological systems. In
brief, computational methods, along with models and
adaptive algorithms for scientific advances show the ca-
pacity of handling large amount of complex datasets. In
addition, significant advancements in computational pro-
cesses aim at addressing the issue from a more versatile and
profound understanding with an interdisciplinary outlook in
various disciplines [36,37]. The related computational
processes with broad applications in integration with frac-
tals, multi-fractals, fractional methods, chaos, nonlinear
dynamical properties, stochastic elements and so on pro-
vide systematic optimized solutions.
Modern scientific thinking has adopted the principles
concerning systemic properties, addressing them by
uncovering the spontaneous processes related to self-
organization in a dynamical system, in a state distant
from the equilibrium point, and in proximity to the
disequilibrium point without any existence of an external
force acting on the system itself. In view of that, evolution,
order and complexity unearth the relationship between
natural and social worlds challenging the dichotomy be-
tween them. This study, vis-a-vis the works in the literature,
for the first time, provides a conceptual outline, brief his-
torical sketches of fractal, multi-chaos, and multi-fractional
AI in different complex systems. In addition, the outlook
provided in the study indicates that the works that employ
AI and computational methods have tracked and will need
to track along an evolutionary process; and if they use
multifarious integrated methods, it would be possible to
reach optimized solutions through developing strategies
within a paradigm shift. This study provides an overview
encompassing multi-chaos, fractal, fractional and Artificial
Intelligence (AI) ways of thinking for the solution of the
complex system problems concerned with natural and so-
cial sciences. In addition, ethical decision-making
frameworks and strategies related to big data and AI ap-
plications assist the identification of the related problems in
different settings and help thinking methodically with a
deliberative compensating process in order that tensions
between different conflicting aspects can systematically be
handled. The values related to ethical issues should also be
addressed, in a way that requires to be practical, flexible
and problem-driven instead of sheerly theory-driven so that
dilemmas can be addressed and critical decision-making
directed beyond theoretical positions focusing on the
applied aspects. In view of that, this study aims at providing
a direction through the topics of data reliability, chaos
thinking, fractal, fractional thinking and artificial intelli-
gence way of thinking as well as processes all revolving
around complexity. Another motivational aspect is the
addressing of ethical issues, which raise thorny questions
for both researchers and practitioners in a way which
adopts practical, flexible, applicable and problem-driven
mode for the solution of ethics-related dilemmas so that
purely theory-driven approaches will not be the only point
to resort to. Furthermore, ethical decision-making frame-
works and strategies related to big data and AI applications
should be developed and applied so that related problems
can be identified correctly in different settings and thinking
can be done methodically in a deliberative way with
compensating processes. Since the impact and ubiquity of
data technologies are the case concerning all aspects of
modern life, it is important to build a balance between data
use and ethical matters. Computational technologies in
different complex systems based on mathematical-driven
informed frameworks can yield more realistic and appli-
cable adaptive models under dynamic and evolving con-
ditions. Through such transformative thinking in
combination with mathematics-informed frameworks that
encompass chaos, fractal, and multi-fractional ways, the
incorporation of technology, with Artificial Intelligence, as
the most practicable component, is a requirement in today’s
world so that we can tackle complexity which shows
nonlinear, dynamic, and chaotic characteristics. Thus,
optimized solutions can be conceived and implemented
efficiently and in a facilitating way with some required
degree of flexibility as well.
The remainder parts of this chapter are organized
as follows. Section 2 addresses the Challenging
Dimensions of Modern Science, Complexity and
Complex Systems with subheadings of 2.1., 2.2., 2.3.,
and 2.4. entitled “Data Reliability and Complexity”,
“Chaos Thinking, Processes and Complexity”, “Fractal
Thinking, Processes and Complexity” and “Fractional
Thinking, Processes and Complexity”, respectively.
Section 3 is concerned with Artificial Intelligence Way of
Thinking, Processes, Complexity and Complex Systems.
Finally, Section 4 of this chapter provides the Concluding
Remarks and Future Directions.
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![2. Challenging dimensions of modern
science, complexity and complex
systems
Complexity seems to be relevant and applicable con-
cerning a vast array of phenomena on all scales. In the
evolution of dynamical systems complexity tackles, the
nature of the system under consideration is often irrele-
vant; yet, the perspective complexity proposed enables the
identification of forms and evolutive characteristics per-
taining to all or nearly all of the systems constituting a
huge number of elements. These elements within the
systems may have reciprocal interactions, positive feed-
back mechanisms, as well as nonlinear interfaces. For this
very reason, such systems are precisely named as complex
systems. Another point to note is the way systems orga-
nize themselves in reaction to an action coming from the
external environment [38]. Although this investigation is
considered to be as old as science itself, the studying of
the way systems make the self-organization spontaneously
is a scope of investigation that is relatively new and per-
tains to modern science whose roots date back to the
important developments of Renaissance [39,40] and
Enlightenment [41]. In addition, traditional science tries to
account for the forms observed by means of a reductionist
outlook such as looking for laws regarding the single
components of the system, the science of complexity, in
modern science sense, in contrast to traditional science,
adopts the systemic properties and considers those by
demonstrating the spontaneous processes of self-
organization and how it occurs in the case a dynamical
system is in a state distant from the equilibrium and close
to disequilibrium without the existence of an external
force acting on the system. Conceptually, in this modern
scientific view, what we are interested in is to develop an
understanding of organisms in their environments, which
points to one aspect of evolution theory taking into
consideration the evolutionary dynamics of populations
of complex organisms. The development of life is
described as evolution, which is the individual organism’s
capability development, which refers to an increase in the
complexity of the organism. Evolution is not regarded as
the development of a collective behavior of many organ-
isms primarily, so the development of species and eco-
systems is also a question to consider. Being a general
approach to the complex organisms’ formation by incre-
mental change, evolution and conceptual incremental
evolutionary processes encompass monotonic evolution
on a fitness inclination as well as traits’ divergence and
extinction. With evolution viewed as a blind fitting pro-
cess, organisms display adaptation to their environment
[42] and theories related to evolution see fitness as the
mere property of the organism that decides on the evolu-
tionary dynamics and in that regard conventional evolu-
tionary theory relies on gradual variations of fitness
[43,44].
To compensate the gaps of the reductionist approach
mentioned briefly above, the use of dynamical equations
which model reproduction and predation can provide the
modeling of a multiple dynamic phenomena in populations.
In other words, a variety of resources with their own
peculiar dynamics need to be analyzed so that the existence
of groups of organisms that have well-distinguished traits
can be explained. While various systems, namely immune
system and artificial computer software, can be employed
as laboratories to develop an understanding toward evolu-
tion, the study of organisms as well as their complex col-
lective behavior and evolution by means of mathematical
tools has become an important area of study of complex
systems in recent years, which mark the evolution of
modern science addressing the construction of models from
the interactions of components through the discussion of
spatial and temporal structures and substructures. Such
interactions can be seen in many different forms like
cooperation, communication, reproduction, competition,
consumption, exploitation and so on. While individual
behavior is often complex, it is not very clear if the
emergent collective behavior of many individuals is com-
plex. For this reason, the behavior of complex individuals
at interaction with their environment becomes a subject of
investigation. Recent developments in artificial neural
networks, in modern scientific view, have provided the
robustness of millions of synaptic weights’ optimization
with observations operating powerfully for many different
phenomena. Correspondingly, the related models make
use of local computations to interpolate task-related mani-
folds within high-dimensional parameter spaces rather
than learning simple rules or world representations. Anal-
ogous to evolutionary processes, models which are over-
parameterized can be parsimonious with their provision of
robust, applicable and versatile solutions to learn a diverse
set of functions and reach optimized outcomes. These
models and outlook the models are derived from establish
links to consider unpredictability and more importantly,
pose a far-reaching challenge to many perspectives in
different areas of science relying merely on theoretical as-
sumptions. In this context, the construction of models
related to life should also include a model of the environ-
ment when behaviors of individuals are measured as reac-
tion to an external stimulus. In addition, the link between
the individual’s capabilities and the environment’s de-
mands can provide a comprehensive description of the or-
ganism. In sum, all these points demonstrate the critical role
the environment plays in the complex dynamics of
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![evolutionary change and that there is a close relationship
between an organism’s complexity and the organism’s
environment.
It is important to establish a balance between providing
rigorousness, pursuit of distinctness as well as clarity and
not abstracting the complexity of the real world so a
junction, rather than a disjunction, is required between
scientific wisdom and reality. One challenge is to stimulate
mathematical intellect without having a reference to the
interests of science that are geared toward the real world.
The mathematical structure, known as nonlinearity, in this
regard, refers to the system’s autonomia which may not
have a single variable but operates in a system of high
dimensions where numerous species and resources are
connected very closely, interacting by producing highly
complex entanglements and/or cooperating at times. In
ecosystems, nonlinearity signifies the coefficient of repro-
ductive rate with time constant not being a priori at all but
one that changes according to the conditions of the
ecosystem in question like the function of population.
Through nonlinearity, the ecosystem can either end up with
stability or it may produce unpredictable or complex os-
cillations [45].
As a system, a simple one provides one single path to
one answer only, providing one solution and one way for
sorting out the problem. In this sense, a complex system has
different and multiple ways toward multiple answers; so
based on the choices made, it is likely to encounter a sys-
tem that changes according to those selections, which also
refer to the adaptiveness of the complex systems. The more
insight is developed, the answers may keep changing and
more learning occurs. While searching for the use of
appropriate mathematical model, the description and anal-
ysis of a system relies on the way the system is perceived
[46]. Thus, when complex systems are at stake, with large
collection of components interacting locally with one
another at smaller scales and having self-organizing to
manifest global behaviors and structures at larger scales
without external intervention from the environment, it is
necessary to understand and/or predict the properties of
such an intricate collection based on the “whole” knowl-
edge pertaining to its constituents. All these elements
require novel mathematical frameworks in ever-changing
current landscape, which was put very well by
Stephen Hawking as: “I think the next [21st] century will
be the century of complexity.” Accordingly, the aim of
complexity and nonlinear science is to gain global under-
standing by taking into consideration the multiple inter-
acting factors of systems, many branches of possible states,
and high-dimensional manifolds while monitoring actuality
regarded as diachrony, which is to say the historical and
evolutionary path that has been through many different
critical points on the manifold.
Within such settings as described and outlined above,
dynamic modeling as a process allows the extension of our
knowledge about reality, with computer being the main
means due to allowing the user to trace the evolution of the
dynamic system, which is represented by the model by
numerically integrating the equations. Therefore, the chief
objectives of dynamic modeling are intended to describe
the flow of the situation and make prediction on its future
evolution. As already mentioned, the construction of the
model is a process, which needs to consider the aspects that
are regarded as secondary so that results for the assump-
tions made are provided, which will help us do the final
analysis by handling the implications of one’s own opin-
ions for the correct and applicable interpretation of related
phenomena [38]. Given that, dynamics of coevolving sys-
tems also need to be taken into account carefully as the
success of natural sciences has reflected in that organisms
do not only evolve but also coevolve with the other or-
ganisms in the environment. The exploration of the re-
quirements of order and capacity to evolve needs networks
that manifest parallel processes where selection is critical in
the emergence of entities which coevolve with each other;
and thus, it is important to comprehend the ways in
which selection achieves systems that are capable of
coevolving [166]. All these elements and intricate qualities
reflect Darwin’s perception of nature, which is described
as “There is grandeur in this view of life” with bewildering
interconnections between laws and amazing organic
beings in endless forms [47]. This vision encompasses
multiple facets like intellectual, sentimental and aesthetic
dimensions, which is a key to dig into the layers of
complexity.
2.1 Data reliability and complexity
Data complexity is a challenging issue and as regards
managing and leveraging data, the problems arise since
significant data may be scattered across different platforms,
some being inaccessible, some incomplete, outdated,
inaccurate, misleading or inconsistent. At times lack of
detailed knowledge about the source system data may be at
the core of the problem, and at other times building data
integration processes may be haphazard due to the lack of
knowledge and skills required. For all these reasons, it is
accepted that data complexity has a huge impact; and
hence, while making critical decisions, it becomes impor-
tant to address such complex problems and barriers to data
reliability, accuracy and quality so that time, efforts and
resources can be used efficiently. As for data management,
complexity and data protection in the digitalized era where
cloud, hybrid, and other type of workloads have been some
common scenarios, the important keys are stated to be
performance, scalability and reliability. To tackle data
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![protection complexity, cut down on disruption, and miti-
gate data loss risk, simple, automated and reliable solutions
are required. Consequently, another issue other than data
complexity, protection and management aspects, reliability
comes as one further noteworthy challenge. Describing the
ability of a component or system to function under speci-
fied circumstances, reliability is tightly connected to
availability, which refers to the temporal ability of a
component or system to function at a certain interval of
time or moment. Reliability and availability are two closely
related terms in data complexity issues to deal with accurate
predictive measurement, uncertainty and risks of failure.
Complexity brings about unique challenges in big data
analytics in which big data does not only refer to huge
volumes of data but its “bigness” has also to do with their
complicated structures. Volume, velocity and variety are
the three common characteristics of big data, as described
by Zhang [48]. The last feature mentioned, namely variety,
is concerned with data complexity, which can refer to
complex relationships, high dimensionality and many other
complications in a given dataset. The challenges involved
in dealing with complexity in big data concerning modeling
and analysis also provide opportunities for the development
and application of related statistical methods, which also
applies for reliability applications. Through engineering
systems that generate big data for reliability analysis, it
becomes possible to deal with degradation measurements,
time to failure and time to recurrence of events [49].
Reliability analyses are important for complicated struc-
tures while focusing on the use of data [50]. Furthermore,
big data also requires new processing means to possess
insight discovery, more robust decision-making power and
better process of optimization ability along with the ways
of effectively excavating useful information as another
challenge faced in the current present big data background
[51]. In this background, data complexity measures can be
classified into three preliminary categories, which are
measures of separability of classes, measures of overlap of
individual feature values and measures of geometry, to-
pology and density of manifolds. Considering these factors
are of central importance while tackling multi-class datasets
[52]. When missing data is the case, this will be a situation
which would degrade performance, and incorrect imputa-
tion of missing values can further cause inaccurate pre-
dictions [120]. Since missing data could have serious
impact when not handled appropriately, analytical methods
can be applied to fix the missingness aspect [53]. Incom-
plete data or information could also have serious impacts
on the subsequent data processing and reasoning, which
could cause one to end up in wrong decision at critical
times. Some reasons of missing data are as follows: the
attribute of the data may not exist, data may be not avail-
able temporarily or data loss or damage might be the case
[54]. All these interrelated problems bring the utilization of
data to the foreground in pursuit of novel and highly effi-
cient solutions, which is sorted out by the proposing of new
schemes based on algorithms in tandem with technological
advancements. As a result of progress in imaging tech-
nologies and integration of systems approaches, quantita-
tive science is also going through a reform. Within this
context, robust conclusions to be drawn from quantitative
data require a measure of their variability in complex sys-
tems with experiments being carried out under intricate and
measured complex processes. When one is exploring a
complex system, it is important not to discard outlier data
points, as those may be relevant as clustered measurements.
In other words, it is important not to fall into the trap of
ignoring data that do not match the related hypothesis
tested, since the subject of interest may not be simple or
straightforward that would provide only black or white
answers. It is recommended not to turn the hypothesis
driven research into a hypothesis forced one [55].
Reliability and robustness of data and its complexity in
collection, storage, sharing and utilization of data in
different areas also bring about another highly controversial
and challenging point which are the ethical considerations.
In that regard, it is important to be cognizant of different
dimensions in various fields, particularly in health care.
Some of the points associated with ethics are informed
consent providing the person in question why data are
collected, from whom and how, its storage way, length of
keeping, and who will have access to it, all of which have
legal dimensions. Another important aspect related to ethics
is data transparency which suggests openness, communi-
cation and accountability. It also refers to the control flow
of the data in machine learning algorithm [56], while in
medicine, transparency helps the enabling of evidence-
based decisions that is critical to foster trust among the
related parties [57]. Used in science, engineering, business
and social sciences, transparency is practiced in many
different systems ranging from communities to adminis-
trations, companies to organizations. As a related compo-
nent, accountability, namely tracking where data come
from, can be taken as an operational issue since different
queries are at stake like how were the data produced, who
produced the data, when were the data extract produced and
what data were produced [58]. Along with these di-
mensions, one argument, in terms of big data, states that big
data algorithm designers must make data sources and pro-
files public to improve transparency and avoid bias [59].
Accountability also includes the related systems, laws, and
regulations, which point to the complexity of the issue, in
other words, even one single dimension of ethics provides
complexity on its own operating within complex systems
that have interacting and variable elements. Clear
communication and data sharing are other important ethical
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![aspects, since they are important to have clear processes to
share data, which is as important as having clear processes
to collect data. This becomes more critical when individual
data are under consideration, which indicate privacy and
sensitivity in widespread data such as mental health, sui-
cide, patient details, biological information like DNA,
personal financial data and so on. Furthermore, data privacy
refers to the relationship between collection and dissemi-
nation of data, which may be sensitive that would not be apt
to reveal for the sake of trust. Also known as information
privacy, data privacy is one branch of data security that
deals with the proper handling of data, including factors
like consent, notice as well as regulatory obligations.
Practically speaking, data privacy concerns focus on
whether or not data will be shared with third parties, if it is
to be shared, then how it will be shared, how data are le-
gally collected and stored, and finally regulatory re-
strictions are to be observed [60]. Controlling access to
personal information is viewed to be a significant aspect of
maintaining privacy, and if information is accessed or
revealed against the wish of the person, this situation has
the potential to impact the well-being and breaching of
rights, which include the respect dimension too [61].
Confidentiality refers to keeping data like medical records
or service records confidential, while anonymous data refer
to information that cannot be traced back to a particular
person. Data linkability is also a related concept, which
refers to the ability for anonymized data to remain linkable
so that the data value will not be diminished; whereas data
composability points to the privacy guarantees which could
be provided when data from multiple sources (to which the
same or different privacy models have been applied) are
integrated to one single fused data-rich source [62].
Confidentiality and anonymity as well as bias include the
ethical considerations linked with the collection and use of
data in nonclinical areas with the relationship between
patients as a group and organizations [63]. In business
settings, online data collection has its ethical dimensions
from the customers’ side, since they need to know that the
data they are using are reliable and sourced ethically. On
the other hand, companies that provide online data collec-
tion technology have to commit themselves to trans-
parency. Therefore, regulators have started to focus on the
background of data collection process, which seems to be
among the other future challenges related to this issue.
More formally, concerning governance and law, legal in-
struments protect privacy in different ways through acts and
data protection laws and limitations [61].
In health and research characterized by evolving land-
scape of big health data, the use, sharing and reusing of big
data have become an important feature, and with the advent
of technological developments, particularly artificial intel-
ligence, the use of big data has become far-reaching.
Although ethics of big data has become a topic of extensive
research, guidance and ethical decision-making frameworks
are required further, which will provide insight into the
values and how decisions will be made in an increasingly
complex environment of health and research. Ethical
decision-making frameworks, in this regard, provide
assistance to identify the related issues in different settings
and help think through in a methodical manner while at the
same time these frameworks integrate a deliberative
compensating process to manage tensions between con-
flicting values step by step and systematically. The values
concerning the ethical issues point to being practical,
explicit, of equal weight, flexible, problem-driven, rather
than theory-driven so that real-world dilemmas can be
addressed beyond theoretical positions and guide decision-
making (see [62] for further details and the key ethical
values defined). In the age of big data, it has become
possible to generate more and more data from growing
number of sources about health and biology. While access
has become easier, challenges also arise in terms of ethics
of data, when the relationship between privacy and public
interest is considered. This is because data science and
computing developments have imposed pressure on tradi-
tional approaches to information governance like seeking
consent or making data anonymous. For these reasons, data
initiatives, governance and design thereof address key
ethical principles so that good practice can be established
[61]. One significant idea about consent is that it needs to
be obtained prior to an intervention, which is to be based on
a sound understanding of its implications and likely out-
comes consequences. Yet, in the modern data world, it is
disputable to what extent this is practicable or relevant [64].
Consent becomes a critical concern and an ethical
requirement while carrying out research with human par-
ticipants who are expected to comprehend the presented
information like the aim of the research, predictable risks,
likely outcomes of any decision to be made based on the
information [65]. In addition, there is a surging interest
toward aggregating biomedical and patient data into big
datasets intended for research purposes and also public
benefits, which causes new ethical issues to emerge with
regard to social justice, human rights and trust that concern
many parties including but not limited to researchers, pol-
icy makers, regulators and healthcare professionals [66].
Moreover, responsible data sharing in health research needs
to be addressed as well since it should have principles and
norms when large-scale of dataset linkage in clinical set-
tings is at stake considering treatment evaluation, disease
etiology, its facilitation, diagnostic purposes, to name just a
few [67]. Consequently, both ethical concepts such as
privacy, informed consent and confidentiality as well as
new ones are likely to arise in the evolving landscape of big
health data, which shows that ethical issues also show
evolution due its dynamic features, all pointing to the
transient and emerging challenges of modern science.
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![Considering the requirements of modern science, which
also necessitates comprehensive and fruitful view of ethics,
it should be noted that the dimensions of ethics need to be
integrated into a complex theory that shows the interplay of
tensions and finding a balance so that none will obscure the
practice of ethics [68,69]. The far-reaching effects of data
technologies and data science due to their intrinsic
complexity are evident in all dimensions of modern life,
which obligates those dealing with data to engage with the
ethical issues. Even though data are available, powerful
tools to extract information and economical storage ca-
pacity seem to be the advantageous aspects, such advanced
technologies also come with their challenges like their
potential misuse [64]. As is the case with everything else,
data are also evolving with time with some datasets accu-
mulating over time, changing dramatically as a result of the
spatial and temporal conditions. For these reasons, various
properties of modern data and the use thereof necessitate
taking care of changing ethical issues some of which are
interconnectedness, real-time decision-making as data
arrive, lack of time, space and social context restriction on
the scope of data as well as use for unexpected reasons and
revealing unexpected information [64].
In this landscape, data ethics has emerged accordingly
as a new branch of ethics that explores and evaluates the
moral problems concerning data, which include but are not
limited to generation, recording, processing, publishing,
sharing and use, along with algorithms, machine learning
and robots all related to corresponding practices with
responsible acting, innovation and programming. To come
up with morally acceptable solutions, right conducts and
values are to be observed. Consequently, the conditions
require the shift from being information-centric to being
data-centric, which also indicate the complexity of ethical
challenges brought by complex data science considering
the collection and analysis of big datasets in many different
fields like biomedical research, social sciences, profiling,
open data, data philanthropy and so forth [70].
With the accumulating and increasing quantity as well
as variety, data with different types and nature pose serious
challenges associated with the complexity of the matters
and considerations revolving around ethics during pro-
cesses concerning critical decision-making, handling
complexity, collecting or sharing data. Accordingly,
different types of data that cause different instances of
complexity and challenges can be briefly categorized as
follows [71].
Data downloaded from databases that are available to
public: These may be country-level data or international as
well. They are exposed to updates all the time, thus,
version and date of retrieval need to be specified. The
typical relational databases have restricted ability to
manage the heterogeneous nature of complex and modern
data; therefore, high complexity of data in those databases
could cause bottleneck in conventional information sys-
tems concerning the efficient and reliable retrieval of in-
formation [72].
(1) Data collected manually and coded accordingly: They
could fall into the category of archival data and original
survey data. Effort, time, and financial resources
contribute to the complexity of such data, since select-
ing and entering them are demanding and arduous.
(2) Data utilized under license coming from a commercial
data provider: Commercial databases are used exten-
sively in many studies focusing on economics, finance,
and strategic works. Copyright becomes a legal issue,
since it is held by a private organization, which ensures
data access through subscription. When data are shared
with others, there would be cases of copyright viola-
tions; and providing public access to the original data
is not viable on legal terms. All these dimensions add
to the complexity of the problems.
(3) Data available on a remote computer: To preserve the
confidentiality of data, some data providers allow re-
searchers to analyze data from the computers of the or-
ganization or researchers submit their codes to the staff
of the organization so that they can analyze the data for
them. Thus, researchers are in a position of being un-
able to share raw data, since they do not have access
to the data.
(4) Data which are generated by laboratory experiments:
These may concern experimental design-based studies,
which have grown in recent times. New standards
come into existence, indicating the dynamic changing
environment of data as well.
(5) Data which are generated by qualitative research:
Such research can be observations and interviews
which cause certain problems and conflicts regarding
transparency and ethical obligations among scholars
doing the research [73]. In contrast with quantitative
research that uses secondary data, qualitative area
brings the confidentiality notion in ethics into fore-
ground. Direct interaction is at stake during interviews,
which is a qualitative research method; hence, a partic-
ipant’s confidence becomes integral to the reliability
of the data [71]. In addition, regarding the secondary
analysis of qualitative data, culture of data archiving
is important to be cultivated in qualitative research,
and since data archiving would include someone’s
personal views, the best practice could be to practice
anonymization during the initial transcription. Thus,
a record or log of aggregations, replacements or re-
movals need to be made and stored from the anony-
mized data files separately. It should be noted
that further ethical concerns may arise during the rein-
terpretation of qualitative data at a later time period
[74].
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![(6) Industrial data: The realization of data collection ar-
chitectures in Industry 4.0 is linked with vast imple-
mentation efforts as a result of the heteregeneous
structure of systems, interfaces, protocols and other
disciplines involved in the projects. Digitization of
processes has increased due to industrial automation
transforms and available data in the production has
also been increasing in amount, which makes it essen-
tial to leverage data to adjust to production plants and
machine parameters so that efficient and flexible pro-
duction can be ensured [75].
Handling of data also requires the accurate categoriza-
tion and comprehension thereof; and in that sense, data can
be categorized as numeric or nonnumeric with qualitative
data belonging to nonnumeric category, which represents
certain descriptive features, and they are referred to as
categorical data. Quantitative data, on the other hand, are
numeric, being further clustered as discrete or continuous
[76]. Complex data types are divided into two descriptive
classes, which are discrete and continuous. While the
former represents the features that exist independently that
have definable boundaries and a finite number of possible
values, being any of the three geometries, that is to say
polylines, polygons and points, continuous data do not have
clear or definable boundaries. Thus, continuous data can be
remembered as data with no defined boundaries that cover a
scale of values and its opposite counterpart, namely discrete
data refers to data that have defined and clear boundaries,
which can be described by values that can be eliminated
from a system without disrupting the whole system. In
addition, while continuous data constitutes the rest of nu-
merical data, usually associated with some kind of physical
measurement; discrete data often occurs in an instance in
which there is certain number of values or whole numbers
are counted. Another difference is that discrete data can be
represented by integer or whole numbers, whereas contin-
uous data are usually regarded as exact and integer with
infinite possibilities, represented by real numbers [77].
The complexity of the systems and data utilized, shared,
handled and managed should involve optimal strategies that
seek the understanding of the interactions relevant to all the
different parts of the system and in this setting, data
collection should be geared toward the assessment of re-
lationships within a system so that the analyses can inte-
grate notions and ideas taken from systems thinking
approach and also complexity science [78]. Systems theory
is important in this context, since it is the interdisciplinary
study of systems with interrelated parts that can be either
manmade or natural, with each system being bounded by
time and space impacted by the related environment,
defined by its purpose and structure; eventually expressed
by its functioning. Since the change in one part of the
system can affect the whole system, predicting the changes
in patterns of behavior becomes a must therein. Systems
thinking helps as an approach with regard to problem
solving, which intends to strike a balance between holistic
thinking and reductionist thinking. The related purposes of
this mode of thinking are significant. It enables one to
define a system in this way and apply the definition to a
variety of complex systems. It also helps one recognize the
features of systems that make them complex ones, while
analyzing a system means identifying problems and
formulating it in systemic way. Finally, it allows us to
apply advanced systems thinking so that we can seek so-
lutions to disorganized management faced in real life. For
systems that are in continuous adaptation and learning, the
goals with respect to systems theory become to model a
system’s limitations, dynamics, and conditions while
elucidating the related principles. These ideas are closely
linked to evolution, which can only take place when
something generates spontaneous heterogeneity. In critical
decision-making and handling of the complex processes,
the situation becomes more complicated when its dynamics
are taken into consideration in conjunction with the internal
structure as well as the environment. Accordingly, the
science of complex systems provides us with the required
conceptual and methodological equipment to deal with
emergence, self-organization, learning, adaptation, path-
dependency, diversity, transformation and evolution
through which microlevel properties lead to macrolevel
behaviors, which make it necessary to explain dynamic
behaviors and novel emerging structure in time. In this
regard, systems thinking enables one to choose and make
use of abstractions to understand those dynamics that un-
derline the individuals’ and elements’ behaviors. Measure
of complexity, being the amount of information required to
describe the behavior of a complex system, complex sys-
tems thinking with its appealing features characterize be-
haviors, allowing the model development possibility that is
capable of capturing richness and diversity of human ex-
istence. Thus, a framework is set that includes a variety of
approaches, enabling us to tackle the notion that a system’s
component parts can optimally be understood within the
context of relationships with one another as well as with
other systems, rather than viewing them in isolation and
with reductionist approach [79].
2.2 Chaos thinking, processes and complexity
Chaos and its study along with the advances in scientific
realm are important roots of modern study of complex
systems, which display nonlinear, dynamic, open qualities
and interconnection with the environment made up of many
components which interact, and new unanticipated patterns
emerge. Chaos, in this context, can be said to have more or
less strict definitions portraying a nonlinear world, and it
deals with deterministic systems with trajectories diverging
30 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-53-320.jpg)
![exponentially in time, which is also among the properties of
behaviors in complex systems. Models of chaos generally
provide the description of the dynamics of one or few
variables that are real and denoted by a decimal number,
and with the utilization of these models, the characteristic
makes the behaviors of their dynamics found and known. In
contrast, it is not always necessary that complex systems
possess a representation of this sort of form or such be-
haviors due to having different degrees of freedom with
many elements not being entirely but partially independent.
Complexity becomes further complicated when aspects of
system-environment interaction are modeled by chaos.
Another different aspect is that while complex systems’
study deals with both the dynamics and structure of the
structure, chaos, referring to randomness and disorder, ad-
dresses a few parameters and the dynamics of their values.
Randomness and disorder also belong to the part of com-
plex systems’ study and the concept of a chaotic environ-
ment can be replaced with complex environment, since
complex refers to both randomness of disorder and also
deterministic chaos [43,80]. Originated in mathematics and
physical sciences and widely used in humanities and social
sciences, complexity theory as a transdisciplinary systems
theory deals with change basically speaking [81]. In terms
of evolution, as Kauffman suggests, the fate of all complex
adapting systems, ranging from single cells to economies,
is to evolve into a natural state that is between order and
chaos, which is considered to be a compromise between
structure and unexpectedness [82]. In industrial, technical
or organizational issues, a pattern inherent therein is also
subtly poised between the pendulum order and chaos
[83,84]. Einstein put it very well that imagination has no
boundaries, which also points to the fact that complexity
theory definitely appeals to the imagination.
Scientific theories are based on the premise that they are
open to refutation by experimental studies, whereas math-
ematical models remain to be noncontradictory relying on
the mathematical logic. Scientists attempt to understand if
events from the real world conform to the given mathe-
matical model or not; and for this, they follow three steps to
construct the model with the first step being the observation
of the phenomenon, afterward, conversion into equations,
and then solving of those equations. Being a mathematical
theory and still in development, chaos theory allows one to
describe a series of phenomena from dynamics that con-
cerns the impact of forces on the objects’ motion, and in
that regard, the archetype of all theories related to dynamics
belongs to Newton, with regard to celestial motions. The
quote by Oliver Sacks summarizes the order of mathe-
matics in chaos: “I liked numbers because they were solid,
invariant; they stood unmoved in a chaotic world” [85].
Developed by a group of scientists who came from
different backgrounds like mathematics, physics and
chemistry working on the complex physical phenomena’s
dynamics, chaos theory which has some significant
cornerstone concepts also points to the roots of modern
science. Causality principle maybe the most fundamental
principle that refers to the premise that each effect has the
antecedent immediate cause. This principle, being non-
refutable is derived from Descartes (1641, Third Medita-
tion) and yet, experience does not confirm it. The
consolidation and simplification of causality principle was
achieved by Newton (1687) who maintained that initial
conditions and law of motion needed to be viewed sepa-
rately from each other, which had the calculations parallel
to the laws of Kepler. Determinism, based on causality, is
another principle which philosophically proposes that each
event is determined physically by a chain of previous oc-
currences and this chain is unbroken in nature. Laplace was
the figure who evidently stated the concept of universal
determinism following d’Holbach. It would be apt to note
that the birth of the chaos theory is related to the work of
Laplace, which posited that past and future of the solar
system could be calculated with precision and relied on the
capacity to be cognizant of the initial conditions of the
system (for further timeline-related details see [86]). Chaos
is stated to have three features which are the primary ones;
these are unpredictability, boundedness, and sensitivity to
initial conditions as put forth by Kaplan and Glass [87].
Unpredictability refers to the fact that a sequence of
numbers generated from a chaotic function does not repeat
itself; while boundedness refers that all points remain
within certain boundaries for the motion’s unpredictability
[88]. Since chaos is assumed to play different functional
roles in living systems, the principles and methods to detect
chaos should be present in the toolkit of the scientist.
Different chaotic functions yield time series that may
change in terms of complexity whose meaning relies on the
nature of the system and its underlying theory. Lower
complexity could hint rigidity in the system, whereas a
higher level of complexity shows a greater adaptability as
per the evolution [88].
Complexity theory, as a related component for complex
systems, in the realm of modern science has no precise
boundaries with no exact definition actually. Portrayed in
two different ways, as general systems theory (GST), the
grand theory and complex adaptive systems (CAS) theory
is the umbrella term. GST includes chaos theory, adaptation
and nonlinear dynamics, with some delineation with
complexity theory. The second portrayal makes a distinc-
tion between GST and complexity theory, placing systems
theory under GST and CAS under complexity theory.
Complexity science was inspired by the original systems
sciences including information theory, cybernetics and
GST. CAS includes pillars such as history, nonlinearity,
emergence, adaptability, self-organization, path de-
pendency, irreducibility, balance between order and chaos
[89]. Nonlinear dynamics, chaos theory, and adaptation/
Multi-chaos, fractal and multi-fractional AI in different complex systems Chapter | 3 31](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-54-320.jpg)
![paved with marble and streams flowing in channels along the floor
and meeting in a great and grand cistern of many-coloured marbles.
Quoth the Emir to the Shaykh Abd al-Samad, “Come, let us visit
yonder pavilion!” So they entered the first and found it full of gold
and silver and pearls and jacinths and other precious stones and
metals, besides chests filled with brocades, red and yellow and
white. Then they repaired to the second pavilion, and, opening a
closet there, found it full of arms and armour, such as gilded helmets
and Davidean[140]
hauberks and Hindi swords and Arabian spears and
Chorasmian[141]
maces and other gear of fight and fray. Thence they
passed to the third pavilion, wherein they saw closets padlocked and
covered with curtains wrought with all manner of embroidery. They
opened one of these and found it full of weapons curiously adorned
with open work and with gold and silver damascene and jewels.
Then they entered the fourth pavilion, and opening one of the
closets there, beheld in it great store of eating and drinking vessels
of gold and silver, with platters of crystal and goblets set with fine
pearls and cups of carnelian and so forth. So they all fell to taking
that which suited their tastes and each of the soldiers carried off
what he could. When they left the pavilions, they saw in the midst of
the palace a door of teak-wood marqueteried with ivory and ebony
and plated with glittering gold, over which hung a silken curtain
purfled with all manner of embroideries; and on this door were locks
of white silver, that opened by artifice without a key. The Shaykh
Abd al-Samad went valiantly up thereto and by the aid of his
knowledge and skill opened the locks, whereupon the door admitted
them into a corridor paved with marble and hung with veil-like[142]
tapestries embroidered with figures of all manner beasts and birds,
whose bodies were of red gold and white silver and their eyes of
pearls and rubies, amazing all who looked upon them. Passing
onwards they came to a saloon builded all of polished marble, inlaid
with jewels, which seemed to the beholder as though the floor were
flowing water[143]
and whoso walked thereon slipped. The Emir bade
the Shaykh strew somewhat upon it, that they might walk over it;
which being done, they made shift to fare forwards till they came to
a great domed pavilion of stone, gilded with red gold and crowned](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-57-320.jpg)
![with a cupola of alabaster, about which were set lattice-windows
carved and jewelled with rods of emerald,[144]
beyond the
competence of any King. Under this dome was a canopy of brocade,
reposing upon pillars of red gold and wrought with figures of birds
whose feet were of smaragd, and beneath each bird was a network
of fresh-hued pearls. The canopy was spread above a jetting
fountain of ivory and carnelian, plated with glittering gold and
thereby stood a couch set with pearls and rubies and other jewels
and beside the couch a pillar of gold. On the capital of the column
stood a bird fashioned of red rubies and holding in his bill a pearl
which shone like a star; and on the couch lay a damsel, as she were
the lucident sun, eyes never saw a fairer. She wore a tight-fitting
body-robe of fine pearls, with a crown of red gold on her head,
filleted with gems, and on her forehead were two great jewels,
whose light was as the light of the sun. On her breast she wore a
jewelled amulet, filled with musk and ambergris and worth the
empire of the Cæsars; and around her neck hung a collar of rubies
and great pearls, hollowed and filled with odoriferous musk. And it
seemed as if she gazed on them to the right and to the left.——And
Shahrazad perceived the dawn of day and ceased to say her
permitted say.
Now when it was the Five Hundred and
Seventy-sixth Night,
She said, It hath reached me, O auspicious King, that the damsel
seemed to be gazing at the folk to the right and to the left. The Emir
Musa marvelled at her exceeding beauty and was confounded at the
blackness of her hair and the redness of her cheeks, which made the
beholder deem her alive and not dead, and said to her, “Peace be
with thee, O damsel!” But Talib ibn Sahl said to him, “Allah preserve
thee, O Emir, verily this damsel is dead and there is no life in her; so
how shall she return thy salam?”; adding, “Indeed, she is but a
corpse embalmed with exceeding art; her eyes were taken out after](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-58-320.jpg)
![her death and quicksilver set under them, after which they were
restored to their sockets. Wherefore they glisten and when the air
moveth the lashes, she seemeth to wink and it appeareth to the
beholder as though she looked at him, for all she is dead.” At this
the Emir marvelled beyond measure and said, “Glory be to God who
subjugateth His creatures to the dominion of Death!” Now the couch
on which the damsel lay, had steps, and thereon stood two statues
of Andalusian copper representing slaves, one white and the other
black. The first held a mace of steel[145]
and the second a sword of
watered steel which dazzled the eye; and between them, on one of
the steps of the couch, lay a golden tablet, whereon were written, in
characters of white silver, the following words: “In the name of God,
the Compassionating, the Compassionate! Praise be to Allah, the
Creator of mankind; and He is the Lord of Lords, the Causer of
Causes! In the name of Allah, the Never-beginning, the Everlasting,
the Ordainer of Fate and Fortune! O son of Adam! what hath
befooled thee in this long esperance? What hath unminded thee of
the Death-day’s mischance? Knowest thou not that Death calleth for
thee and hasteneth to seize upon the soul of thee? Be ready,
therefore, for the way and provide thee for thy departure from the
world; for, assuredly, thou shalt leave it without delay. Where is
Adam, first of humanity? Where is Noah with his progeny? Where be
the Kings of Hind and Irak-plain and they who over earth’s widest
regions reign? Where do the Amalekites abide and the giants and
tyrants of olden tide? Indeed, the dwelling-places are void of them
and they have departed from kindred and home. Where be the Kings
of Arab and Ajem? They are dead, all of them, and gone and are
become rotten bones. Where be the lords so high in stead? They are
all done dead. Where are Kora and Haman? Where is Shaddad son
of Ad? Where be Canaan and Zu’l-Autád,[146]
Lord of the Stakes? By
Allah, the Reaper of lives hath reaped them and made void the lands
of them. Did they provide them against the Day of Resurrection or
make ready to answer the Lord of men? O thou, if thou know me
not, I will acquaint thee with my name: I am Tadmurah,[147]
daughter of the Kings of the Amalekites, of those who held dominion
over the lands in equity and brought low the necks of humanity. I](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-59-320.jpg)
![possessed that which never King possessed and was righteous in my
rule and did justice among my lieges; yea, I gave gifts and largesse
and freed bondsmen and bondswomen. Thus lived I many years in
all ease and delight of life, till Death knocked at my door and to me
and to my folk befel calamities galore; and it was on this wise. There
betided us seven successive years of drought, wherein no drop of
rain fell on us from the skies and no green thing sprouted for us on
the face of earth.[148]
So we ate what was with us of victual, then we
fell upon the cattle and devoured them, until nothing was left.
Thereupon I let bring my treasures and meted them with measures
and sent out trusty men to buy food. They circuited all the lands in
quest thereof and left no city unsought, but found it not to be
bought and returned to us with the treasure after a long absence;
and gave us to know that they could not succeed in bartering fine
pearls for poor wheat, bushel for bushel, weight for weight. So,
when we despaired of succour, we displayed all our riches and things
of price and, shutting the gates of the city and its strong places,
resigned ourselves to the deme of our Lord and committed our case
to our King. Then we all died,[149]
as thou seest us, and left what we
had builded and all we had hoarded. This, then, is our story, and
after the substance naught abideth but the trace.” Then they looked
at the foot of the tablet and read these couplets:—](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-60-320.jpg)
![place, that she whom thou seest here was not deluded by the world
and its frail delights, for it is faithless, perfidious, a house of ruin,
vain and treacherous; and salutary to the creature is the
remembrance of his sins; wherefore she feared her Lord and made
fair her dealings and provided herself with provaunt against the
appointed marching-day. Whoso cometh to our city and Allah
vouchsafeth him competence to enter it, let him take of the treasure
all he can, but touch not aught that is on my body, for it is the
covering of my shame[150]
and the outfit for the last journey;
wherefore let him fear Allah and despoil naught thereof; else will he
destroy his own self. This have I set forth to him for a warning from
me and a solemn trust to be; wherewith, peace be with ye and I
pray Allah to keep you from sickness and calamity.”——And
Shahrazad perceived the dawn of day and ceased saying her
permitted say.
Now when it was the Five Hundred and
Seventy-seventh Night,
She said, it hath reached me, O auspicious King, that when the Emir
Musa read this, he wept with exceeding weeping till he swooned
away and presently coming to himself, wrote down all he had seen
and was admonished by all he had witnessed. Then he said to his
men, “Fetch the camels and load them with these treasures and
vases and jewels.” “O Emir,” asked Talib, “shall we leave our damsel
with what is upon her, things which have no equal and whose like is
not to be found and more perfect than aught else thou takest; nor
couldst thou find a goodlier offering wherewithal to propitiate the
favour of the Commander of the Faithful?” But Musa answered, “O
man, heardest thou not what the Lady saith on this tablet? More by
token that she giveth it in trust to us who are no traitors.” “And shall
we,” rejoined the Wazir Talib, “because of these words, leave all
these riches and jewels, seeing that she is dead? What should she
do with these that are the adornments of the world and the](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-62-320.jpg)
![Emir,” replied the King, “that there appeared to us whilere from out
the sea a man, from whom issued a light that illumined the horizons
and he cried out, in a voice which was heard of men far and near,
saying:—O children of Ham, reverence to Him who seeth and is not
seen and say ye, There is no god but the God, and Mohammed is
the messenger of God! And he added:—I am Abu al-Abbás al-Khizr.
Before this we were wont to worship one another, but he summoned
us to the service of the Lord of all creatures; and he taught us to
repeat these words, There is no god save the God alone, who hath
for partner none, and His is the kingdom and His is the praise. He
giveth life and death and He over all things is Almighty. Nor do we
draw near unto Allah (be He exalted and extolled!) except with these
words, for we know none other; but every eve before Friday[151]
we
see a light upon the face of earth and we hear a voice saying, Holy
and glorious, Lord of the Angels and the Spirit! What He willeth is,
and what He willeth not, is not. Every boon is of His grace and there
is neither Majesty nor is there Might save in Allah, the Glorious, the
Great!” “But ye,” quoth the King, “who and what are ye and what
bringeth you to this land?” Quoth Musa, “We are officers of the
Sovereign of Al-Islam, the Commander of the Faithful, Abd al-Malik
bin Marwan, who hath heard tell of the lord Solomon, son of David
(on whom be peace!) and of that which the Most High bestowed
upon him of supreme dominion; how he held sway over Jinn and
beast and bird and was wont when he was wroth with one of the
Marids, to shut him in a cucurbite of brass and, stopping its mouth
on him with lead, whereon he impressed his seal-ring, to cast him
into the sea of Al-Karkar. Now we have heard tell that this sea is
nigh your land; so the Commander of the Faithful hath sent us
hither, to bring him some of these cucurbites, that he may look
thereon and solace himself with their sight. Such, then, is our case
and what we seek of thee, O King, and we desire that thou further
us in the accomplishment of our errand commanded by the
Commander of the Faithful.” “With love and gladness,” replied the
black King, and carrying them to the guest-house, entreated them
with the utmost honour and furnished them with all they needed,
feeding them upon fish. They abode thus three days, when he bade](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-64-320.jpg)
![his divers fetch from out the sea some of the vessels of Solomon. So
they dived and brought up twelve cucurbites, whereat the Emir and
the Shaykh and all the company rejoiced in the accomplishment of
the Caliph’s need. Then Musa gave the King of the blacks many and
great gifts; and he, in turn, made him a present of the wonders of
the deep, being fishes in human form,[152]
saying “Your
entertainment these three days hath been of the meat of these fish.”
Quoth the Emir, “Needs must we carry some of these to the Caliph,
for the sight of them will please him more than the cucurbites of
Solomon.” Then they took leave of the black King and, setting out on
their homeward journey, travelled till they came to Damascus, where
Musa went in to the Commander of the Faithful and told him all that
he had sighted and heard of verses and legends and instances,
together with the manner of the death of Talib bin Sahl; and the
Caliph said, “Would I had been with you, that I might have seen
what you saw!” Then he took the brazen vessels and opened them,
cucurbite after cucurbite, whereupon the devils came forth of them,
saying, “We repent, O Prophet of Allah! Never again will we return to
the like of this thing; no never!” And the Caliph marvelled at this. As
for the daughters of the deep presented to them by the black King,
they made them cisterns of planks, full of water, and laid them
therein; but they died of the great heat. Then the Caliph sent for the
spoils of the Brazen City and divided them among the Faithful,——
And Shahrazad perceived the dawn of day and ceased to say her
permitted say.
Now when it was the Five Hundred and
Seventy-eighth Night,
She said, It hath reached me, O auspicious King, that the Caliph
marvelled much at the cucurbites and their contents; then he sent
for the spoils and divided them among the Faithful, saying, “Never
gave Allah unto any the like of that which he bestowed upon
Solomon David-son!” Thereupon the Emir Musa sought leave of him](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-65-320.jpg)
![CRAFT AND MALICE OF WOMEN,[153]
OR THE
TALE OF THE KING, HIS SON, HIS CONCUBINE
AND THE SEVEN WAZIRS.
There was, in days of yore and in ages and times long gone before,
a puissant King among the Kings of China, the crown of crowned
heads, who ruled over many men of war and vassals with wisdom
and justice, might and majesty; equitable to his Ryots, liberal to his
lieges and dearly beloved by the hearts of his subjects. He was
wealthy as he was powerful, but he had grown old without being
blessed with a son, and this caused him sore affliction. He could only
brood over the cutting off of his seed and the oblivion that would
bury his name and the passing of his realm into the stranger’s
hands. So he secluded himself in his palace, never going in and out
or rising and taking rest till the lieges lost all tidings of him and were
sore perplexed and began to talk about their King. Some said, “He’s
dead”; others said, “No, he’s not but all resolved to find a ruler who
could reign over them and carry out the customs of government.” At
last, utterly despairing of male issue, he sought the intercession of
the Prophet (whom Allah bless and keep!) with the Most High and
implored Him, by the glory of His Prophets and Saints and Martyrs
and others of the Faithful who were acceptable to Heaven that he
would grant him a son, to be the coolth of his eyes and heir to the
kingdom after him. Then he rose forthright and, withdrawing to his
sitting-saloon, sent for his wife who was the daughter of his uncle.
Now this Queen was of surpassing beauty and loveliness, the fairest
of all his wives and the dearest to him as she was the nearest: and
to boot a woman of excellent wit and passing judgement. She found
the King dejected and sorrowful, tearful-eyed and heavy-hearted; so](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-73-320.jpg)
![she kissed ground between his hands and said, “O King, may my life
ransom thy life! may Time never prove thy foe, nor the shifts of
Fortune prevail over thee; may Allah grant thee every joy and ward
off from thee all annoy! How is it I see thee brooding over thy case
and tormented by the displeasures of memory?” He replied, “Thou
wottest well that I am a man now shotten in years, who hath never
been blessed with a son, a sight to cool his eyes; so I know that my
kingdom shall pass away to the stranger in blood and my name and
memory will be blotted out amongst men. ‘Tis this causeth me to
grieve with excessive grief.” “Allah do away with thy sorrows,” quoth
she: “long ere this day a thought struck me; and yearning for issue
arose in my heart even as in thine. One night I dreamed a dream
and a voice said to me:—The King thy husband pineth for progeny: if
a daughter be vouchsafed to him, she will be the ruin of his realm; if
a son, the youth will undergo much trouble and annoy but he will
pass through it without loss of life. Such a son can be conceived by
thee and thee only and the time of thy conception is when the moon
conjoineth with Gemini! I woke from my dream, but after what I
heard that voice declare I refrained from breeding and would not
consent to bear children.” “There is no help for it but that I have a
son, Inshallah,—God willing!” cried the King. Thereupon she soothed
and consoled him till he forgot his sorrows and went forth amongst
the lieges and sat, as of wont, upon his throne of estate. All rejoiced
to see him once more and especially the Lords of his realm. Now
when the conjunction of the moon and Gemini took place, the King
knew his wife carnally and, by order of Allah Almighty she became
pregnant. Presently she announced the glad tidings to her husband
and led her usual life until her nine months of pregnancy were
completed and she bare a male child whose face was as the rondure
of the moon on its fourteenth night. The lieges of the realm
congratulated one another thereanent and the King commanded an
assembly of his Olema and philosophers, astrologers and
horoscopists, whom he thus addressed, “I desire you to forecast the
fortune of my son and to determine his ascendant[154]
and whatever
is shown by his nativity.” They replied “‘Tis well, in Allah’s name, let
us do so!” and cast his nativity with all diligence. After ascertaining](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-74-320.jpg)
![his ascendant, they pronounced judgement in these words, “We see
his lot favourable and his life viable and durable; save that a danger
awaiteth his youth.” The father was sorely concerned at this saying,
when they added “But, O King, he shall escape from it nor shall
aught of injury accrue to him!” Hereupon the King cast aside all cark
and care and robed the wizards and dismissed them with splendid
honoraria; and he resigned himself to the will of Heaven and
acknowledged that the decrees of Destiny may not be countervailed.
He committed his boy to wet nurses and dry nurses, handmaids and
eunuchs, leaving him to grow and fill out in the Harim till he reached
the age of seven. Then he addressed letters to his Viceroys and
Governors in every clime and by their means gathered together
Olema and philosophers and doctors of law and religion, from all
countries, to a number of three hundred and three score. He held an
especial assembly for them and, when all were in presence, he bade
them draw near him and be at their ease while he sent for the food-
trays and all ate their sufficiency. And when the banquet ended and
the wizards had taken seats in their several degrees, the King asked
them, “Wot ye wherefore I have gathered ye together?”; whereto all
answered, “We wot not, O King!” He continued, “It is my wish that
you select from amongst you fifty men, and from these fifty ten, and
from these ten one, that he may teach my son omnem rem scibilem;
for whenas I see the youth perfect in all science, I will share my
dignity with the Prince and make him partner with me in my
possessions.” “Know, O King,” they replied, “that among us none is
more learned or more excellent than Al-Sindibad,[155]
hight the Sage,
who woneth in thy capital under thy protection. If such be thy
design, summon him and bid him do thy will.” The King acted upon
their advice and the Sage, standing in the presence, expressed his
loyal sentiments with his salutation, whereupon his Sovereign bade
him draw nigh and thus raised his rank, saying, “I would have thee
to know, O Sage, that I summoned this assembly of the learned and
bade them choose me out a man to teach my son all knowledge;
when they selected thee without dissenting thought or voice. If,
then, thou feel capable of what they claimed for thee, come thou to
the task and understand that a man’s son and heir is the very fruit of](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-75-320.jpg)
![his vitals and core of his heart and liver. My desire of thee is thine
instruction of him; and to happy issue Allah guideth!” The King then
sent for his son and committed him to Al-Sindibad conditioning the
Sage to finish his education in three years. He did accordingly but, at
the end of that time, the young Prince had learned nothing, his mind
being wholly occupied with play and disport; and when summoned
and examined by his sire, behold, his knowledge was as nil.
Thereupon the King turned his attention to the learned once more
and bade them elect a tutor for his youth; so they asked, “And what
hath his governor, Al-Sindibad, been doing?” and when the King
answered, “He hath taught my son naught;” the Olema and
philosophers and high officers summoned the instructor and said to
him, “O Sage, what prevented thee from teaching the King’s son
during this length of days?” “O wise men,” he replied, “the Prince’s
mind is wholly occupied with disport and play; yet, an the King will
make with me three conditions and keep to them, I will teach him in
seven months what he would not learn (nor indeed could any other
lesson him) within seven years.” “I hearken to thee,” quoth the King,
“and I submit myself to thy conditions;” and quoth Al-Sindibad,
“Hear from me, Sire, and bear in mind these three sayings, whereof
the first is:—Do not to others what thou wouldest not they do unto
thee;[156]
and second:—Do naught hastily without consulting the
experienced; and thirdly:—Where thou hast power show pity.[157]
In
teaching this lad I require no more of thee but to accept these three
dictes and adhere thereto.” Cried the King, “Bear ye witness against
me, O all ye here assembled, that I stand firm by these conditions!”;
and caused a procès verbal to be drawn up with his personal
security and the testimony of his courtiers. Thereupon the Sage,
taking the Prince’s hand, led him to his place, and the King sent
them all requisites of provaunt and kitchen-batteries, carpets and
other furniture. Moreover the tutor bade build a house whose walls
he lined with the whitest stucco painted over with ceruse,[158]
and,
lastly, he delineated thereon all the objects concerning which he
proposed to lecture his pupil. When the place was duly furnished, he
took the lad’s hand and installed him in the apartment which was
amply furnished with belly-timber; and, after stablishing him therein,](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-76-320.jpg)
![he spake not. Whereupon the courtiers were astounded and the
monarch, sore concerned for his son, summoned Al-Sindibad. But
the tutor so hid himself that none could hit upon his trace nor gain
tidings of him; and folk said, “He was ashamed to appear before the
King’s majesty and the courtiers.” Under these conditions the
Sovereign heard some of those present saying, “Send the lad to the
Serraglio where he will talk with the women and soon set aside this
bashfulness;” and, approving their counsel, gave orders accordingly.
So the Prince was led into the palace, which was compassed about
by a running stream whose banks were planted with all manner of
fruit-trees and sweet-smelling flowers. Moreover, in this palace were
forty chambers and in every chamber ten slave-girls, each skilled in
some instrument of music, so that whenever one of them played,
the palace danced to her melodious strains. Here the Prince passed
one night; but, on the following morning, the King’s favourite
concubine happened to cast eyes upon his beauty and loveliness, his
symmetrical stature, his brilliancy and his perfect grace, and love gat
hold of her heart and she was ravished with his charms.[159]
So she
went up to him and threw herself upon him, but he made her no
response; whereupon, being dazed by his beauty, she cried out to
him and required him of himself and importuned him; then she
again threw herself upon him and clasped him to her bosom kissing
him and saying, “O King’s son, grant me thy favours and I will set
thee in thy father’s stead; I will give him to drink of poison, so he
may die and thou shalt enjoy his realm and wealth.” When the
Prince heard these words, he was sore enraged against her and said
to her by signs, “O accursed one, so it please Almighty Allah, I will
assuredly requite thee this thy deed, whenas I can speak; for I will
go forth to my father and will tell him, and he shall kill thee.” So
signing, he arose in rage, and went out from her chamber; whereat
she feared for herself. Thereupon she buffeted her face and rent her
raiment and tare her hair and bared her head, then went in to the
King and cast herself at his feet, weeping and wailing. When he saw
her in this plight, he was sore concerned and asked her, “What aileth
thee, O damsel? How is it with thy lord, my son? Is he not well?”;
and she answered, “O King, this thy son, whom thy courtiers avouch](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-78-320.jpg)
![THE KING AND HIS WAZIR’S WIFE.[160]
There was once a King of the Kings, a potent man and a proud, who
was devoted to the love of women and one day being in the privacy
of his palace, he espied a beautiful woman on the terrace-roof of her
house and could not contain himself from falling consumedly in love
with her.[161]
He asked his folk to whom the house and the damsel
belonged and they said, “This is the dwelling of the Wazir such an
one and she is his wife.” So he called the Minister in question and
despatched him on an errand to a distant part of the kingdom,
where he was to collect information and to return; but, as soon as
he obeyed and was gone, the King contrived by a trick to gain
access to his house and his spouse. When the Wazir’s wife saw him,
she knew him and springing up, kissed his hands and feet and
welcomed him. Then she stood afar off, busying herself in his
service, and said to him, “O our lord, what is the cause of thy
gracious coming? Such an honour is not for the like of me.” Quoth
he, “The cause of it is that love of thee and desire thee-wards have
moved me to this.” Whereupon she kissed ground before him a
second time and said, “By Allah, O our lord, indeed I am not worthy
to be the handmaid of one of the King’s servants; whence then have
I the great good fortune to be in such high honour and favour with
thee?” Then the King put out his hand to her intending to enjoy her
person, when she said, “This thing shall not escape us; but take
patience, O my King, and abide with thy handmaid all this day, that
she may make ready for thee somewhat to eat and drink.” So the
King sat down on his Minister’s couch and she went in haste and
brought him a book wherein he might read, whilst she made ready
the food. He took the book and, beginning to read, found therein
moral instances and exhortations, such as restrained him from
adultery and broke his courage to commit sin and crime. After
awhile, she returned and set before him some ninety dishes of
different kinds and colours, and he ate a mouthful of each and found](https://image.slidesharecdn.com/23116795-250512094255-4d3a7f13/85/Multichaos-Fractal-And-Multifractional-Artificial-Intelligence-Of-Different-Complex-Systems-Yeliz-Karaca-80-320.jpg)
Multichaos Fractal And Multifractional Artificial Intelligence Of Different Complex Systems Yeliz Karaca
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- 6. Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 7. This page intentionally left blank
- 8. Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems Edited by Yeliz Karaca University of Massachusetts Medical School, Worcester, MA, United States Dumitru Baleanu Çankaya University, Ankara, Turkey and Institute of Space Sciences, Magurele-Bucharest, Romania Yu-Dong Zhang University of Leicester, Leicester, United Kingdom Osvaldo Gervasi Perugia University, Perugia, Italy Majaz Moonis University of Massachusetts Medical School, Worcester, MA, United States
- 9. Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2022 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www. elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-90032-4 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisitions Editor: Chris Katsaropoulos Editorial Project Manager: Maria Elaine D. Desamero Production Project Manager: Niranjan Bhaskaran Cover Designer: Yeliz Karaca Typeset by TNQ Technologies
- 10. Contents List of contributors xi Preface xiii Acknowledgment xvii 1. Introduction Yeliz Karaca and Dumitru Baleanu 2. Theory of complexity, origin and complex systems Yeliz Karaca 1. Introduction 9 2. Theory of complexity, origin and complex systems 13 2.1 A brief history of complexity and the related areas of different complex systems 13 2.2 Theories pertaining to complexity and their historical account 14 3. Complex order processes toward modern scientific path: from Darwin and onwards 15 3.1 A conceptual outline: complexity and complex systems 17 4. Concluding remarks and future directions 17 References 18 3. Multi-chaos, fractal and multi- fractional AI in different complex systems Yeliz Karaca 1. Introduction 21 2. Challenging dimensions of modern science, complexity and complex systems 25 2.1 Data reliability and complexity 26 2.2 Chaos thinking, processes and complexity 30 2.3 Fractal thinking, processes and complexity 33 2.4 Fractional thinking, processes and complexity 35 3. Artificial intelligence way of thinking, processes, complexity and complex systems 40 4. Concluding remarks and future directions 49 References 50 Further reading 54 4. High-performance computing and computational intelligence applications with a multi-chaos perspective Damiano Perri, Marco Simonetti, Osvaldo Gervasi and Sergio Tasso 1. Introduction 55 2. Related works 56 3. High-performance computing approaches to solving complex problems 56 3.1 Cloud containers 56 3.2 Container insights 57 3.3 GPGPU computing 58 3.4 GPGPU insights 59 3.5 GPGPU and neural networks 59 4. Quantum computing to treat multi-chaos scenarios 60 4.1 Bits and qubits 61 4.2 Quantum register 62 4.3 Relevant quantum algorithms 63 4.4 Quantum computing insights 65 5. Techniques enabling the solution of complex problems based on computational intelligence 69 5.1 Approaches based on machine learning 69 5.2 Machine learning insights 70 6. The dilemma of respecting privacy in multi-chaos situations 70 6.1 GDPR 70 6.2 AI and privacy 72 7. Conclusions 72 8. Acronyms 74 References 74 v
- 11. 5. Human hypercomplexity. Error and unpredictability in complex multichaotic social systems Piero Dominici 1. Introduction 77 2. The complexity of living energy and living beings 78 3. Complicated, complex, and hypercomplex systems 79 4. Taking a step back: a brief history of complexity 80 5. An epistemology of error 84 6. “Objects” are relations 85 7. Everything depends on everything else 87 8. Cognitive cages 88 9. è troppo, o troppo ravvicinato? 90 References 90 6. Multifractal complexity analysis- based dynamic media text categorization models by natural language processing with BERT Yeliz Karaca, Yu-Dong Zhang, Ahu Dereli Dursun and Shui-Hua Wang 1. Introduction 95 2. Data and methodology 99 2.1 Complex media text data 99 2.2 Fractal complexity analysis 99 2.3 Natural language processing 103 3. Experimental results and discussion 104 4. Conclusion and future directions 111 References 113 7. Mittag-Leffler functions with heavy-tailed distributions’ algorithm based on different biology datasets to be fit for optimum mathematical models’ strategies Dumitru Baleanu and Yeliz Karaca 1. Introduction 117 1.1 The motivation of the integrative method proposed 119 2. Complex biological datasets and methodology 120 2.1 Complex biological datasets 120 2.2 Methodology 121 3. Experimental results and discussion: com- putational application of Mittag-Leffler function based on heavy-tailed distribu- tions for different biological datasets 123 3.1 Computational applications for fit- ting Mittag-Leffler function based on heavy-tailed distributions to the can- cer cell dataset 124 3.2 Computational applications for fit- ting Mittag-Leffler function based on heavy-tailed distributions to the diabetes dataset 125 4. Conclusion and future directions 127 References 131 8. Artificial neural network modeling of systems biology datasets fit based on Mittag-Leffler functions with heavy-tailed distributions for diagnostic and predictive precision medicine Yeliz Karaca and Dumitru Baleanu 1. Introduction 133 1.1 The motivation of the integrative method proposed 135 2. Complex biological datasets and methodology 136 2.1 Complex biological datasets 136 2.2 Methodology 136 3. Experimental results and discussions: artificial neural network modeling of complex biological datasets to be fit based on Mittag-Leffler function with heavy-tailed distributions for diagnosis and prediction 140 3.1 Artificial neural network modeling of cancer cell datasets to be fit based on Mittag-Leffler function with heavy-tailed distributions for diagnosis and prediction 140 3.2 Artificial neural network modeling of diabetes datasets to be fit based on Mittag-Leffler function with heavy- tailed distributions for diagnosis and prediction 141 4. Conclusion and future directions 146 References 147 9. Computational fractional-order calculus and classical calculus AI for comparative differentiability prediction analyses of complex-systems-grounded paradigm Yeliz Karaca and Dumitru Baleanu 1. Introduction 149 1.1. The motivation and challenges of the integrative method proposed 152 vi Contents
- 12. 2. Datasets and methodology 153 2.1 The modeling of different complex datasets 153 2.2 Methods 154 2.3 Artificial neural networks 156 3. Experimental results and discussion 157 3.1 Computational application of Caputo fractional-order derivative models 157 3.2 Computational application of Caputo fractional-order derivative and classical derivative models for comparative prediction analyses of cancer cell and stroke with FFBP algorithm 160 4. Conclusion and future directions 162 References 166 10. Pattern formation induced by fractional-order diffusive model of COVID-19 Naveed Iqbal and Yeliz Karaca 1. Introduction 169 2. Model 171 2.1 Stability analysis of E2 j 1; j 2; j 3 172 3. Spatiotemporal model 172 3.1 Conditions for turing instability 173 4. Weakly nonlinear analysis 174 5. Numerical simulation 179 6. Conclusion 182 References 184 11. Prony’s series and modern fractional calculus Jordan Hristov 1. Introduction 187 2. Prony’s method 187 3. Exponential sums approximation of functions 188 3.1 Exponential sum approximation for tb 188 3.2 Exponential sums approximation of Mittag-Leffler function 189 3.3 Exponential sums approximation of the Kohlrausch function 189 4. Fractional operators in applied rheology 190 4.1 Caputo derivative 190 4.2 Caputo-Fabrizio fractional operator 190 5. Modeling linear viscoelastic responses 191 5.1 Constitutive equations: time domain 191 5.2 Frequency domain: sinusoidal responses 192 5.3 Response function 192 6. Prony’s series in linear viscoelasticity 192 6.1 Example 1. completely monotone responses as Prony’s series and related discrete spectra 192 6.2 Example 2: KWW as a stress relax- ation function 194 6.3 Example 3. Mittag-Leffler function as stress relaxation modulus 195 6.4 Example 4. The Bagley-Torvik equation 197 7. Final comments 198 References 198 12. A chain of kinetic equations of BogoliuboveBorneGreene KirkwoodeYvon and its application to nonequilibrium complex systems Nikolai (Jr) Bogoliubov, Mukhayo Yunusovna Rasulova, Tohir Vohidovich Akramov and Umarbek Avazov 1. Introduction 201 2. Formulation of the problem 202 3. The solution of the BBGKY hierarchy for many-type particle systems 204 3.1 Introduction 204 3.2 Formulation and solution of the problem 204 4. Derivation of the GrossePitaevskii equation from the BBGKY hierarchy 206 4.1 Formulation of the problem 207 4.2 Derivation of hierarchy of kinetic equations for correlation matrices 207 4.3 For the case s ¼ 1 209 4.4 Another method for deriving the GrossePitaevskii equation 210 5. Summary 211 References 211 Further reading 213 13. Hearing loss detection in complex setting by stationary wavelet Renyi entropy and three-segment biogeography-based optimization Yabei Li, Junding Sun and Chong Yao 1. Introduction 215 Contents vii
- 13. 2. Dataset 216 3. Methods 216 3.1 Feature extractiondstationary wavelet Renyi entropy 218 3.2 Single hidden layer feedforward neural network 219 3.3 Three-segment biogeography-based optimization 220 4. Implementation 222 5. Measure 222 6. Experiment results and discussions 224 6.1 Statistical analysis of the proposed method 224 6.2 Biogeography-based optimization versus three-segment bio- geography-based optimization 224 6.3 Optimal decomposition level 224 6.4 Comparison to state-of-the-art approaches 225 7. Conclusions 226 Appendix 227 References 228 14. Shannon entropy-based complexity quantification of nonlinear stochastic process: diagnostic and predictive spatiotemporal uncertainty of multiple sclerosis subgroups Yeliz Karaca and Majaz Moonis 1. Introduction 231 2. Materials and methods 234 2.1 Materials 234 2.2 Methods 234 2.3 k-Nearest neighbor and decision tree algorithms 237 3. Experimental results 238 4. Conclusion and future directions 241 References 243 15. Chest X-ray image detection for pneumonia via complex convolutional neural network and biogeography-based optimization Xiang Li, Mengyao Zhai and Junding Sun 1. Introduction 247 2. Dataset 248 3. Methodology 249 3.1 Complex convolutional neural network 249 3.2 Biogeography-based optimization 251 3.3 Implementation 254 3.4 Measure 256 4. Experiment results and discussions 256 4.1 Confusion matrix of the proposed method 256 4.2 Statistical results 257 4.3 Optimal number of fully connected layers 258 4.4 Comparison to state-of-the-art approaches 258 5. Conclusions 260 References 260 Appendix 261 16. Facial expression recognition by DenseNet-121 Bin Li 1. Introduction 263 2. Dataset 264 3. Methodology 264 3.1 Convolution 264 3.2 Pooling 265 3.3 Batch normalization 266 3.4 Rectified linear unit 267 3.5 K-fold cross-validation 268 3.6 DenseNet-121 270 4. Experiment result and discussions 271 4.1 Statistical analysis 271 4.2 Comparison with state-of-the-art approaches 273 5. Conclusions 275 References 275 17. Quantitative assessment of local warming based on urban dynamics Lucia Saganeiti, Angela Pilogallo, Francesco Scorza, Beniamino Murgante, Valentina Santarsiero and Gabriele Nolè 1. Introduction 277 2. Study areas 277 3. Materials and methods 278 3.1 Urbanization dynamics 279 3.2 Land surface temperature 280 4. Results and discussion 281 5. Conclusions 286 References 287 viii Contents
- 14. 18. Managing information security risk and Internet of Things (IoT) impact on challenges of medicinal problems with complex settings: a complete systematic approach Eali Stephen Neal Joshua, Debnath Bhattacharyya and N. Thirupathi Rao 1. Introduction to information security 291 1.1 Various vulnerabilities in healthcare 292 2. Information security in healthcare 296 2.1 Background of health information privacy and security 296 2.2 State of information security research in healthcare 298 2.3 Threats to information privacy 298 3. Impact of IoT in medical problems 300 3.1 Internet of Things in healthcare 300 3.2 Challenges of IoT in medical problems 301 3.3 Applications of IoT in healthcare 302 4. Medical problems with complex settings 303 4.1 The challenge of interoperability 303 4.2 Keeping up with old technology 303 4.3 User-unfriendly interfaces 303 4.4 Exacerbating malpractice claims 303 4.5 Overcomplicated asset tracking 304 4.6 Overall implementation 304 5. IoT and information security 304 5.1 Understanding the needs of IoT security 304 5.2 Data interoperability and information security 305 5.3 Information security issues of e-health 306 5.4 Healthcare information system with complex settings 306 5.5 Providers’ perspective of regulatory compliance 307 5.6 Information-access control 308 6. Challenges of medicinal problems using IoT: a case study 309 7. Conclusion 309 References 310 19. An extensive discussion on utilization of data security and big data models for resolving healthcare problems N. Thirupathi Rao, Debnath Bhattacharyya and Eali Stephen Neal Joshua 1. Information security 311 1.1 Confidentiality 311 1.2 Integrity 311 1.3 Availability 311 1.4 Information security policy 312 1.5 Information security measures 312 1.6 Managing information security 312 2. Internet of Things 312 2.1 Connecting with the IoT 313 2.2 IoT for physicians 313 2.3 IoT for hospitals 313 2.4 IoT for health insurance companies 314 2.5 IoT for patients 314 2.6 Redefining healthcare 314 3. Information security and IoT 315 3.1 Information security threats 315 3.2 Information security threats? 315 4. Data security and IoT in medicine 316 4.1 Benefits of IoT healthcare 316 4.2 Challenges in information security and IoT with respect to medicine 317 5. Big data and its applications 318 6. IoT and big data applications in medicine 319 7. Complex system in healthcare 321 8. Role of IoT and big data applications in medicine 323 9. Conclusion 323 References 323 Index 325 Contents ix
- 15. This page intentionally left blank
- 16. List of contributors Tohir Vohidovich Akramov, Nuclear Physics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan; Na- tional University of Uzbekistan, Tashkent, Uzbekistan Umarbek Avazov, Nuclear Physics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan Dumitru Baleanu, Çankaya University, Ankara, Turkey; Institute of Space Science, Magurele, Bucharest, Romania Debnath Bhattacharyya, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, Andhra Pradesh, India Nikolai (Jr) Bogoliubov, Steklov Institute of Mathematics of the Russian Academy of Sciences, Moscow, Russia Piero Dominici, CHAOSeInternational Research and Ed- ucation Programme “Complex Human Adaptive Orga- nizations and Systems”, Perugia University, Italy; Department of Philosophy, Social, Human and Educa- tional Sciences, University of Perugia, Italy; WAAS - World Academy of Art and Science, Rome, Italy Ahu Dereli Dursun, Institute of Social Sciences, Com- munication Studies, Istanbul Bilgi University, Istanbul, Turkey Osvaldo Gervasi, University of Perugia, Perugia, Italy Jordan Hristov, University of Chemical Technology and Metallurgy, Sofia, Bulgaria Naveed Iqbal, University of Ha’il, Ha’il, Saudi Arabia Yeliz Karaca, University of Massachusetts Medical School, Worcester, MA, United States Xiang Li, Henan Polytechnic University, Jiaozuo, Henan, PR China Yabei Li, Henan Polytechnic University, Jiaozuo, Henan, PR China Bin Li, Henan Polytechnic University, Jiaozuo, Henan, PR China Majaz Moonis, University of Massachusetts Medical School, Worcester, MA, United States Beniamino Murgante, University of Basilicata, Via del- l’Ateneo Lucano, Potenza, Italy Eali Stephen Neal Joshua, Vignan’s Institute of Infor- mation Technology (A), Visakhapatnam, Andhra Pra- desh, India Gabriele Nolè, CNR-IMAA, C.da Santa Loja Zona Industriale Tito Scalo, Potenza, Italy Damiano Perri, University of Florence, Firenze, Italy; University of Perugia, Perugia, Italy Angela Pilogallo, University of Basilicata, Via dell’Ateneo Lucano, Potenza, Italy N. Thirupathi Rao, Vignan’s Institute of Information Technology (A), Visakhapatnam, Andhra Pradesh, India Mukhayo Yunusovna Rasulova, Nuclear Physics, Acad- emy of Sciences of Uzbekistan, Tashkent, Uzbekistan Lucia Saganeiti, University of L’Aquila, L’Aquila, Italy Valentina Santarsiero, University of Basilicata, Via del- l’Ateneo Lucano, Potenza, Italy; CNR-IMAA, C.da Santa Loja Zona Industriale Tito Scalo, Potenza, Italy Francesco Scorza, University of Basilicata, Via dell’Ate- neo Lucano, Potenza, Italy Marco Simonetti, University of Florence, Firenze, Italy; University of Perugia, Perugia, Italy Junding Sun, Henan Polytechnic University, Jiaozuo, Henan, PR China Sergio Tasso, University of Perugia, Perugia, Italy Shui-Hua Wang, University of Leicester, Leicester, United Kingdom Chong Yao, Henan Polytechnic University, Jiaozuo, Henan, PR China Mengyao Zhai, Hebi Polytechnic, Hebi, Henan, PR China Yu-Dong Zhang, University of Leicester, Leicester, United Kingdom xi
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- 18. Preface Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems is an edited book that addresses different uncertain processes inherent in the complex systems, attempting to provide global and robust optimized solutions distinctively through multifarious methods, technical analyses, modeling, optimization processes, numerical simulations, case studies and appli- cations not excluding theoretical aspects of complexity. Based on advanced mathematical foundation, our edited book foregrounds multichaos, fractal, multifractional, fractional calculus, fractional operators, quantum, wavelet, entropy-based applications and artificial intelligence (AI) mathematics-informed and data-driven processes. The primary focus and purpose, herein, is related to the needs and solutions for new analytic strategies and mathematical modeling to attain accurate, timely and optimized solutions. Appealing to an interdisciplinary network of scientists and researchers to disseminate the theory and application of multichaos, fractal and multifractional AI of different complex systems in medicine, neurology, mathematics, physics, biology, chemistry, information theory, engineer- ing, computer science, social sciences and other far- reaching domains, the overarching aim is to enable the provision of global and optimized robust solutions distinctively with a perspective through multifarious methods, different from the conventional perspective, as directed toward paradoxical situations, different uncertain processes, nonlinear dynamic systems inherent in complex systems. Elaborating on the most intriguing theoretical as- pects, modeling and applications of multichaos, fractal, multifractional, fractional calculus, fractional operators, quantum, wavelet, entropy-based applications and AI mathematics-informed and data-driven processes around the common theme of complexity and nonlinearity under consideration, current applications, future directions and perspectives, limitations, strengths and opportunities are provided in our edited book for scientists, researchers, students, and anyone who is interested in the enigma of complexity. The invaluable inputs of 31 experts worldwide specialized in mathematics, physics, biology, chemistry, neurology, information theory, computer science, engi- neering, applied sciences, sociology, philosophy and communication, among others, from 11 countries, are sig- nificant to establish a holistic body of work and spectrum, owing to their personal contributions in their respective fields. The edited book includes a total of 19 chapters, as has been inspired by the aforesaid considerations; the chapters along the book are outlined in terms of their content as follows. Chapter 1 is the “Introduction” (by Yeliz Karaca and Dumitru Baleanu), which provides the basic motivations underlying complexity, complexity thinking and theory along with the important role of computational processes with extensive applications in integration with fractals, multifractals, fractional methods, chaos, nonlinear dynam- ical properties and stochastic elements. Computational technologies, with machine learning as the core component of AI, is stated to have broad use and transformative im- pacts, enabling the training of complex data to automate or augment some of the critical human skills. Thus, it is presented that our edited book foregrounds multichaos, fractal, and multifractional in the era of AI, which requires the integration of advanced mathematical models and mathematics-informed frameworks as well as AI address- ing fractal, fractional calculus, fractional operators, quan- tum, wavelet, entropy-based applications aside from the means of modeling, technical analyses and numerical simulations as some of the most broadly employed methods for the solution of multifaceted problems characterized by nonlinearity, nonregularity, self-similarity and many other properties, frequently encountered in different complex systems. Accordingly, the chapter presents the overarching aim of the edited book of ours, its key objectives, moti- vational aspects and the detailed content of all other chapters presented herein. Chapter 2 entitled “Theory of Complexity, Origin and Complex Systems” (by Yeliz Karaca) attempts to touch on the possible dimensions of complex systems in different fields with a focus on origin-related, historical, evolutionary and epistemological viewpoints of complexity by taking into consideration the various multiple interacting factors of systems with the goal of providing a global un- derstanding between variables, sensitivity to initial control, and strange, nonperiodic and unpredictable time evolution. The detailed presentation in the chapter tries to ensure that xiii
- 19. the foundation for the complex systems’ interpretations can be explored in different related areas of complexity. Chapter 3 “Multichaos, Fractal and Multifractional AI in Different Complex Systems” (by Yeliz Karaca) provides an overview including multichaos, fractal, fractional and AI way of thinking with regard to the solutions of the complex system problems concerned with natural and social sci- ences. Ethical decision-making frameworks and strategies related to big data and AI applications are also presented in detail to enable assistance for the identification of the related problems in different settings and thinking methodically so that tensions between conflicting aspects can be managed systematically. Chapter 4, “High-Performance Computing and Computational Intelligence Applications with Multichaos Perspective” (by Damiano Perri, Marco Simonetti, Osvaldo Gervasi and Sergio Tasso), addresses the experience of the COVID-19 pandemic, which has accelerated many chaotic processes in modern society besides revealing the need to understand complex processes to achieve common well- being in a very serious and emergent way. A set of best practices and case studies, which provide assistance to the researchers while handling computationally complex problems, are presented in the chapter, providing a general sketch of various topics, which could be of help to re- searchers and developers to deal with complex and chaotic situations within the scope of machine learning and the issue of privacy including the recent related regulations. Chapter 5 “Human Hypercomplexity. Error and Unpredictability in Complex Multichaotic Social Systems” (by Piero Dominici) has the perspective that traditional linear models and deterministic approaches can no longer be capable of the analyzing the dynamics of unstable dynamics. The chapter provides perspectives on the complexity of living energy and living beings, along with 12 essential planes of awareness, the characteristics of complicated, complex and hypercomplex systems, episte- mology of error and complex and chaotic characteristics of social systems. Chapter 6 “Multifractal Complexity Analysis-Based Dynamic Media Text Categorization Models by Natural Language Processing with BERT” (by Yeliz Karaca, Yu-Dong Zhang, Ahu Dereli Dursun and Shui-Hua Wang) addresses the challenges and complexity inherent in digital- based complex media texts. The study puts forth the significance of the fractal behavior while articulating the distinguishing quality of BERT owing to its capability of classification accuracy and adaptiveness into integrated methodologies. Chapter 7 (Part I) “Mittag-Leffler Functions With Heavy-Tailed Distributions’ Algorithm Based on Different Biology Datasets to be Fit for Optimum Mathematical Models’ Strategies” (by Dumitru Baleanu and Yeliz Karaca) addresses the challenges of integrating fractional calculus in cases of complexity, which necessitates an effective use of empirical, numerical, experimental, and analytical methods to tackle complexity. The proposed in- tegrated approach in this chapter uses the MittageLeffler function with two parameters (a, fl) for the purpose of investigating the dynamics of two diseases: cancer cell and diabetes. Chapter 8 (Part II) “Artificial Neural Network Modeling of Systems Biology Datasets Fit Based on Mittag-Leffler Functions with Heavy-Tailed Distributions for Diagnostic and Predictive Precision Medicine” (by Yeliz Karaca and Dumitru Baleanu) obtains the generation of optimum model strategies for different biology datasets along with the Mittag-Leffler functions with heavy-tailed distri- butions. The integrative modeling scheme proposed in the chapter is concerned with the applicability and reliability of the solutions obtained by the two-parametric Mittag-Leffler functions with heavy-tailed distributions. Accordingly, the proposed integrated approach in this chapter investigates the dynamics of diseases related to biological elements. The application of multilayer perceptron, as one of the Artificial Neural Network (ANN) algorithms, is directed for the diagnostic and predictive purpose of the disease. The content of the chapter intends to enable the building of precise models to avoid unpredictable risks and identify opportunities in nonlinear complex situations, along with the integration of precision medicine. Chapter 9 “Computational Fractional Order Calculus and Classical Calculus AI for Comparative Differentiability Prediction Analyses of Complex Systems-grounded Para- digm” (by Yeliz Karaca and Dumitru Baleanu) intends to provide an intermediary facilitating function for both the physicians and individuals through establishing an accurate and robust model based on the integration of fractional order calculus and ANN in terms of the diagnostic and differentiability predictive purposes with the diseases, which display highly complex properties. The integrative and multistaged approach proposed includes the application of the Caputo fractional derivative with two-parametric Mittag-Leffler function on the stroke dataset and cancer cell dataset. The chapter reveals that modeling many complex systems can be possible by fractional order de- rivatives based on fractional calculus and computational complexity is shown to provide us with applicable sets of ideas or integrative paradigms to understand the intricate properties of complex systems. Chapter 10 “Pattern Formation Induced by Fractional Order Diffusive Model of COVID-19” (by Naveed Iqbal and Yeliz Karaca) presents the investigation of the Turing instability produced by fractional diffusion in a COVID-19 model. Differential equations with complex order fractional derivatives enable the regulation of complicated fractional systems, positive equilibrium points have been initially specified, and Routh-Hurwitz criteria have been used to xiv Preface
- 20. assess the stability of positive equilibrium point. Local equilibrium points and stability analysis have been employed to find the conditions for Turing instability. The analysis, by exploring the system’s dynamical behavior and the bifurcation point centered on the death rate, anticipates to serve as a leverage in different disciplines concerning COVID-19 model through the lenses of distinct viewpoints; and within that framework, fractional calculus is known to unfold the fundamental mechanisms and multiscale dynamic phenomena. Chapter 11 “Prony’s Series in Time and Frequency Domains and Relevant Fractional Models” (by Jordan Hristov) deals with Prony’s series approximation of mono- tonically responses in material viscoelastic rheology and the possibilities of implementation on this sort of basis relying on modern fractional operations with nonsingular kernels, which is to say the Caputo-Fabrizio operator. The chapter provides the origins of Prony’s series in time and frequency domains together with the relevant approxima- tion and calculation techniques. In this way, contributions in pure mathematics and experimental aspects are put forth, while the elaboration and application of Prony’s series are said to have the extension possibility to modeling problems emerging in mechanical engineering, chemical engineering and other related disciplines. Chapter 12 “A Chain of Kinetic Equations of Bogoliu- bov-Born-Green-Kirkwood-Yvon and Its Application to Nonequilibrium Complex Systems” (by Nicolai (Jr) Bogo- liubov, Mukhayo Yunusovna Rasulova, Tohir Akramov and Umarbek Avazov) is directed to the study of the Bogoliubov- Born-Green-Kirkwood-Yvon chain of kinetic equations (BBGKYchke) and its applications to modern problems of physics. The chapter has the focus on the need of creating a mathematical apparatus fulfilling the existing theory of one- particle systems and systems made up of a huge number of particles. Two types of BBGKY chains are addressed for both classical and quantum particle systems. The solution of the BBGKYchqke for generalized Yukawa potential (gYp) is provided, solving of the BBGKYchqke with the gYp for systems of many type particles is also elaborated on, and the Gross-Pitaevskii equation derived based on the BBGKYchqke is presented. Chapter 13 “Hearing Loss Detection in Complex Setting by Stationary Wavelet Renvi Entropy and Three- Segment Biogeography-Based Optimization” (by Yabei Li) addresses hearing loss with the main objective of improving the accuracy and efficiency of detecting images in sensorineural hearing loss through a new solution. To this end, an improved feature extraction method stationary wavelet Renvi entropy as well as optimization algorithm for model and feature extraction, namely three-segment biogeography-based optimization have been proposed. Chapter 14 “Shannon Entropy-Based Complexity Quantification of Nonlinear Stochastic Process: Diagnostic and Predictive Spatio-temporal Uncertainty of Multiple Sclerosis Subgroups” (by Yeliz Karaca and Majaz Moonis) aims at facilitating the accurate classification and course of three subgroups of multiple sclerosis (MS) (relapsing remitting MS, secondary progressive MS, primary pro- gressive MS), which is a debilitating neurological disease. An entropy-based feature selection method (Shannon entropy and minimum redundancy maximum relevance) as well as linear transformation methods (principal component analysis and linear discriminant analysis) have been applied. Each new dataset obtained has been addressed as input for the training procedure of k-nearest neighbor and decision tree algorithms. The accuracy rates for the MS subgroups’ classification have also been analyzed comparatively based on the optimized experimental results, which demonstrate that Shannon entropy, as a distinctive entropy method, has proven to be higher in terms of ac- curacy compared with the other feature selection methods. Accordingly, a new perspective with a multilevel aspect has been presented to cope with the complex dynamic systems where uncertainty and heterogeneity prevail for critical decision-making and manageable tracking in medicine and relevant fields. Chapter 15 “Chest X-ray Image Detection for Pneu- monia via Complex Convolutional Neural Network and Biogeography-Based Optimization” (by Xiang Li, Meng- vao Zhai and Junding Sun) proposes a novel chest X-ray image detection for pneumonia. The detection model proposed is reliant on the combination of complex con- volutional neural network (CNN) and biogeography-based optimization. It has been proven that the model has higher sensitivity and accuracy in terms of detecting the pneumonia-related chest X-ray images with a detection performance being significantly better than that of advanced approaches in complex medical settings. The utilization of BBO, employed as the global optimization algorithm of the related model, also provides the benefit of optimizing the stride size of the convolution kernel on CNN to obtain better detection effects with less model training cost. Chapter 16 “Complex Facial Expression Recognition by DenseNet-121” (by Bin Li) is concerned with facial expression recognition system, which has gradually been integrated into different fields of our lives with the advent of AI era. The application prospects of intelligent face recognition via computer technology are very broad, which can also be applied to the diagnosis of facial paralysis in medicine. Handling the complex nature of facial expression since it involves emodiversity and emotional complexity, the chapter shows that facial expression recognition is a difficult task bringing about some problems such as low accuracy and poor generalization ability of network model recognition. To address these challenges, the authors have proposed a DenseNet-121 image feature extraction method, Preface xv
- 21. combined with CNN for facial expression recognition. The presentation of an improved face emotion recognition system proposed employing a method based on densely connected neural network also facilitates the avoiding of complex feature extraction required by traditional deep learning while saving on the training time. Chapter 17 “Quantitative Assessment of Local Warm- ing Based on Complex Urban Dynamics Using Remote Sensing Techniques” (by L. Saganeiti, Angela Pilogallo, Francesco Scorza, Valentina Santarsiero, Gabriele Nole and Beniamino Murgante) is concerned with urban growth, which is one of the cornerstones of sustainable develop- ment policies that require to be implemented at initial states for a well-managed urbanization process and experience. The chapter provides a simultaneous analysis of the vari- ations of land surface temperature and urbanized environ- ment over a period of 15 years within two regions that differ in size, population density, and growth dynamics. The research also provides an appealing and innovative contribution to grasp the relationships between urban growth spatial patterns and the urban thermal environment. Detailed analyses presented in the chapter are beneficial in supporting decision-making processes underlying future urban policies and assessment of development scenarios with regard to quality of life, environmental sustainability and preservation of ecosystems. Chapter 18 “Managing Information Security Risk and Internet of Things Impact on Challenges of Medicinal Problems with Complex Settings: a Complete Systematic Approach” (by Eali Stephen Neal Joshua, Debnath Bhatta- charyya and N. Thirupathi Rao), discovers the crossway of healthcare and significant data, providing details with respect to information security, different vulnerabilities in health- care, data breaches, distributed denial of service assaults, insider threats, information security in healthcare, health information privacy and security, and various information threat elements regarding medical health reports. The chapter also points out the impact of IoT in medical problems, IoT in healthcare, and challenges in IoT in medical problems. The information threats are outlined in detail in the chapter, which presents the challenges of medicinal problems using IoT through a case study that shows the efficiency of IoT owing to exponentially increasing patient monitoring (blood pressure monitoring, glucose monitoring, and pulse rate monitoring) in the healthcare plans. Chapter 19 is entitled “An Extensive Discussion on Uti- lization of Data Security and Big Data Models for Resolving Complex Healthcare Problems” (by N. Thirupathi Rao, Debnath Bhattacharyya and Eali Stephen Neal Joshua), and it is concerned with the utilization of technology in the health- care settings with a focus on the employment of the IoT technology, providing an extensive elaboration of its oppor- tunities, benefits, impacts, existing gaps, security threats and adaptive frameworks that need to be developed. The chapter, with updated information for our current time, presents detailed discussions on big data in healthcare, information security, confidentiality, integrity, and availability by considering the related stakeholders in the area that are the physicians, patients, hospitals and insurance companies. The chapter presents the complex system with its components in various healthcare domains, and this attribute concerns many different disciplines including but not limited to medicine, microbiology, biomedical engineering, computer science and big data analytics. Awareness into and efficient management of all the components involved is noted to have benefits for the patients who will be knowledgeable in terms of pertinent medical resources and faith in healthcare professionals. In addition, access into a variety of medical services based on technological devices will be of great benefit to all the stake- holders and complex settings. We are of the opinion and anticipation that our edited book will provide new dimensions into layers of complexity thinking, momentum to progressive ideas into complexity, complexity thinking and processes, and above all out-of-the-box way of thinking for everyone interested in the theory, applications and modeling of complexity and different complex systems. September, 2021 Yeliz Karaca University of Massachusetts Medical School, Worcester, United States Dumitru Baleanu Çankaya University, Ankara, Turkey and Institute of Space Sciences, Magurele-Bucharest, Romania Yu-Dong Zhang University of Leicester, Leicester, United Kingdom Osvaldo Gervasi University of Perugia, Perugia, Italy Majaz Moonis University of Massachusetts Medical School, Worcester, MA, United States xvi Preface
- 22. Acknowledgment Yeliz Karaca would like to express her deep respect and gratitude to her family members: her mother, Fahrive Ekecik Karaca; her father, Emin Karaca; and her brother, Mehmet Karacaandhisfamilywhohavealwaysprovidedunconditional truelove,offeringallkindsofsupportallthewaythrough.Yeliz Karaca is also sincerely indebted to her ancestor, late grand- father, Hasan Hüseyin Ekecik, holding the superiority service award by the Turkish Grand National Assembly for his bene- ficial contributions in public welfare, education and social development both at national and international scales, whom she has taken as an esteemed role model in her life. Dumitru Baleanu would like to thank his wife Mihaela-Cristina for her continuous support. Yu-Dong Zhang would like to express his acknowl- edgment to all his family members, including his wife and son, who support his research work all the time. Osvaldo Gervasi would like to express his deepest thanks for the continuous support in the course of his work to his wife Lorella Giovannelli and his children Marta, Andrea and Damiano and to his parents Loretta Pucci and Angelo Gervasi for the profound values that Osvaldo Gervasi was able to transmit to his children. Majaz Moonis is deeply grateful to his father Professor Moonis Raza who taught and encouraged the idea of research, his mother and wife who in all adversities stood behind him and made it possible to continue his work. xvii
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- 24. Chapter 1 Introduction Yeliz Karaca1 and Dumitru Baleanu2,3 1 University of Massachusetts Medical School, Worcester, MA, United States; 2 Çankaya University, Ankara, Turkey; 3 Institute of Space Science, Magurele, Bucharest, Romania Complexity, having existed since antiquity, entails the understanding of the complex components’ origin, with meticulous computations and causal processes. Nonline- arity, self-organization, adaptation, synchronization, noise, a high number of descriptive variables or dimensions involved in the description of differential equation systems, and reaction to responses in the external environment are some of the numerous characteristics of a complex system in which multiple interactions emerge. Along these lines, complexity thinking and theory, one of the basic premises of which is the acknowledging of the existence of a hidden order to the behavior and evolution of complex systems, requires a horizon that takes the subtle and hidden prop- erties of different domains into account, necessitating their own means of optimized solutions and applicability. Bearing in mind the quote by Stephen Hawking: I think the next [21st] century will be the century of complexity is critically significant not only for this era but also for on- wards. Accordingly, the idea of complexity is stated to be part of a new unifying framework for science and a revo- lution in our understanding of systems the behavior of which has proved to be difficult in terms of prediction, management and control. In a complex system, different and multiple ways need to be contemplated for the provision of solutions and sorting out the problems. The system is likely to change depending on these selections, which shows us the complex systems’ adaptiveness. And, the more insight is developed, the answers to the problems keep changing which enables more learning in the process. Given this, modern science has embarked on the attempts for a thorough, holistic, multifaceted and accurate interpretation of natural and physical phenomena, which has proven to provide suc- cessful models for the analysis of complex systems and harnessing of control over the various related processes. Computational complexity, in this regard, comes to the foreground by providing the applicable sets of ideas and/or integrative paradigms to recognize and understand the intricate properties and dynamics of many different com- plex systems. The lenses of such transformative thinking in conjunction with mathematics-informed frameworks encompass chaos, fractal and multi-fractional ways as well as the indispensable incorporation of technology, with Artificial Intelligence, as a far-reaching leg, which are all essentially required to be capable of addressing and tack- ling complexity manifesting chaotic, nonlinear, and dy- namic characteristics. Chaos refers to irregular and unpredictable behavior characterized by sensitive reliance on initial conditions. The tendency of nature toward pattern formation, iteration and creation of order out of chaos all point to the generation of expectations of predictability. Chaos and its study in con- sort with the advances in scientific realm are important roots of modern study of complex systems that display dynamic, nonlinear, open qualities and interconnection with the environment constituting many interacting com- ponents, with new unanticipated patterns emerging. Chaos, in this context, is said to have somehow strict definitions portraying a nonlinear world, addressing deterministic systems with trajectories diverging exponentially in time, which is also among the properties of behaviors in complex systems. In mathematics and physics, chaos theory is concerned with the nonlinear dynamical systems’ behavior, which under certain circumstances exhibits a phenomenon referred to as chaos marked by sensitivity to initial conditions. Fractals are also components of dynamic systems, being the images thereof, driven by recursion, which is to say the image of chaos. Accordingly, fractals are used for modeling structures where patterns recur repeatedly and describe random or chaotic phenomena. For the handling of complex systems, the concept of progressive smoothness on finer scales may not always prove to be useful as a starting point from mathematical point of view. This Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems. https://doi.org/10.1016/B978-0-323-90032-4.00013-4 Copyright © 2022 Elsevier Inc. All rights reserved. 1
- 25. acknowledgment is important as a fundamental change in outlook when traditional geometry studying the properties of objects and spaces with integral dimensions is not useful. Effective fractional dimensions of objects, named as frac- tals, are integrated into an integral dimension space. Being never-ending patterns, fractals can be curves or geometric figures, with each part appearing to be the same as the whole pattern, which is called self-similarity brought about by a process or function’s iterative repetition. Fractals are, in other words, images of dynamic systems driven by recur- sion, namely the image of chaos. Fractals are employed to model structures in which patterns recur in a repeated way and to describe random or chaotic phenomena. The advent of increasing capacity of computational processes in numerical methods, interest in fractional de- rivative equations (FDEs) has been on the rise to be able to represent complex physical courses where dynamics may not be as accurately detected through classical differential equations. Fractional dynamics, in this regard, refers to such systems for which derivatives and integrals of frac- tional orders are employed to describe objects likely to be characterized by power-law nonlocality, fractal properties, or long-range dependence. For this reason, fractional-order system model can be regarded as a key for describing the system performance in a better way, with predictive reli- ability and applicability. In view of these concepts and challenges, it is important not to disregard data reliability, chaos thinking and processes, fractal thinking and pro- cesses, as well as artificial intelligence way of thinking and processes around complexity as the common theme under consideration. The related computational processes with broad applications in integration with fractals, multi- fractals, fractional methods, chaos, nonlinear dynamical properties, stochastic elements and so forth can provide systematic optimized solutions. Furthermore, computa- tional technologies, with machine learning as the core component of AI, enjoy the broad use and transformative impacts enabling us to train complex data to automate or augment some of the critical human skills. Hence, the crosscutting nature of AI provides motivational power to formulize research in a systematic way. Artificial neural networks (ANNs), which are networks of computer systems inspired by the human brain and biological neural networks have the capability of learning and modeling complex, dynamic and nonlinear relationships. As the simplification, abstraction and simulation of the human brain, ANNs also reflect the related fundamental characteristics of this com- plex organ. Thus, optimized solutions need to be conceived and applied in a facilitating way and efficiently with some required degree of flexibility, too. Considering the impact of data technologies vis-à-vis all aspects of conditions of modern era and life, it becomes highly important to establish a balance between data use and ethical matters. Computational technologies in different complex systems based on mathematical-driven informed frameworks can enable the generation of more realistic, applicable, adaptive models open to learning and flexibility under transient, dynamic, chaotic and ever-evolving conditions of different complex systems. To put it differently, complexity along with all the variations in networks and systems demonstrates that the decisions made are not based on one single parameter per se, but also on multiple numbers of parameters with hid- den and subtle information being at stake. To this end, multifarious adaptive methods within mathematics- informed frameworks have gained prominence for the optimized solution of complex problems. This will enable us to ensure that solution is not superficial or pretentious but reliable, robust, and smooth enabling the maintenance of quality, sustainability and meritocracy. The overarching aim of this book is to address the need concerning novel analytic strategies and mathematical modeling to achieve reliable and optimized global solutions with regard to Multi-chaos, Fractal, and Multi-fractional in the era of Artificial Intelligence, which requires the indis- pensable integration of advanced mathematical models and AI for a much smarter level of blended systems in complex settings. Appealing to an interdisciplinary network of sci- entists and researchers to disseminate the theory and application of Multi-chaos, Fractal, and Multi-fractional AI of Different Complex Systems in medicine, neurology, mathematics, physics, biology, chemistry, information theory, engineering, computer science, social sciences and other far-reaching domains, the primary focus is to enable the provision of global and optimized robust solutions distinctively with a perspective through multifarious methods, different from the conventional perspective, as directed toward paradoxical situations, different uncertain processes, nonlinear dynamic systems inherent in complex systems. Based on these ideas and consideration, the prominent objectives of our edited book can be outlined as follows: - Constructing and presenting a multifarious approach for critical decision-making processes embodying para- doxes and uncertainty, - Combining theory and applications with regard to multi-chaos, fractal and multi-fractional AI of different complex systems and many-body systems, - Enabling the provision of global and optimized robust solutions distinctively with a perspective through multifarious methods and mathematics-informed frameworks, as different from the conventional perspective, - Providing an outlook directed toward the prediction and management of paradoxical situations, different uncer- tain processes, and nonlinear dynamic components inherent in a given complex system, 2 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 26. - Facilitating the dissemination of theory and application of multi-chaos, fractal, and multi-fractional AI in different complex systems of various areas, - Establishing a balance between data use and ethical matters while employing computational technologies in different complex systems of numerous domains, - Acting as a bridge between application of advanced computational mathematical methods and AI based on comprehensive analyses and broad theories. Accordingly, each chapter of this edited book addresses different uncertain processes inherent in the complex sys- tems and attempts to provide accurate, flexible, global, and robust optimized solutions distinctively, with a perspective through the related multifarious methods fit for the content. To this end, this edited book of ours foregrounds Multi- chaos, Fractal and Multi-fractional in the era of Artificial Intelligence, which definitely requires the integration of advanced mathematical models and mathematics-informed frameworks as well as AI addressing fractal, fractional calculus, fractional operators, quantum, wavelet, entropy- based applications apart from the means of modeling, technical analyses, and numerical simulations as some of the most extensively used methods for the solution of related multifaceted problems characterized by nonline- arity, nonregularity, self-similarity and many other prop- erties, frequently encountered in different complex systems. Motivated by the aforementioned considerations, the content of the chapters along with their novel aspects are outlined as follows. Chapter 2 entitled “Theory of Complexity, Origin and Complex Systems” (by Yeliz Karaca) attempts to encom- pass the possible dimensions of complex systems in different fields focusing on origin-related, historical, evolutionary and epistemological viewpoints of complexity with the goal of providing a global understanding thereof, taking into consideration the various multiple interacting factors of systems. In addition, through the presentation of complex order processes toward modern scientific path, it aims to understand the related conditions and demands for handling complex problems of the 21st century and on- wards. It, furthermore, intends to elaborate on accounts of past, present and future in different complex systems, which can help us adopt a deeper understanding and implement the steps along the way. By providing the complex order processes toward modern scientific path, from Darwin and onwards, a conceptual outline is also presented along with the details of complexity and complex systems. Complex systems, complexity thinking and the- ory, in fact, can broaden the horizon and scope of modern way of thinking, which needs to depend on transition from evolutionary dimension as a revolutionary stage and as a new paradigm for natural sciences and social sciences. Therefore, the characterization, definition, analysis and understanding of complex systems include a powerful relation between variables, sensitivity to initial control as well as strange, nonperiodic and unpredictable time evo- lution. Overall, this detailed presentation aims to ensure that the foundation for the complex systems’ interpretations can be explored in different related areas of complexity. Chapter 3 named “Multi-chaos, Fractal and Multi- fractional AI in Different Complex Systems” (by Yeliz Karaca) provides an overview that includes multi-chaos, fractal, fractional and Artificial Intelligence (AI) way of thinking regarding the solution of the complex system problems concerned with natural and social sciences. Moreover, ethical decision-making frameworks and strate- gies related to big data and AI applications are provided in detail for the purpose of enabling assistance to identify the related problems in different settings and thinking methodically in order that tensions between conflicting aspects can be managed in a systematic way. Data reli- ability and complexity, chaos thinking and processes and complexity, fractal thinking and processes and complexity, fractional thinking and processes and complexity, finally, AI way of thinking and processes and complexity are among the points elaborated in the chapter. Thus, the chapter is directed toward modern scientific thinking which has to adopt the systemic properties, addressing them by revealing the spontaneous processes pertaining to self- organization in a dynamical system in a state far from the equilibrium point and close to the disequilibrium point with no existence of an external force acting on the system. This way of thinking, naturally, poses a challenge against reductionist way of thinking and the dichotomy between the natural world and social world, by considering the concepts around complexity, evolution and order in detail. Chapter 4 named “High Performance Computing and Computational Intelligence Applications with Multi-chaos Perspective” (by Damiano Perri, Marco Simonetti, Osvaldo Gervasi and Sergio Tasso) addresses the experi- ence of the COVID-19 pandemic which has actually accelerated many chaotic processes in modern society be- sides pronouncing the urge to understand complex pro- cesses to achieve common well-being in a very serious and emergent way. The main contribution of the chapter is directed to the set of best practices and case studies, which provide assistance to the researchers while handling computationally complex problems. By analyzing different technologies and applications, complex phenomena are sought to be understood in the environment with ever increasing complexity bearing in mind different elements such as technology, algorithms and changing lifestyles, while striving to achieve maximum efficiency as well as outcomes besides protecting the integrity of individuals’ personal data and, above all, respecting the human being as a whole. The chapter considers that all these challenges impose a radical change in many different areas, including Introduction Chapter | 1 3
- 27. ones related to computational resources, which makes it very important to manage complex problems brought about by multi-chaotic situations. One section of the chapter is on computational intelligence, with the description of some of the techniques that enable the acceleration of complex problems’ resolution by exploiting the potential provided by machine learning techniques (like Multi-layer Percep- tron and Convolutional Neural Network) that can attain dimensions which used to be unimaginable in the past. The chapter also deals with the features of a quantum computer, which can process data at a rate exponentially faster than a classical computer. Taken together, the chapter provides a general sketch of various topics which could be of help to researchers and developers to deal with complex and chaotic situations within the scope of machine learning and the issue of privacy including the recent related regulations. Chapter 5 bears the title of “Human Hypercomplexity, Error and Unpredictability in Complex Multi-Chaotic So- cial Systems” (by Piero Dominici), which has the outlook that traditional linear models and deterministic approaches can no longer be capable of the analysis of reality’s un- stable dynamics. The chapter provides perspectives on the complexity of living energy and living beings; 12 essential planes of awareness; the characteristics of complicated, complex and hypercomplex systems; epistemology of error as well as complex and chaotic characteristics of social systems. The author of the chapter provides insights into the ambivalent nature of complexity, cognitive, subjective, social, ecological and ethical aspects of complexity including linguistics and communication as well as a “culture of communication.” Given that, hypercomplexity is not an option; but a fact of life. However, the problem- atics is related to the condition that we have not been trained and educated to recognize it, much less to inhabit it. Thus, it is important to bear in mind that complexity is a structural characteristic of human groups, relations, social systems and the biological world. Chapter 6 entitled “Multifractal Complexity Analysis- Based Dynamic Media Text Categorization Models by Natural Language Processing with BERT” (by Yeliz Karaca, Yu-Dong Zhang, Ahu Dereli Dursun and Shui-Hua Wang) addresses the challenges and complexity pertaining to media texts. Due to properties like being unstructured, noisy and nonstandard, accurate conveyance of meaning becomes problematic and against this background, the study aims at ensuring regularity and self-similarity within the digital-based complex media text by multi-fractal methods, which are multifractal Bayesian, multifractal regularization and multifractal wavelet shrinkage. Bidirec- tional Encoder Representations from Transformers (BERT) as the Natural Language Processing (NLP) method is employed to attain the accurate classification and catego- rization of the words within texts in the dataset. The related steps of the integrative method proposed in the study in- cludes regularity enhancement by the application of the three aforementioned multifractal methods to the text dataset. By obtaining the significant, self-similar and reg- ular attributes, new datasets were generated with the respective application of the multifractal methods. Subse- quently, BERT, as the NLP technique, was employed to the text dataset and the three new datasets were obtained for the classification purposes. In this way, accurate word detection within the text for the category classification was ensured for the analyses. The analysis results for the text dataset and the new datasets were compared by BERT and the most optimal result could be attained by multifractal Bayesian method. The study enunciates the significance of the behavioral patterns of fractal while setting forth the distinctive quality of BERT owing to its capability of classification accuracy and adaptiveness into integrated methodologies. Chapter 7 (Part I) entitled “Mittag-Leffler Functions with Heavy-tailed Distributions’ Algorithm based on Different Biology Datasets to be Fit for Optimum Mathematical Models’ Strategies” (by Dumitru Baleanu and Yeliz Karaca) is motivated by the challenge of integrating fractional cal- culus in cases of complexity, which requires an effective use of empirical, numerical, experimental and analytical methods to tackle complexity. One of the most noteworthy tools in the fractional calculus context is noted to be the Mittag-Leffler (ML) functions whose distributions have extensive application domains while dealing with irregular and nonhomogeneous environments for the solutions of dynamic problems. The proposed integrated approach in this chapter addresses the Mittag-Leffler (ML) function with two parameters for the purpose of investigating the dynamics of two diseases: cancer cell and diabetes. The following are the steps of the study: ML function with two parameters was applied to the biological datasets, namely the cancer cell dataset and diabetes dataset. It was aimed to obtain new datasets (ml_cancer cell dataset and ml_diabetes dataset) with significant attributes for the diagnosis, prognosis and classification of diseases. Next, heavy-tailed distributions, which are Mittag-Leffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution, were applied to the new datasets obtained. The comparison of them was done relating to the performances by employing the log likelihood value and the Akaike Information Criterion (AIC). Following these steps, the ML functions that repre- sent the cancer cell and diabetes data were identified so that the two parameters Ea;bðzÞ which yield the optimum value based on the distributions fit could be found. By finding the most significant attributes with heavy-tailed distributions 4 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 28. (The Mittag-Leffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution) based on Mittag- Leffler function with two parameters ða; bÞ, the diagnosis, prognosis, and classification of the diseases were ensured in the chapter. The integrative scheme proposed along with the optimal strategical means were for the accurate and robust mathematical models’ strategies concerning the diagnosis and progress of the diseases. Accordingly, the results ob- tained demonstrate that the integrative approach with Mittag-Leffler with heavy-tailed distribution algorithm is applicable, fitting very well to the related data with the robust parameter values observed and estimated in transient chaotic and unpredictable settings. Chapter 8 (Part II) has the title “Artificial Neural Network Modeling of Systems Biology Datasets Fit Based on Mittag-Leffler Functions with Heavy-tailed Distributions for Diagnostic and Predictive Precision Medicine” (by Yeliz Karaca and Dumitru Baleanu), which obtains the generation of optimum model strategies for different biology datasets along with the Mittag-Leffler functions with heavy-tailed distributions. The integrative modeling scheme proposed in the chapter is concerned with the applicability and reli- ability of the solutions obtained by the two-parametric Mittag-Leffler functions with heavy-tailed distributions. Accordingly, the proposed integrated approach in this chapter investigates the dynamics of diseases related to biological elements. Emerging in the different solutions of varying complex biological systems, the ML function with two parameters was applied to the biological dataset, namely cancer cell and diabetes and the new datasets were generated. The heavy-tailed distributions (The Mittag- Leffler distribution, Pareto distribution, Cauchy distribu- tion and Weibull distribution) were applied to the new datasets obtained with their comparison performed in rela- tion to the performances (by employing the log likelihood value and the Akaike Information Criterion (AIC)). ML functions that represent the cancer cell and diabetes data were identified so that the two parameters Ea;bðzÞ yielding the optimum value based on the distributions fit could be found. Subsequently, Multilayer Perceptron (MLP), as one of the ANN algorithms, was applied for the diagnostic and predictive purpose of the disease related to the optimized ML functions that represent the cancer cell and diabetes datasets obtained and the performances of the ML functions with heavy-tailed distributions were compared with ANN training functions (Levenberg-Marquart, Bayes Regulari- zation and BFGS-Quasi-Newton). The results based on mathematical models demonstrate that the integrative approach with Mittag-Leffler and ANN applications is applicable and also fits very well to the related data with the robust parameter values observed and estimated. The inte- gration of ANN with the self-organization and self-learning capability in pattern identification and recognition along with the rational thinking and acting ability while making inferences and decisions based on past experience has also been shown to be critical. Since AI enables the building of precise models to avoid unpredictable risks and identify opportunities in nonlinear complex situations, its integration in precision medicine is also foregrounded in this chapter. Chapter 9 named “Computational Fractional-Order Calculus and Classical Calculus AI for Comparative Differentiability Prediction Analyses of Complex-systems- grounded Paradigm” (by Yeliz Karaca and Dumitru Baleanu) aims to provide an intermediary facilitating function both for the physicians and individuals through establishing an accurate and robust model based on the integration of fractional-order calculus and ANN for the diagnostic and differentiability predictive purposes with the diseases which display highly complex properties. The integrative and multi-staged approach proposed in the chapter includes the application of the Caputo fractional derivative with two-parametric Mittag-Leffler function on the stroke dataset and cancer cell dataset. The establishing of new fractional models with varying degrees is performed and the reason why the Mittag-Leffler function has been opted is for its distributions of extensive application do- mains, which can enable it to handle irregular and hetero- geneous environments for the solution of dynamic problems. Subsequently, the new datasets related to cancer cell and stroke were obtained by employing Caputo frac- tional derivative with the two-parametric Mittag-Leffler function. Furthermore, classical calculus is applied to the raw datasets; and the performance of the new datasets as obtained from the Caputo fractional derivative with the two-parametric Mittag-Leffler function, the datasets ob- tained from the classical calculus application and the raw datasets is compared by using Feed Forward Back Propa- gation (FFBP), as one of the algorithms of ANN. As per the accuracy rate results obtained, the FFBP application, the suitability of the Caputo fractional-order derivative model for the diseases has been demonstrated. The experimental results obtained by this chapter also point to the applica- bility of the complex-systems-grounded paradigm scheme as has been proposed. It should also be noted that modeling many complex systems can be possible by fractional-order derivatives based on fractional calculus so that related syntheses can be realized robustly and effectively. Conse- quently, computational complexity is shown to provide us with applicable sets of ideas or integrative paradigms to recognize and understand the intricate properties of com- plex systems. Entitled “Pattern Formation Induced by Fractional-order Diffusive Model of COVID-19,” Chapter 10 (by Naveed Iqbal and Yeliz Karaca) provides the investigation of the Turing instability produced by fractional diffusion in a COVID-19 model. Considering that differential Introduction Chapter | 1 5
- 29. equations with complex order fractional derivatives enable the regulation of complicated fractional systems, positive equilibrium points have been initially specified and Routh- Hurwitz criteria are used for the assessment of the positive equilibrium point’s stability. Local equilibrium points and stability analysis have been employed to find the conditions for Turing instability. The analysis, by looking into the system’s dynamical behavior and the bifurcation point centered on the death rate, aims to serve as a leverage for further studies in different disciplines concerning COVID- 19 model through the lenses of distinct viewpoints. The results of the analyses reveal the highly complex connec- tion between COVID-19 and fractional order diffusion, the turing bifurcation point, and weakly nonlinear analysis used in the fractional-order dynamics discussed in the chapter. The Turing bifurcation point and weakly nonlinear analysis used throughout the complex fractional-order dynamics handled in the chapter are particularly relevant experi- mentally and computationally since the related effects can be examined and utilized in numerous mathematical, chemical, and ecological models, along with engineering, computer science, bioengineering, information science, applied sciences and virology as well as other related areas. Within this scale, fractional calculus is known to unfold the fundamental mechanisms and multi-scale dynamic phe- nomena in biological tissues. The results of the chapter are important in terms of showing that, on a quantitative basis, they can be extended to a variety of statistical, physical, engineering, biological and further related models. Chapter 11 whose title is “Prony’s Series in Time and Frequency Domains and Relevant Fractional Models” (by Jordan Hristov) addresses Prony’s series approximation of monotonical responses in material viscoelastic rheology as well as the possibilities of implementation on such a basis depending on modern fractional operations with non- singular kernels, namely the Caputo-Fabrizio operator. The chapter also provides the outline of the origins of Prony’s series in time and frequency domains along with the rele- vant approximation and calculation techniques. The results of the study expose the mutual relationships between the operators with singular and nonsingular kernels. The chapter sheds light on what type of operators are applicable in models fitting and modeling their experimental data. In this way, contributions in pure mathematics and experi- mental aspects are put forth. Consequently, the elaboration and application of Prony’s series are said to be extended to modeling problems emerging in mechanical engineering, chemical engineering as well as other related disciplines. Chapter 12 is entitled “A Chain of Kinetic Equations of Bogoliubov-Born-Green-Kirkwood-Yvon and Its Applica- tion to Nonequilibrium Complex Systems” (by Nicolai (Jr) Bogoliubov, Mukhayo Rasulova, Tohir Akramov and Umarbek Vazov) which is devoted to the study of the Bogoliubov-Born-Green-Kirkwood-Yvon chain of kinetic equations (BBGKYchke) and its applications to modern problems of physics. The chapter focuses on the need of creating a mathematical apparatus which fulfills the exist- ing theory of one-particle systems and systems made up of a huge number of particles. A unique object which satisfies the related conditions is the BBGKYchke as obtained from the Liouville equation for many particles. Two types of BBGKY chains are addressed for both classical and quantum particle systems. And, in contrast with the Liou- ville equation, the BBGKYchke has collision integrals. The first approximation coincides with the well-known Boltz- mann, Vlasov and Landau equations, while the last equa- tions provide the description of the evolution of one or two particles in modern physics. In the chapter, the example of quantum many-particle systems has been provided, which shows how the use of the BBGKYchqke, one-particle problems can be generalized for the case of nonequilib- rium systems that consist of interacting particles within a kinetic theory framework. The chapter concerns such nonequilibrium particle systems interacting with the generalized Yukawa potential as well. Overall, the solution of the BBGKYchqke for generalized Yukava potential (gYp) is provided, and solving of the BBGKYchqke with the gYp for systems of many type particles is elaborated on. Finally, the Gross-Pitaevsky equation is derived based on the BBGKYchqke. Chapter 13 named “Hearing Loss Detection in Complex Setting by Stationary Wavelet Rényi Entropy and Three- segment Biogeography-based Optimization” (by Yabei Li and Junding Sun) addresses another health problem which is hearing loss that decreases the life quality of individuals. The main objective of the research is to improve the ac- curacy and efficiency of detecting images in sensorineural hearing loss through a new solution. The chapter includes the proposal of an improved feature extraction method stationary wavelet Rényi entropy (SWRE) as well as opti- mization algorithm for model and feature extraction, namely three-segment biogeography-based optimization (3SBBO). It is noted that the current hearing loss detection methods have only a fixed scheme of feature extraction process and optimization mostly for classifiers. The ex- periments conducted demonstrate high rates of sensitivities, which corroborate the fact that the approach adopted in the research has attained a state-of-the-art performance and can be applied in the diagnosis of hearing loss. Chapter 14 entitled “Shannon Entropy-based Complexity Quantification of Nonlinear Stochastic Pro- cess: Diagnostic and Predictive Spatio-temporal Uncer- tainty of Multiple Sclerosis Subgroups” (by Yeliz Karaca and Majaz Moonis) considers the growth of complexity, which in more nonlinear and complicated instances, evolves with increasing information and entropy in a monotonous way. Complex dynamic characteristics of systems based on entropy require a detailed specification 6 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 30. and synthesis of the intricate elements as the system gets more and more complex. Thus, the chapter carries the aim of facilitating the accurate classification and course of three subgroups of Multiple Sclerosis (MS), namely Relapsing Remitting (RRMS), Secondary Progressive MS (SPMS), Primary Progressive MS (PPMS), which is a debilitating neurological disease. For this particular aim, an entropy- based feature selection method (Shannon Entropy and Minimum Redundancy Maximum Relevance) as well as linear transformation methods (Principal Component Analysis and Linear Discriminant Analysis) were applied to the MS dataset, from which four new datasets with sig- nificant attributes were generated. In addition, each new dataset obtained was addressed as input for the training procedure of k-Nearest Neighbor (k-NN) and decision tree algorithms. Finally, the accuracy rates for the MS sub- groups’ classification were analyzed comparatively based on the optimized experimental results which demonstrate that Shannon Entropy, as a distinctive entropy method, proved to be higher in terms of accuracy compared to the other feature selection methods. The chapter, therefore, intends to point a new perspective, with a multi-level aspect, for critical decision-making and manageable tracking in medicine and relevant fields, which all need to cope with the complex dynamic systems in which uncer- tainty and heterogeneity prevail. Chapter 15 entitled “Chest X-ray Image Detection for Pneumonia via Complex Convolutional Neural Network and Biogeography-based Optimization” (by Xiang Li, Mengyao Zhai and Junding Sun) proposes a novel chest X- ray image detection for pneumonia, which is stated to be a leading reason for death among children and afflict the elderly worldwide. The detection model proposed by the authors is based on the combination of complex convolu- tional neural network (CNN) and biogeography-based optimization (BBO). It is proven that the model has higher sensitivity and accuracy in terms of detecting the pneumonia-related chest X-ray images with a detection performance being significantly better than that of advanced approaches within complex medical settings. The utilization of BBO, which is employed as the global opti- mization algorithm of the related model, has the benefit of optimizing the stride size of the convolution kernel on CNN to obtain better detection effects with less model training cost. Chapter 16 entitled “Complex Facial Expression Recognition by DenseNet-121” (by Bin Li) is on facial expression recognition system, which has gradually been integrated into different fields of our lives with the advent of artificial intelligence era. The application prospects of intelligent face recognition via computer technology are very extensive, and can also be applied to the diagnosis of facial paralysis in the field of medicine. The chapter han- dles the complex nature of facial expression since it involves emodiversity and emotional complexity and makes the point that facial expression recognition is a difficult task which may also bring about some problems such as low accuracy and poor generalization ability of network model recognition. To address such challenges, the author of the chapter has proposed a DenseNet-121 image feature extraction method, combined with convolutional neural network (CNN) for facial expression recognition. For this, the principle and method that this model can quickly and accurately recognize human facial expressions have been analyzed. Afterward, the experimental analysis has been carried out. The experimental results prove that the network model proposed has high precision and robustness with a good ability of generalization. The pre- sentation of an improved face emotion recognition system proposed employing a method based on densely connected neural network also helps in avoiding complex feature extraction required by traditional deep learning and also saving the training time. Chapter 17 is named “Quantitative Assessment of Local Warming Based on Complex Urban Dynamics Using Remote Sensing Techniques” (by L.Saganeiti, Angela Pilogallo, Francesco Scorza, Valentina Santarsiero, Gabri- ele Nolè and Beniamino Murgante), which is concerned with urban growth, which is one of the cornerstones of sustainable development policies that require to be put into practice at initial states for a well-managed urbanization process and experience. Demographic qualities and city’s growth as well as the consequent need to densify urban aggregates are noted to be increasingly conflicting with the theme of livability of urban spaces and the services they provide for the well-being of the citizens. Accordingly, the chapter provides a simultaneous analysis of the variations of land surface temperature and urbanized environment over a period of 15 years within two regions that differ in size, population density, and growth dynamics. The results reveal a much more marked increase in all of these com- ponents as regards minimum temperatures in areas where urbanization has been matched by a decrease in the number of aggregates. The research, as presented in this chapter, provides an appealing and innovative contribution to un- derstand the relationships between urban growth spatial patterns and the urban thermal environment. Detailed ana- lyses with this sort of approach presented in the chapter are considered to be beneficial in supporting decision-making processes that underlie future urban policies and assess- ment of development scenarios with regard to quality of life, environmental sustainability and preservation of ecosystems. Chapter 18 is entitled “Managing Information Security Risk and Internet of Things (IoT) Impact on Challenges of Medicinal Problems with Complex Settings: A Complete Systematic Approach” (by Eali Stephen Neal Joshua, Debnath Bhattacharyya and N. Thirupathi Rao), which Introduction Chapter | 1 7
- 31. discovers the crossway of healthcare and significant data. The chapter provides details regarding information security, various vulnerabilities in healthcare, data breaches, distributed denial of service (DDoS) assaults, insider threats, information security in healthcare, health informa- tion privacy and security as well as various information threat elements regarding medical health reports. The chapter also points out the impact of IoT in medical problems, IoT in healthcare, challenges in IoT in medical problems (data security and privacy, integration: multiple devices and protocols, data overload and accuracy and cost) as well as applications of IoT in healthcare. The informa- tion threats are outlined in detail in the chapter which presents the challenges of medicinal problems using IoT through a case study that demonstrates the efficiency of IoT as a result of exponentially increasing patient monitoring (blood pressure monitoring, glucose monitoring, and pulse rate monitoring) in the healthcare plans. The application created is also important since it integrates personal and enterprise medical IoT applications for the centralization of medical statistics and providing a unified dashboard. Last but not least, Chapter 19 is entitled “An Extensive Discussion on Utilization of Data Security and Big Data Models for Resolving Complex Healthcare Problems” (by N. Thirupathi Rao, Debnath Bhattacharyya and Eali Ste- phen Neal Joshua); and it is concerned with the utilization of technology in the healthcare settings with a focus on the employment of the Internet of Things (IoT) technology, providing an extensive elaboration of its opportunities, benefits, impacts, existing gaps, security threats as well as adaptive frameworks that need to be developed. The chap- ter, with updated information for our current time, presents detailed discussions on big data in health care, information security, confidentiality, integrity and availability by considering the related stakeholders in the area that are the physicians, patients, hospitals and insurance companies. The chapter presents the complex system with its compo- nents in various healthcare domains, and this attribute concerns many different disciplines including but not limited to medicine, microbiology, biomedical engineering, computer science and big data analytics. Awareness into and efficient management of all the components involved are noted to have benefits for the patients who will be knowledgeable in terms of pertinent medical resources and faith in healthcare professionals. In addition, access into a variety of medical services based on technological devices will be of great benefit to all the stakeholders and settings. By addressing different uncertain processes inherent in complex systems and providing global and robust opti- mized solutions distinctively, with a perspective through multifarious methods, this comprehensive book attempts to bridge some gaps by the presentation of novel methods, integrative adaptive methods within mathematics-informed frameworks and related applications which have all become prominent to solve complex problems. The fundamental aspects of complexity require to be addressed in a way to capture the universal features, which requires the transcending of particular domains relying on medical, neurological, mathematical, physical, biological, chemical, information-based, engineering, social, sociological, phil- osophical and epistemological perspectives as elaborated theoretically and exemplified through technical analyses, modeling, optimization processes, numerical simulations, case studies, as well as applications in our edited book. Nevertheless, the identification of those determining fea- tures may cause the sharp differences to arise. When in- dividual complex systems are handled as the objects of study in different disciplines, it is stated that little com- mon ground is seen between the abstractions, methods and models of them. As a matter of fact, complexity science transcends and expands on the reductionist framework of traditional understanding, which enables us to understand that the components that make up the whole along with the comprehending of the way each part in- teracts with all the remaining parts as they emerge into a new entity. Optimized solutions based on multi-chaos, fractal, multi-fractional and AI in different complex systems have become sine qua non vis-à -vis all aspects of conditions of modern era and artificial life. Thus, computational tech- nologies in different complex systems based on mathematical-driven informed frameworks will be able to provide the generation of more realistic, applicable, adap- tive models open to learning and flexibility under transient, dynamic, chaotic and ever-evolving conditions of complex systems. All these can suggest many beneficial and prac- tical paths to be discovered while presenting a promising paradigm with potential insights waiting to be gained, which we have tried to cover in the chapters of our edited book. 8 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 32. Chapter 2 Theory of complexity, origin and complex systems Yeliz Karaca University of Massachusetts Medical School, Worcester, MA, United States 1. Introduction Complexity, as an idea and scientific concept that has existed since antiquity, requires the understanding of the origin of complex components, entails lengthy and metic- ulous computations as well as causal processes. Multiple interactions emerge among the components in a complex system whose research address characteristics like adapta- tion, self-organization, noise, synchronization, high number of descriptive variables or dimensions, nonlinearity in description of differential equation systems and reaction to responses in the external environment. Given that, the characterization, definition, analysis and understanding of complex systems encompass powerful relation between variables and discrimination by noisy nondeterministic phenomena, sensitivity to initial control as well as strange, nonperiodic and unpredictable time evolution. As physicist Nigel Goldenfeld put very aptly, “Complexity starts when causality breaks down.” Having structure with variations and far-reaching conditions like spontaneous order, nonlinearity, feedback, robustness, lack of central control, numerosity, hierarchical organization and emergence are considered, complexity reveals many deep layers involved in complexity. The large number of independent interacting components and multiple pathways by which the complex system can evolve further indicate some of the reasons why causality breaks down as complexity starts. Along these lines, causality is relative, being prone to fundamental variations depending on perception, external factors, the environment, space, time and so on. Time is a flying and flowing phenomenon; and if we, as the human agents, are the pilots of the time which is on an infinitive continuum, the decision-making processes need to be prompt aside from being efficient and robust. It would not be possible to proceed along one way on this continuum since constancy causes nonstationary and steady state; for this reason, multifarious and integrative way of thinking and methods unifying the elements of different disciplines would address the different parameters of complex systems and their related problems with optimized solutions. Multifarious, meaning the possession of many varied parts and aspects as well as happening in a great variety, in fact, can help identify the solutions of the complex problems taking into account nearly all possible parameters of complex systems. To this end, it is necessary to identify the optimal model and by providing cross-validations, multifarious methods consider the fact that each complex problem has a different nature and the solution to each problem needs custom- ization. Complexity along with all the variations in net- works demonstrates that our decisions are not based on only one single parameter, but on multiple numbers of parameters with hidden information being at stake as well. Multifarious adaptive methods within mathematics- informed frameworks come into prominence for the solu- tion of complex problems, which enable that the solution is not superficial or pretentious but reliable, robust and smooth ensuring the maintenance of quality, sustainability and meritocracy. Time is not only comprised of de- notations, it is beyond the traditional thinking of science, with representations and reflections on life, which apply to all domains of life. Evolutionary processes, nonlinearity, and all the other dimensions of complexity rest on time, reveal time and occur within time. In the ever-changing current landscape and variations, with causality breaking down, Stephen Hawking’s quote, “I think the next [21st] century will be the century of complexity” is critically significant. The idea of complexity is sometimes stated to be part of a new unifying framework for science and a revolution in our understanding of systems the behavior of which has proved difficult to predict and control thus far. The goal of complexity science is to achieve a global un- derstanding by considering the multiple interacting factors Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems. https://doi.org/10.1016/B978-0-323-90032-4.00003-1 Copyright © 2022 Elsevier Inc. All rights reserved. 9
- 33. of systems, many branches of possible states and high- dimensional manifolds while keeping abreast of actuality along the historical and evolutionary path, which itself has also been through different critical points on the manifold. Changing or removing the causes will not necessarily change or remove the outcomes, and thus, modern scientific way of thinking is geared toward the development of models that benefit from local computations, interpolate task-related manifolds in spaces with high dimensions instead of just learning the rules or representations of the world. Similar to evolutionary processes, models that are overparameterized can be thrifty in terms of providing applicable, versatile and robust solutions so that multifar- ious set of functions can be learned and optimized out- comes can be achieved. These applicable models and way of thinking from which the models are derived pose broad challenges to different areas of science if they rely on just theoretical assumptions. Evolution, order and complexity reveal the relationship between natural and social worlds, which reflects a modern way of thinking that definitely challenges the dichotomy between the natural and the social. This study, accordingly, provides a conceptual outline and historical account of complexity and complex systems along with complex order processes toward modern scientific path from Darwin and onwards with a focus directed toward natural, applied and social sciences. The capacity of managing the modern societies ulti- mately rests on a communication network that runs effec- tively. Just like the neural nets of biological brains, those networks determine the learning capability, which will eventually help with survival. The dynamics of information technologies in the framework of complex systems need to be modeled within their natural, social, economic and cultural environments, so as to say informational and computational ecologies [1]. Under such complexity, fundamental principles of complex thinking, such as being holistic, human-oriented, and transdisciplinary, are all dependent on the concepts of modern theory of evolution and self-organization of complex systems. Along these lines, nonlinearity of evolution, chaos, space-time elements and complexity emerge as important aspects, which is very well reflected by the quote of Ilya Prigogine: “Our vision of nature is undergoing a radical change toward the multiple, the temporal, and the complex” [2]. Aside from these no- tions mentioned, the following concepts also come along with complexity: openness, nonlinearity, chaos, self- organization and synergetics. When something complex is handled and defined, then all the other aspects including its nature, structure, evolution as well as its principles should be considered, which all point toward the definition of modern science of complexity. In essence, the science of complex systems has provided us with the conceptual and methodological tools so that issues of evolution, self- organization, emergence, and transformation can be tackled and mechanisms of micro and macro levels can be explained in terms of their behaviors over time [3]. Mechanistic thinking posits that the universe is under- standable, analytical method is the only way of research, and causality explains everything in the world. In contrast, complexity way of thinking is totally different, stating that different matters are interconnected and interaction shows nonlinearity and noncausal determinism. Besides having the self-generation and self-organization properties, this strand of thinking includes uncertainty, noncontinuity, inseparability and unpredictability [4]. Consequently, the- ory of complexity enables us to gain powerful evidence and elucidation to challenge traditional and mechanistic thinking so that humanity can adopt a new way of thinking. Complexity thinking requires a horizon that takes into account the subtle properties of different domains, which require their own means of solutions and applicability. In neurological system complexity, “evolvability” is concerned with the species owing their existence to the capability of their ancestors with regard to evolving and adapting. Another important point has to do with the correlation between the complexity of brain design and optimality. The progress made in the neurosciences has shown the complexity of even very simple nervous systems, and complexity is manifested in their structure, function, coding schemes used to represent information as well as in their evolutionary history. These viewpoints are of critical importance in the future science of brain complexity as well [5]. The present and future science of complexity relies on application aspects for optimal development and strategy generation to solve the complex problems. To cite relevant works, [6] introduced a measure, named as neural complexity, to capture the relationship of two aspects of brain organization, which are anatomy and physiology. The complexity measure is applied in computer simulations regarding the cortical parts of the brain so that the basic principles of neuroanatomical organization limiting brain dynamics can be examined. That approach is said to be useful for the analysis of complexity in other areas related to biology like embryogenesis and gene regulation as well. Another work [7] is on the resting-state functional magnetic resonance imaging (rs-fMRI) to record and analyze the brain’s neural activity. Noting that different regions of the brain show different levels of complexity, the study ex- amines individuals, correlated entropy/complexity changes, concluding that there are important differences at the level of subject when there is a memory task which is linked to performance opposed to being at rest. Finally, the study by [8] employs functional magnetic resonance imaging to explore the functional networks that underlie cognitive reasoning in humans with an anatomical abnormality, called corpus callosum dysgenesis. The findings of the study demonstrate that resting-state functional brain 10 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 34. network that supports cognitive control is preserved partially in people with corpus callosum dysgenesis. Medicine and healthcare systems also display highly complex properties, particularly during uncertain situations where promptness of response, accuracy of diagnosis, drug regimens, interactions between the related parties and so forth are of vital importance. Furthermore, humans them- selves are complex, considering their mind, spirit, body and social contexts. Healthcare settings’ complexity adds further elements to the situation like the organizational infrastructure and multi-faceted dynamics there. Complexity thinking and science can assist all persons involved in health issues so that simplistic linear thinking can be avoided and multi-dimensional processes related to patient care and healthcare service organizations can be considered within a nonlinear and complex framework. Consequently, in the study [9], it is stated that complex systems approaches use a collection of advanced analytical and methodological tools to be involved in innovative theory testing and development. The exploration of dynamically complex systems is done in virtual labora- tories, which would not be possible to realize by traditional methodologies. With regard to neuroimaging, the study [10] illustrates the basic properties characterizing complex systems while evaluating the way they relate to brain structure and function based on the experiments of neuro- imaging. Neural systems are reviewed from the complexity science perspective by matching the complex systems’ properties with evidence obtained from contemporary neuroscience. On the topic of law, complexity and medi- cine, the study [11] discusses the strategic decision-making in healthcare system by systems methodology based on the fact that society is a complex system. The models devel- oped in the frame of systems dynamics as well as the other means are presented in the related study with a focus on the importance of feedback loops. As can be seen, complexity has its implications with other theoretical assumptions, applications and methods in the field of neurology and medicine [12e23]. Being staggeringly complex, biological systems including cells, organisms and ecosystems at large cannot be analyzed by abstracting them from the whole. Extending across space, biological networks go across the dimension ranging from the microscopic scale of intracellular orga- nization to the global scale of planet ecology. It is acknowledged that integrated system of complexity along the evolutionary path is a significant feature of living sys- tems. Thus, it can be stated that everything biological needs to be viewed in an evolutionary framework. In addition, one example of the reason why reductionist approach would fail is related to the function of one cell in a multi- cellular organism. Even though one understands the func- tion of the cell, that would not necessarily mean that the organism’s physiology can be understood completely [24]. As each cell’s activity would be affected by that of other cells in the organs, tissues and organ systems within the organism itself abstraction and isolation would not be of help. Rather complexity thinking would show that inte- gration into the system enables us to understand that system has emergent properties and should be interpreted from that vision. If we are to note some of the works where complexity and biology are addressed, it can be said that the study by [25] is on complexity measures for the com- parison of genomic characteristics. The authors demon- strate that the fluxes, block entropy content, conditional probability and exit distance distributions can be utilized as markers, which can help the discrimination between eukaryotic and prokaryotic DNA so that in many cases details about finer classes can be discerned. The complex measures handled in the study highlight the discerning power of genomic observables for the prediction of the evolutionary position of DNA sequence even if its origin is not known. Another study [26] handles degeneracy, which is a ubiquitous biological characteristic related to genetic code and immune systems. Degeneracy is stated to be required for and result of natural selection. Deeper under- standing of the way degenerate systems become associated and synchronized across levels is also said to be an important challenge to deal with in modern evolutionary biology. [27] is a study which extends self-organizing approach for a bacterial genome for the purpose of analyzing the raw sequencing of human data. The authors indicate that metagenomics allows for the genomic study of uncultured microorganisms, and more economical and faster technologies can help the sequencing of uncultured microbes, which are sampled directly from their habitats can provide transformation related to the view for the mi- crobial world. Finally, the authors of the paper [28] developed a notion for a model to understand stage tran- sition systems through hierarchical coordinate systems. Through this way, an algebraic definition for biological systems’ complexity is handled along with the comparison with genome size and number of cell types. The complexity measure of the authors is said to be unique for maximal complexity to fulfill a natural set of axioms, which also shows a strong connection between hierarchical complexity in biological systems and global semigroup theory, which is one area of algebra. Biological complexity has its extensions in other studies with different theoretical assumptions, models and applications [29e33]. Engineering systems’ uncertainty increases with the exponentially growing complexity of the system, which makes tolerance to the uncertainty an essential factor. En- gineering processes involve many facets and stages, so complexity is indispensable for engineering as well. [34] is a study concerned with risk assessment of complex systems in a supervised dynamic probability manner. The method developed with this vision is said to improve execution Theory of complexity, origin and complex systems Chapter | 2 11
- 35. time of dynamic probabilistic risk assessment models, and the optimization model is employed for the generation of failure scenarios apart from the comparison of appropriate optimization solution algorithms. The aim of the method used in the study is that operators can monitor the risk level of all probable failure scenarios in real time while helping with the better decisions in situations that require emer- gency, which are all properties of complex systems. Regarding health and engineering, the work [35] proposes a nonlinear data fusion method for composite health indicator and derives the reliabililty measures in a computationally efficient way. Another study [36] is on the integration environment for engineering and science, which requires multidisciplinary efforts from a multitude of specializations working in harmony. For accuracy and efficient use of time, the authors present a remote component environment to allow the users to integrate disciplinary tools. The study is important in terms of displaying the complex factors at play during engineering processes like design and analysis. With regard to extreme events, which require interdisciplinary interaction, the study [37] reviews the existing approaches for the definitions of extreme events along with the case study that emphasizes the intricate properties in the defi- nition of extreme events. Since addressed in a broad variety of disciplines like climatology, mathematics, meteorology as well as social sciences, extreme events due to their im- pacts also require a complex systems understanding. Complexity and engineering have provided studies with different perspectives [38e41]. Being adaptive and responsive to changing circum- stances, science is an evolutionary process itself, which also applies the dynamics within the complex social systems. The call for the adoption of complexity theory has also been the case for social sciences so that it would be possible to get away from reductionist frameworks. In addition, more recent connectionist methods are sought to address complexity and open social systems in a better way. The study [42] provides the differentiation between general systems theory (GST) and complexity theory along with the benefits of them for social sciences. Complexity theory is identified as a theory that can provide a new perspective and also a novel method of theorizing as well as addressing complexity by linking it with advances in technology, markets, globalization, cultural changes and all the other related future challenges as well as opportunities consid- ering today’s complex problems. The paper [43] provides a new definition of social complexity based on the number of differentiated relationships individuals have. This definition is argued to be an objective and cognitive one, stemming from the view that social complexity is used widely but measured poorly. The authors, who review previous defi- nitions, posit that the number o differentiated relationships is a flexible measure as well. The paper [44] handles social complexity around the study of animal and human societies. This author of this work also states that the concept is defined and understood in a poor way. The definitions for vertebrate and invertebrate societies are reviewed with a critical outlook; terms like social structure, social organization, care system and mating system are defined and characterized along with the provision of an outline of the different aspects related to evolution of social complexity. The work [45] is on complexity and archeol- ogy, which has complex elements like social roles, eco- nomic roles, big permanent settlements, large populations of people and other marker criteria. It is also maintained by the author that computational and systems dynamics modeling can provide the systematic study of complex adaptive systems. A small-scale society is given as a computational model to exemplify the complexity of even simple societies, which shows that complexity is regardless of size or organizational structure. New modeling methods are also said to be of assistance to archeologists in their studies. A much older work [46], on the examination of complexity concept in sociology, shows that complexity has been discussed for quite a long time period The author discusses and shows that the complexity of social data are no more complex than other sorts of natural phenomena. The work [47] is on the complexity applications in lan- guage and communication sciences, and the authors point out the interdisciplinary approaches for human sciences. For instance, the paradigm of complexity in sociology, the reason and the way to model the complexity of thought systems, the impact of social reputation in language evolution, to name some aspects in the relevant fields. The diversity of social science lends itself to the use of many varied theoretical stances and methods within the framework of complexity in different studies [48e52]. One of the basic premises of complexity theory is that there exists a hidden order to the behavior and evolution of complex systems. And that system can be an ecosystem, an organization or economy of a country. Along this line of thinking, complex adaptive systems are stated to be revising and rearranging their building blocks in a constant way based on their experience. Examples include pro- motions for some employees in businesses and concluding new trading agreements for countries. Complex adaptive systems, including ecosystems, brains and economies are governed by anticipation and feedback from the environ- ment based on which models are improved. Their shared properties are nonlinearity, aggregation, diversity and flowing [53]. Moreover, there is a coherent system behavior generated by cooperation and competition between actors in complexity theory, which is another important governing concept. Informed by natural and mathematics sciences, complex systems approach is also stated to be a useful tool for political economists to focus on the dynamic adapta- tions and their nonlinear implications at different levels of a social organization [54]. The survey [55] provides a 12 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 36. discussion of behavioral and experimental macroeconomics through the emphasis on complex systems perspective. Heterogenous expectations and heuristics switching models also match the micro and macro behaviors in economics as well. The work [56] is on the generalization of model of economic growth with constant pace considering the im- plications of memory. Fractional differential equations and their solutions are obtained for the description of the output dynamics brought about by the changes of net investments and effects of power-law fading memory. The application of fractional calculus for mathematical economics is addressed in [57] together with the recent mathematical and conceptual developments in the field focusing on memory and nonlocality. Economic complexity has also relationship with regard to phenomena related to climate; accordingly, the study [58] posits that economic complexity leads to intensity of greenhouse emission, which highlights that the knowledge situated in the productive structure of complex economies includes the knowledge needed for cleaner technologies of production. Aside from business, including finance and other fields that strive for the solution of the complex problems, related studies have been carried out [59e63]. Taken together, complexity, as an idea and scientific concept, has been in existence since antiquity and discussed along different origin-related, historical, philosophical and epistemological viewpoints as addressed in this study [53,64e69]. Complexity requires the understanding of interacting components in the complex systems, along with lengthy and meticulous computations as well as causal processes for the solution of different complex problems in different systems [18,21,23e53,63e69]. Different from the previous works, this study attempts to encompass all possible dimensions of complex systems in different fields, not disregarding varying origin-related, historical and epistemological viewpoints. Accordingly, the goal of pre- senting the brief history of complexity in this study is intended toward a global understanding of complexity, taking into account various multiple interacting factors of systems, many branches of possible states and high- dimensional manifolds while keeping up with reality and actuality along the historical and evolutionary path, which itself, has also been through different critical points on the path. Subsequently, the presentation of complex order processes toward modern scientific path aims at under- standing the conditions and requirements to handle com- plex problems of the 21st century and onwards. The reason for handling these details, in the parlance of complex sys- tems, is for the sake of modern way of thinking depending on the transition from evolutionary dimension as a revo- lutionary stage, as a new paradigm for natural sciences and social sciences so that the foundation for the complex systems’ interpretations can be explored by related areas. Succinctly, accounts of past, present and future in different complex systems help a deeper understanding and guidance along the way. The rest of this study is structured as follows. Section 2 provides the A Brief History of Complexity and the Related Areas of Different Complex Systems with the Section 2.1., which is entitled “Theories Pertaining to Complexity and Their Historical Account”. Section 3 addresses Complex Order Processes Toward Modern Scientific Path: From Darwin and Onwards by providing Section 3.1 entitled “A Conceptual Outline: Complexity and Complex Systems.” Last but not least, Section 4 of the study provides the concluding remarks and future directions. 2. Theory of complexity, origin and complex systems Complexity theory shows that units in large populations can be involved in self-organization process, producing patterns, storing information and engaging in collective behaviors. In natural landscapes, natural patterns are derived from nonlinear processes with properties that are modified; and nonlinearity, in this regard, is usually viewed to be one of the essential elements related to complexity [70]. In social systems, complexity refers to the amount of data or information required to be able to completely describe the system which exhibits complex features. Nonphysical systems like social structures are said to have similarity in terms of behavior, character, or rules to be conformed to; and within such settings, social hierarchies can combine into systems getting ever larger via the mechanism of representation [70]. Most of the things related to the behavior of social systems refer to the inter- action of its members instead of the individuality of those members; and each social system exhibits specific charac- teristics that may remain although all of its individual members are replaced [71,72]. 2.1 A brief history of complexity and the related areas of different complex systems Having existed as a term since antiquity, complexity as an idea and scientific concept gained its explicit definition in the late 1980s although its presence in mathematical sys- tems had already been noted by Henri Poincaré in regard to three-body problem. The 1920s saw the quantification of complexity of simple mathematical formulae for statistical models, whereas biological, social systems as well as other systems were characterized by high complexity during the 1940s. A huge number of components that had different types and behaviors with interconnections and in- terdependencies used to be the common definition terms. A decade later, complexity started to be handled in pure Theory of complexity, origin and complex systems Chapter | 2 13
- 37. mathematics, followed by information theory and the DNA structure discovery, which underlined the information content of complexity. The content of algorithmic infor- mation came into existence in the 1960s and this pointed toward the shaping of the definition of complexity. Different areas used several other definitions in the 1960 and 1970s when computational complexity theory started to head for a varied direction, with complexity defined in terms of resources required to carry out the computational tasks. Afterward, the 1980s saw the growth of research in complex systems, with a focus placed on numerical mea- sure of complexity [69]. Another related development is the creation of the Santa Fe Institute whose founders wanted to react to the specialization and reductionism in science, and thus, to enhance the development of the science of complex systems, in other words, complexity science [53,64e68]. The notion of complexity in different areas has many dimensions. To start with, mathematics is concerned with the study of arbitrarily general abstract systems and a very high level of complexity in the behavior of many systems which have rules that are actually simpler than the rules of most systems in traditional sense. It can be said that the traditional mathematical approach to science has contrib- uted to physics, as another area, and it is nearly acknowl- edged universally that physical theory is to be based on mathematical equations. In theoretical physics, existing methods are around continuous numbers and calculus, probability as well at some times. Nevertheless, a greater simplicity in that structure yields the identification of new phenomena. In computer science, computational systems established to carry out specific tasks have been the focal point, and within this purpose, even the simplest con- struction is capable of yielding a behavior that is immensely complex. Computational ideas, in this sense, can include all kinds of core questions regarding mathe- matics and nature. As another field, biology encompasses vast and profound details about living organisms and bio- logical elements, with evolution by natural selection being one of the most classical realms thereof since general observations on living systems are customarily analyzed based on evolutionary history instead of abstract theories. Social sciences, varying from psychology to economics, philosophy and sociology, also offer complexity with the ever changing, adapting, and evolving features over time as a function of people’s preferences and attitudes. Although physical sciences require the formulation of solid theories in terms of equations and numbers, social complexity reflects behaviors of humans as ongoing and broader as a result of complicated conditions of individual and group existence through many different arrangements, patterns and movements. For philosophy, on the other hand, issues regarding the universe and the role of human beings therein, besides the uniqueness of humans’ conditions, limit to knowledge and the inevitable position of mathematics are positioned at the core [69]. Engineering, as another discipline, has its obvious association with complexity, which also shows that even simple underlying rules can be put into practice to carry out a sophisticated task. This means construction of a system with complicated basic rules is not always required in engineering. This is because for the design and operation of engineering systems, the aim is to reduce complexity so that the system can be rendered robust, which assures long-term stability, system reliability and cost minimization [39]. Rather than the classical approach of “dividing and conquering,” complexity engineering tackles adaptive, self-managing, self-organizing and emergent features [73]. The important changes in the foundations of complex and sophisticated technology yield the application for purposes related to humans, which enables the imagination of a whole new sort of technology that can attain the same sophistication as nature whose essential mechanism can be captured by rules even in simple programs. 2.2 Theories pertaining to complexity and their historical account The investigation of studying complexity theory on its own as a separate phenomenon dates back to the early years of the 1980s upon the suggestion of Wolfram [69]. Afterward, its popularity grew in time and theory has enabled the development of the fundamental understanding of the complexity as a general phenomenon as well as its origins. Further, computational complexity theory, developed in the 1970s, intends to characterize the extent of difficulty regarding certain computation tasks. Starting about a cen- tury ago as a branch of mathematics, dynamical systems theory concerns itself with the investigation of systems that evolve over time in line with specific sorts of equations, and mathematical and geometrical methods for the char- acterization of the probable forms of behavior that can be produced by related complex systems. Accordingly, the field of fractal geometry, unlike conventional science and mathematics, which deal with regular and smooth kinds of shapes, starting in the late 1970s, underlined the signifi- cance of embedded and nested shapes, which include arbitrarily intricate pieces common in nature, since many systems produce shapes that are inexplicably complex. Dynamic system theory, on the other hand, stems intel- lectually from mathematics, physics, meteorology, astron- omy and biology explains the developments as the probabilistic consequence of the processes’ interactions at different multiple levels, with interactions of multiple factors as well as systems on different levels and timescales [74,75]. Relatedly, chaos theory is reliant on the observa- tion of specific mathematical systems, which display behaviors depending on the details of initial conditions. This was spotted at the end of the 19th century and became 14 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 38. prominent following the computer simulation works of the 1960 and 1970s afterward. One of the most contentious theories is evolution the- ory, the Darwinian part of which with natural selection is usually assumed to explain the complexity observed in biological systems. The theory, which has no clear-cut lines about why it should imply that complexity has generated, has had applications outside biology as well in recent years [69]. Darwin saw the path to the evolution of complexity; individuals’ varied traits were also seen in each generation; and new variations’ emergence and spread came to mean the production of complex structures. While some of the variations contributed to the increase in their survival, some others enabled them to produce more offspring. On the other hand, some scholars have recently suggested that complexity can emerge through other routes. McShea and Brandon claim there is a more basic biological law than natural selection, which is diversity and complexity increasing in evolutionary systems. The authors, including Gould, Kauffman, and Lineweaver et al. also put forth that complexity, increasing even without the action of natural selection, refers to the number of parts of the amount of differentiation between parts in an individual and it is important that we do not consider only the number of parts that constitute living things, but also look at the types of those parts. Correspondingly, Kauffman [65] explains that complexity emerges without natural selection to help it along [76,77]. This argument, also famously put by Richard Dawkins as “the blind watchmaker” postulates that complexity is not the outcome of fine-tuning through nat- ural selection which took millions of years. In other words, it just happens [78]. The schematic depiction of evolution process with complexity over time can be interpreted as the shape of the tree of life. This suggests that there is a cu- mulative type of increase in complexity, which provides evolution with a direction and an arrow of time, as a concept of asymmetry of time developed by the British astrophysicist and mathematician Arthur Eddington in 1927 [79,80]. Entropy is another related concept in this context, which is one of the few quantities in physical sciences that require arrow of time. The entropy of an isolated system is known to increase, not decrease, and the increase of dis- order, or entropy, is what enables one to distinguish the past from the future [66] (for further views on complexity and how it differs in terms of its increasing quality, see [77]). 3. Complex order processes toward modern scientific path: from Darwin and onwards The interaction of selection and self-organization is an important theme in developmental and evolutionary biology, which should include Darwin in a broader context. Both simple and complex systems can manifest powerful self-organization and spontaneous order is at stake for natural selection, which is very aptly incorporated by evolutionary theory. As for the adaptation processes, while some systems can adapt readily, others may be exposed to disruptions by minor modifications, which make the adaptive improvement by selection and random mutation occur hardly. Darwin was of the assumption that that kind of an improvement was viable and the complex systems’ adaptation capability, as shown one century later, made us be aware of the construction requirements to permit adap- tation [65]. The first captivating view of Darwin is that natural se- lection and branching tree of life spread from major phyla to minor genera as well as species, reaching out to humans. Although being enchanting, Darwin’s answer to the sources of order appeals only to one force, that is to say natural selection. Hence the reason why the view is considered to be inadequate, since it is thought that the view does not integrate the possibility that simple and complex systems show spontaneous emergence of order. That’s why, the mingling of self-ordering with Darwin’s evolution-natural selection mechanism produces much unfolding and makes the order we see sensible albeit the complexity of organ- isms. The current stage of science has become so extraor- dinary that while molecular biology leads us to the inmost ultimate mechanisms, complexity and evolution capacity of cells, studies carried out in mathematics, biology, chemis- try, physics and so forth expose the powers of self- organization [65]. These far-reaching effects help us explore the sources of order and understand the order inherent in complexity, which means that complexity is associated with self-organization and spontaneity; and natural selection is not the only reason thereof. An inte- gration of this knowledge with Darwin’s basic insight is recommended so that self-organization and selection themes can be combined and evolutionary theory can be expanded to stand on a broader structure, which is stated to have three tiers according to Kauffman (1993): first of all, a delineation is necessary for the spontaneous sources of order as well as the simple and complex systems’ self- organized properties. Secondly, it is noted that self- ordered properties both enable and limit the efficacy of natural selection. For this reason, organisms need to be viewed in a different and new perspective with a striking of balance and collaboration. Natural selection molds the or- der which already exists, so selection is not the only source of order in organisms, yet it needs to be acknowledged somehow. Finally, adaptation capacity of the complex living systems should be understood. In Darwin’s view, accumulation of mutations was probable; however, the capacity of doing that is not very clear, since some systems cannot adapt at all. Thus, the fact that selection enables the organisms adapting successfully needs to be considered Theory of complexity, origin and complex systems Chapter | 2 15
- 39. carefully, which leads us to the concept of adaptation capacity in a co-evolutionary process with organisms whose selection operates on complex co-evolving systems. As a universal process and dynamics, evolution leads to diverse phenomenology of life and its theory brings about rich phenomenology of life on Earth for modern biology having been subject to modifications in terms of its nature over the years. Although it is recognized that process of evolutionary change does not necessarily cause more complex organisms, evolution, in fact, is a process which can give rise to more complex organisms. For this reason, it is important to understand the theory of evolution as related to the phenomena of life and the way complex systems generally arise. Extremely improbable combinations of nature’s building blocks are possible due to the current complexity of living organisms. Therefore, the explanation of their existence and scope of evolution, with the parts, former being the formation of simple self-replicating or- ganisms from molecules and the latter being the formation of complex organisms from simple organisms, are still issues under consideration and analysis [68]. Considering the fact that evolution can produce more complex organ- isms with selection, it is important to note that complexity is seen in the co-evolution of hosts and pathogens, each one developing more sophisticated adaptations. When we have a glance at other fields, like economy, evolutionary theory, in terms of individuals and institutions, and complexity science, viewing economies as complex adaptive systems, is seen to have their integration to enhance the under- standing toward economics. In that regard, multi-level selection, causation, human psychology and cultural changes as evolutionary processes are products of gene- culture co-evolution [81]. Evolution, order and complexity reveal the relationship between social worlds and natural worlds, which also reflects a modern way of thinking that challenges the dichotomy of natural/social, presenting how the ideas from biology can be put into practice [82]. It is well noted that life requires structural complexity; yet, there is the chaotic mixture of organic compounds, which are highly complex. This points to the fact that life also requires a specific de- gree of structural order. At this point, the dilemma shows that neither complexity itself nor order on its own is able to characterize a living organism. Thus, combination of these two requirements, namely complexity and structural order, can characterize the difference between living things and nonliving ones, which also marks the course and results of Darwinian way of evolution. Evolution as a robust mech- anism drives order and complexity in several natural pro- cesses, and this cycle leads to an increase of the structural order in a system. This line of thinking makes it necessary to define complexity and order as the integral characteris- tics of life, which also point toward their use as parameters to evaluate the related processes along the course [83]. This perspective, with the combination of high order and high complexity forming a functional unit and indicating a universal type of biosignature, verifies the importance of balance and integration, as has been discussed above. From the Darwinian view, natural selection, in terms of balance, is the idea of nature producing more individuals that have variety than what is necessary. Those that are fit or well- adapted get selected by nature and get to procreate, which reveals highly dynamic and complex features of such a process. In other words, nature itself contributes to this complexity. Speaking once again of nature, in this context, from the smallest atomic particle to the biggest galaxies, the past, present and future of every living or nonliving thing in the universe is marked with its connection to everything. Thus, nature is characterized by its connection to anything else in this sense where complexity, as the real part of nature, comes to the fore as a significant notion with the paradoxical combination of order and randomness, which characterizes the multiple processes in nature in various areas. To put it differently, complexity is not just an abstract or philosophical concept but in fact a real part of nature as well as the societies of all beings [84]. Order, as one of the most fundamental concepts in science, is defined in mathematics in terms of the total and partial orderings of a set, the order of a differential equation or a group and so on, while order or disorder is generally associated with entropy in physics. The minimal descrip- tion of a pattern’s random aspects refers to the measure of its primary disorder and the average primary disorder of an ensemble of patterns is tied to the entropy of that ensemble; with secondary order being beyond the entropy concept. Additionally, the primary order has the tendency to decrease in evolution, while the secondary one tends to increase [85]. Regarding the temporal scale, when a randomness-finding complexity is at stake as the measure of complexity, the first-order complexity is proposed as a measure of randomness of original time series, whereas the second-order complexity is a measure and indicator of its nonstationarity degree. To elaborate, it may be expected that the second-order or macroscopic complexity of time- series is higher, and its value could be reliant on the de- gree of its nonstationarity; and so, a stationary random time-series could possess a high value of the first-order complexity measure, yet a low value of the second-order complexity measure [86]. Order, according to Kruger (1979), has two aspects, which are homogeneity and symmetry [87]. Complexity, on the other hand, as a unique way of estimating the information content of a pattern, is viewed as means to measure disorder or randomness, defined as the length of the shortest probable description of a pattern or the shortest probable computer program to generate a pattern. When selection, in this regard, is taken as the only source of order, it becomes necessary to assume that there would be only chaos without selection in 16 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 40. accordance with this basic Darwinian approach. Therefore, selective aspect is required to attain and sustain order [65] (for further details on these issues see [88e90]. 3.1 A conceptual outline: complexity and complex systems The study of complex systems struggles to enhance our capability of understanding universality that emerges when systems are extremely complex with interconnected parts. Across this line, if one attempts to understand the behavior of a complex system, it is required to understand how the behavior of the parts act together to form the behavior of the whole. This means, it would be insufficient to try to understand only the behavior of the parts. This consider- ation underpins the significance of “whole” in complex systems, which makes it evident that each part needs to be described in regard to other parts and relation thereof; just like the case of being unable to describe the whole without describing each part. All these intricate elements and subtle details make the understanding and analysis of “complex” a difficult one [68]. Going on with evolutionary theory, “wholeness” stands as another contentious aspect in view of the evolutionary origin of complex “wholes.” The consideration of self-organized collective properties of both simple and complex systems provides an accurate account of analysis. Apart from selection, another idea happens to be self-organization. The balance of the self-organized properties which is typical in selection and ensemble is reliant on the level selection can move the population to ensemble’s parts where typical order is not exhibited any longer. In systems which are adequately complex, selection cannot avoid the order which is displayed by the majority of the members; and thus, such an order is at stake not due to selection, rather despite selection, which suggests that sorts of collective self-organization correspond to some of the order which are displayed by organisms. As stated above, consideration of self-ordering proves to be benefi- cial owing to the concerted action of a huge number of constituents in systems and when evolution is the question, it is possible to face the question of how difficult it might have been to capture a particular property or structure [65]. The interrelated or interwoven elements and self- organization are two of the important qualities of com- plex systems, as mentioned briefly above. In this section, let us have a deeper glance at the other important properties for a thorough understanding. Although not easy to define, as stated previously, complexity characterizes many evolving parts interacting with one another in varied forms and displaying nonlinear patterns as aggregate. This set of characteristics is referred to as a complex system, which displays certain properties. One of the properties is cardi- nality, which refers to many parts varying from particles to agents making up the system. For example, a particular pattern in the sand might be viewed as complex because of having many regularities, not being symmetric perfectly. The reason why it is considered complex is not due to its imperfection, but rather those kinds of regular patterns form regardless of the way wind blows, which shows that an interplay exists between regularities and a sort of robust- ness [70]. Diversity is another important characteristic which shows that parts are different from one another, and dimensionality shows that parts differ from one another in many ways across different dimensions. A further quality is the acting, interacting, and adapting of the parts through networks, which is called connectivity. Adaptive interac- tion is a characteristic which refers to the interacting agents’ modification of their strategies in diverse manners along the accumulation of experience. Chaotic behavior, where small changes in initial conditions produce substantial changes afterward can also be noted along with fat-tailed behavior where rare events happen more frequently than what would be predicted by a normal, or bell-curve, distribution [19e21,23e53,63e69]. Another noteworthy attribute of complex systems is nonlinearity, which represents the relationship between variables in a nonlinear fashion and the aggregate of parts is not actually equal to the sum of their actions or characteristics. Other attributes can be listed as irreducibility, feedback, emergence, adaptiveness, oper- ating between order and chaos and self-organization, the last two of which have been briefly mentioned above [42]. Considering all these elements of complex systems, the core of the problem that needs to be analyzed is how complex systems self-organize their structures and/or how they self-regulate their dynamics. 4. Concluding remarks and future directions Complexity is both an old and new scientific concept and idea, which should entail the understanding of origin of complex components. It is so profound that the inherent complexity of the related phenomena and elements in the related fields should exceed the reductionist outlook of traditional science and mechanistic way of thinking. For this reason, as has been discussed and pointed out, complexity obliges us to adopt an understanding that ex- tends across a class of complex problems with many subtle and intricate attributes with working through more inno- vative and novel ways of thinking as well as applicable laws showing critical importance. In this setting, evolution, order and complexity help the revealing of the relationship between natural and social worlds, portraying a modern way of thinking that challenges the dichotomy between natural and social. Accordingly, this study has provided a conceptual outline and historical account of complexity and complex systems along with complex order processes to- ward modern scientific path from Darwin and onwards Theory of complexity, origin and complex systems Chapter | 2 17
- 41. concerned with natural, applied and social sciences. When compared with the previous works done up until now [18,21,23e53,63e69], this study has tried to include all the possible dimensions of complex systems in varied fields, taking into account the details of origin, history and episte- mology. The reason why those aspects have been handled, within the parlance of complex systems, is to emphasize the transition from evolutionary dimension as a revolutionary stage, and as a new paradigm for natural sciences and social sciences, so that the foundation for the interpretation of complex systems can be explored meticulously by related areas. Concisely, the accounts of past, present and future in different complex systems will facilitate toward a deeper understanding and guidance along the way. Based on these considerations, some of the following future directions can be presented as such: - Complexity thinking entails a broad horizon that con- siders the subtle properties of the domains in question and also other domains’ properties, which all require their own means of solutions and applicability. These will play an important role in the future science of all sorts of complexity, - The properties of evolution and adaptation can shed light on the understanding of past so that the present can be interpreted in a holistic way and future plans and schemes can be designed appropriately and timely, - When stuck in between two extremes of order and chaos under uncertain conditions, complexity thinking and theory can make the systems be adaptive, react to the world and act spontaneously. This line of thinking can also be of help to organizations, nations, scientific research and all other related parties for systematic and adaptive way, - Being cognizant of complex systems enables to analyze the core of the problem by understanding how systems self-organize their structures and how they self-regulate their dynamics, - Theory of complexity ensures powerful evidence and provides elucidation to challenge traditional and mech- anistic thinking to steer humanity toward adopting a new way of thinking. In conclusion, complex systems, complexity thinking and theory actually broaden the horizon and scope of modern way of thinking that relies on transition from evolutionary dimension as a revolutionary stage and as a new paradigm for natural sciences and social sciences. 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- 44. Chapter 3 Multi-chaos, fractal and multi-fractional AI in different complex systems Yeliz Karaca University of Massachusetts Medical School, Worcester, MA, United States Reality is more complicated beyond what is thought at the initial condition. 1. Introduction Complex and nonlinear dynamical systems are thriving as models of natural phenomena, usually characterized by unpredictable behavior whose analysis is hard to do as it takes place, like the incidents in chaotic systems. The essence of the problem lies in comprehending which sort of information, particularly concerning their long-term evo- lution, can be expected to be extracted from those systems. Correspondingly, complexity, order and evolution all manifest in the relationship between natural and social worlds, representing a modern way of thinking that certainly poses challenges against the dichotomy between the natural and the social. Computational complexity, thus, focuses on the amount of computing resources required for certain sorts of tasks, and its theory enables the assessment of resources that will be needed for the class of task at hand for a classification to be made into different levels of complexity. All these details and intricate elements pose challenges, which are also the components of dimensions of modern science and computational complexity, which require innovative methods and unusual ways of thinking. Given these, the idea of complexity is stated to be a component of a new unifying framework for science and a revolutionary way of thinking so that we can understand the complex systems whose behaviors are difficult to predict and control; and also, be able to come up with solutions to complex problems. Through these developments, the use of mathematical methods has become an indispensable method and tool for the improvement of diverse disciplines. The application of correct mathematical analysis methods can enable the accurate extraction of important information and prediction of future inclinations. Even though classical calculus can be powerful as a tool to tackle many dynamic processes in applied sciences, the existence of varied and huge number of complex systems in nature cannot be all characterized by classical integer-order calculus models, particularly with regard to information processing and its analysis. It is acknowledged that fractional-order system model can be the key to describing the system performance in a better way with predictive credibility and viability. Computational technologies in different complex sys- tems, which are based on mathematical-driven informed frameworks, can also generate realistic and applicable adaptive models under dynamic and evolving conditions. Based on such transformative understanding in conjunction with mathematics-informed frameworks embracing chaos, fractal, and multi-fractional ways, the incorporation of technology, with Artificial Intelligence, as the most prac- ticable component, has become an essential necessity in the current era. Through these means, we can tackle complexity which shows nonlinear, dynamic, and chaotic characteristics in addition to conceiving and implementing optimized solutions in an efficient and facilitating way with some required degrees of flexibility at the same time. Chaos theory is concerned with the behavior of nonlinear dynamical systems which under circumstances exhibit a phenomenon referred to as chaos that is marked by sensitivity to initial conditions. Three significant prop- erties of chaotic systems are ergodicity, initial value sensitivity, and unpredictability [1]. Related to chaos and complexity, the study [2] handles finance and economics as complex nonlinear systems affected by many external factors, ranging from human action to conflicts, policy to bilateral relations. By taking the time delays in a financial Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems. https://doi.org/10.1016/B978-0-323-90032-4.00016-X Copyright © 2022 Elsevier Inc. All rights reserved. 21
- 45. system into account, it is noted that fractional-order cal- culus alleviates the shortcomings which integer-order cal- culus cannot express sufficiently and precisely. The authors examine the dynamics and complexity in a fractional-order financial system with time delays while observing capti- vating transitions to deterministic chaos that incudes cascading period doubling and high-level complexity. The paper [3] explores the theoretical issues on drive-response synchronization of a class of fractional order uncertain complex valued neural networks (FOUCNNs) along with time varying delay as well as impulses. Banach contraction mapping principles, robust analysis techniques, and Riemann-Liouville derivatives are used in the study so that a new set of sufficient conditions regarding the uniqueness and existence of equilibrium point of such neural network system can be derived. In addition, Lyapunov functional approach is applied to obtain the global stability of the equilibrium solutions. The study, thus, provides an example of a multifarious approach. Another study [4] aims at improving the randomicity behaviors of some chaotic maps, which represent chaotic systems, and proposes two integrated chaotic systems for the generation of different chaotic maps, by developing a new image encryption al- gorithm using integrated chaotic systems whose theoretical aspects are also provided. Concerning the unpredictability of chaotic systems, the authors of [5] handle COVID-19 pandemic, which has had massive impact on tourism sector through six illustrative examples about how a research agenda should resemble, pointing out a paradigm shift in tourism because of pandemic conditions. The chaotic conditions in such a complex system show that a new understanding is needed in a “new-normal” tourism world by taking into account the greater global economic and political context, by going beyond the purely descrip- tive means. The research paper [6] deals with chaos, brain, and divided consciousness as modern trends of psychology and cognitive neuroscience stating that applications of self- organization, nonlinear dynamics and chaos are important for problems related to mind and brain relationship. Chaotic self-organization is noted to provide a unique tool both in terms of theory and experimental aspects for a more pro- found understanding of dissociative phenomena. The nonlinear methods employed in the study are for the anal- ysis of marked changes in electroencephalography and bilateral electrodermal activity during the experiencing of dissociated traumatic and stressful memories as well as psychopathological states. The analysis of the study cor- roborates the possible role of chaotic transitions, and the author suggests that self-organizing theory of dreaming is significant related to memory formation and processing as cognitive processes. The study is important in terms of showing the dynamic ordering factors and self-organization which lie under the brain physiology and psychological processes, which are different examples of complex systems. The study of [7] is on the key features of complexity from different perspectives with relevance to strategies proposed to combat the COVID-19 pandemic. The author argues that critical systems thinking is the approach to complexity that offers the most fitting under- standing of the phenomenon and suggests that multi- perspectival and multi-methodological approach of critical systems should be adopted by decision makers to get ready for and respond to similar crises during the turbulent times. The author concludes that a variety of systems methodol- ogies and different perspectives with alternative explana- tions can be used to instigate informed ways. Again related to the pandemic, chaotic dynamics is focused on [8] through the mathematical examination of COVID-19 data in different countries, which shows similarity to the chaotic behavior of many dynamics systems just like logistic maps. Apart from providing direction for public policy makers, the study remarks based on the use of an interactive data map that the pandemic’s scale and behavior are unpre- dictable because of the chaotic systems’ properties. Chaos, referring to irregular and unpredictable behavior charac- terized by sensitive reliance on initial conditions, can be shown by some of the nonlinear differential equations. The tendency of nature toward pattern formation, iteration, and creation of order out of chaos generates expectations of predictability. Due to varying degrees of interaction be- tween chance and choice as well as the nonlinearity of systems, nature escapes the boredom of predictability [9]. The studies handling chaos and multi-chaos in various disciplines [10,11] provide important insights into its operation, dynamics, and properties when we consider that characteristics of living systems play a prominent role for surviving and thriving through chaos. Fundamental change in the outlook of traditional ge- ometry is essential to handle complex systems. Conse- quently, fractional dimensions of objects, known as fractals as never-ending patterns, are integrated into an integral dimension space, and geometric fractals are described by a procedure or an algorithm that generates them explicitly self-similar. Fractals are, in fact, images of dynamic sys- tems driven by recursion, which is to say the image of chaos; and fractals are used in order to model structures where patterns recur repeatedly and describe random or chaotic phenomena. One of the related studies is on fractals in the nervous system regarding neurodynamics [12] doc- umenting the prevalence and showing the credence of fractals at all levels of the nervous system along with their functionality as well as paying attention to the relationships between power-law scaling, self-similarity and self- organized criticality. The overview by [13] provides the philosophical, historical and basic concepts related to fractal geometry discussing the ways neurosciences can make use of computational fractal-based analyses. The comparison of fractal with Euclidean approaches to analyze 22 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 46. and quantify the brain across its whole physiopathological spectrum is also provided. The study of [14] is on the predictive optimization of the classification of stroke sub- types based on fractal and multi-fractional methods. Box- counting and wavelet transform are the other ones used and Feed Forward Back Propagation (FFBP) algorithm is employed for stroke subtype classification. The study aims at providing a new direction concerning complex dynamic systems and structures that display multi-fractional prop- erties. [15] is a study on Rescaled Range (R/S) fractal analysis with wavelet entropy characterization for fore- casting purposes based on self-similar time series modeling. The proposed novel method with its multifarious methodology can be applied in different applied sciences to reach optical solutions in complex, dynamic and nonlinear settings for critical decision-making processes. Concerned with financial environments, the study [16] handles char- acterization of complexity and self-similarity using fractal and entropy analyses for stock market forecast modeling. The study shows the critical importance of Hurst exponent (HE) as computed by R/S fractal analysis employed as an indicator along with Shannon entropy (SE) and Renyi entropy (RE) related to future forecasting capability with regard to the stock indices. Finally, [17] is a study on self- similarity and multi-fractality in human brain activity through a wavelet-based analysis regarding scale-free brain dynamics. The authors propose a novel method to enrich the characterization of scale-free brain activity via a robust wavelet-based assessment; and for this they used magne- toencephalography (MEG) to analyze human brain activity. The study of [18] is on fractal stochastic processes on thin Cantor-like sets, reviewing the basics of fractal calculus, defining fractal Fourier transformation on thin Cantor-like sets, introducing fractal generalizations of Brownian mo- tion as well as of fractional Brownian motion. The fractal derivative is said to be the progenitor of the fractional de- rivative, which emerges if a certain fractal distribution of events on the time axis is used. While fractals refer to objects or quantities that manifest self-similarity at every scale, dynamical systems describe the way a complex system changes in time. Thus, research into these aspects provides significant input and contributions for different fields [19,21] to understand complexity deeper. Manifesting evolution in time cumulatively, complex processes are based on the occurrence of both huge and number of phenomena with the mean time interval between two subsequent critical events being infinite. Nonlocality is one of the reasons for interest in applications of fractional calculus and many interesting physical phenomena with memory effects also suggest that their state is not only dependent on time and position but also on the prior states. Consequently, fractional differential equations (FDEs) are seen as alternative models to nonlinear differential equa- tions. In view of that, the study of [22] investigates the behavior of dynamical process of complex systems within two types of ideal ways. FDE is constructed to describe the time evolution of the complex systems. The dynamical behaviors are conveyed universally into a power law dis- tribution with exponents determined by parameters that characterize the system. The authors also demonstrate how slow relation and super slow relaxation processes take place in hybrid systems, with time intervals having been measured at the logarithmic and double logarithmic scale. The incorporation of fractional calculus models for the description of diffusion in heterogeneous and complex materials is steady, and [23] handles biological tissues whose features can be encoded in the attenuation of the MRI signal by the fractional order of time and space de- rivatives. The authors describe different exponential and fractional order models applied in MRI and investigate the connection between model parameters and underlying tis- sue structure. Fractional calculus is stated to provide new functions like Mittag-Leffler and Kilbas-Saigo, which characterize tissues concisely, pointing toward success in treatment. Fractional differential equations and fractal analysis are accepted to be useful means to describe the dynamics of complex phenomena characterized by spatial heterogeneity and long memory. To construct the mathe- matical modeling of many nonlinear phenomena, fractional differential equations (FDEs), being viewed as alternative models to nonlinear differential equations have varieties of them, serve as important tools in various fields including but not limited to mathematics, physics, fluid flow, biology, control theory, signal processing, fractional dynamics and systems identification [24,29]. Computational technologies, with machine learning as the core component of AI, enjoy broad use and trans- formative impacts, which enable us to train complex data to automate or augment some human skills. Thus, the cross- cutting nature of AI provides motivational power to formulize research in a systematic way. Artificial neural networks (ANNs), which are networks of computer systems inspired by the human brain and biological neural networks have the ability of learning and modeling complex and nonlinear relationships. In the simplification, abstraction, and simulation of the human brain, ANNs reflect the related basic characteristics of this complex organ [30]. Among the related studies, the paper by [31] is on the deep fractional max pooling neural network (DFMPNN) with 12 layers for the recognition and better diagnosis of COVID-19. The model proposed by the authors is demonstrated to be su- perior to 10 state-of-the-art models, and three more im- provements are also proposed accordingly. Another study [32] is on the adaptive fractional-order backpropagation neural network for handwritten digit recognition problems through the combination of population extremal optimiza- tion algorithm, named as evolutionary algorithm, and a learning mechanism based on fractional-order gradient Multi-chaos, fractal and multi-fractional AI in different complex systems Chapter | 3 23
- 47. descent. The study emphasizes the significance of the optimization of the initial connection weight parameters. [33] is another work where a fractional-order deep back- propagation neural network model is proposed with L2 regularization. The fractional gradient descent method with Caputo derivative is used to optimize the network proposed whose necessary conditions for convergence are also illustrated, with the conclusion that the proposed network is deterministically convergent and capable of avoiding overfitting in an effective way. Moreover, [34] is a study concerned with swarm intelligence technique employed to solve FDEs. Feedforward ANNs are used to define the unsupervised error, and particle swarm optimization is performed for the learning of the errors’ weights. The scheme proposed by the authors point to conceptual simplicity, easy implementation, and extensive scope of applications. Last but not least, the study by [35] shows the way to use data-driven similar measures in an effective way in standard learning algorithms in the context of life science as well as other related complex biological systems. In brief, computational methods, along with models and adaptive algorithms for scientific advances show the ca- pacity of handling large amount of complex datasets. In addition, significant advancements in computational pro- cesses aim at addressing the issue from a more versatile and profound understanding with an interdisciplinary outlook in various disciplines [36,37]. The related computational processes with broad applications in integration with frac- tals, multi-fractals, fractional methods, chaos, nonlinear dynamical properties, stochastic elements and so on pro- vide systematic optimized solutions. Modern scientific thinking has adopted the principles concerning systemic properties, addressing them by uncovering the spontaneous processes related to self- organization in a dynamical system, in a state distant from the equilibrium point, and in proximity to the disequilibrium point without any existence of an external force acting on the system itself. In view of that, evolution, order and complexity unearth the relationship between natural and social worlds challenging the dichotomy be- tween them. This study, vis-a-vis the works in the literature, for the first time, provides a conceptual outline, brief his- torical sketches of fractal, multi-chaos, and multi-fractional AI in different complex systems. In addition, the outlook provided in the study indicates that the works that employ AI and computational methods have tracked and will need to track along an evolutionary process; and if they use multifarious integrated methods, it would be possible to reach optimized solutions through developing strategies within a paradigm shift. This study provides an overview encompassing multi-chaos, fractal, fractional and Artificial Intelligence (AI) ways of thinking for the solution of the complex system problems concerned with natural and so- cial sciences. In addition, ethical decision-making frameworks and strategies related to big data and AI ap- plications assist the identification of the related problems in different settings and help thinking methodically with a deliberative compensating process in order that tensions between different conflicting aspects can systematically be handled. The values related to ethical issues should also be addressed, in a way that requires to be practical, flexible and problem-driven instead of sheerly theory-driven so that dilemmas can be addressed and critical decision-making directed beyond theoretical positions focusing on the applied aspects. In view of that, this study aims at providing a direction through the topics of data reliability, chaos thinking, fractal, fractional thinking and artificial intelli- gence way of thinking as well as processes all revolving around complexity. Another motivational aspect is the addressing of ethical issues, which raise thorny questions for both researchers and practitioners in a way which adopts practical, flexible, applicable and problem-driven mode for the solution of ethics-related dilemmas so that purely theory-driven approaches will not be the only point to resort to. Furthermore, ethical decision-making frame- works and strategies related to big data and AI applications should be developed and applied so that related problems can be identified correctly in different settings and thinking can be done methodically in a deliberative way with compensating processes. Since the impact and ubiquity of data technologies are the case concerning all aspects of modern life, it is important to build a balance between data use and ethical matters. Computational technologies in different complex systems based on mathematical-driven informed frameworks can yield more realistic and appli- cable adaptive models under dynamic and evolving con- ditions. Through such transformative thinking in combination with mathematics-informed frameworks that encompass chaos, fractal, and multi-fractional ways, the incorporation of technology, with Artificial Intelligence, as the most practicable component, is a requirement in today’s world so that we can tackle complexity which shows nonlinear, dynamic, and chaotic characteristics. Thus, optimized solutions can be conceived and implemented efficiently and in a facilitating way with some required degree of flexibility as well. The remainder parts of this chapter are organized as follows. Section 2 addresses the Challenging Dimensions of Modern Science, Complexity and Complex Systems with subheadings of 2.1., 2.2., 2.3., and 2.4. entitled “Data Reliability and Complexity”, “Chaos Thinking, Processes and Complexity”, “Fractal Thinking, Processes and Complexity” and “Fractional Thinking, Processes and Complexity”, respectively. Section 3 is concerned with Artificial Intelligence Way of Thinking, Processes, Complexity and Complex Systems. Finally, Section 4 of this chapter provides the Concluding Remarks and Future Directions. 24 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 48. 2. Challenging dimensions of modern science, complexity and complex systems Complexity seems to be relevant and applicable con- cerning a vast array of phenomena on all scales. In the evolution of dynamical systems complexity tackles, the nature of the system under consideration is often irrele- vant; yet, the perspective complexity proposed enables the identification of forms and evolutive characteristics per- taining to all or nearly all of the systems constituting a huge number of elements. These elements within the systems may have reciprocal interactions, positive feed- back mechanisms, as well as nonlinear interfaces. For this very reason, such systems are precisely named as complex systems. Another point to note is the way systems orga- nize themselves in reaction to an action coming from the external environment [38]. Although this investigation is considered to be as old as science itself, the studying of the way systems make the self-organization spontaneously is a scope of investigation that is relatively new and per- tains to modern science whose roots date back to the important developments of Renaissance [39,40] and Enlightenment [41]. In addition, traditional science tries to account for the forms observed by means of a reductionist outlook such as looking for laws regarding the single components of the system, the science of complexity, in modern science sense, in contrast to traditional science, adopts the systemic properties and considers those by demonstrating the spontaneous processes of self- organization and how it occurs in the case a dynamical system is in a state distant from the equilibrium and close to disequilibrium without the existence of an external force acting on the system. Conceptually, in this modern scientific view, what we are interested in is to develop an understanding of organisms in their environments, which points to one aspect of evolution theory taking into consideration the evolutionary dynamics of populations of complex organisms. The development of life is described as evolution, which is the individual organism’s capability development, which refers to an increase in the complexity of the organism. Evolution is not regarded as the development of a collective behavior of many organ- isms primarily, so the development of species and eco- systems is also a question to consider. Being a general approach to the complex organisms’ formation by incre- mental change, evolution and conceptual incremental evolutionary processes encompass monotonic evolution on a fitness inclination as well as traits’ divergence and extinction. With evolution viewed as a blind fitting pro- cess, organisms display adaptation to their environment [42] and theories related to evolution see fitness as the mere property of the organism that decides on the evolu- tionary dynamics and in that regard conventional evolu- tionary theory relies on gradual variations of fitness [43,44]. To compensate the gaps of the reductionist approach mentioned briefly above, the use of dynamical equations which model reproduction and predation can provide the modeling of a multiple dynamic phenomena in populations. In other words, a variety of resources with their own peculiar dynamics need to be analyzed so that the existence of groups of organisms that have well-distinguished traits can be explained. While various systems, namely immune system and artificial computer software, can be employed as laboratories to develop an understanding toward evolu- tion, the study of organisms as well as their complex col- lective behavior and evolution by means of mathematical tools has become an important area of study of complex systems in recent years, which mark the evolution of modern science addressing the construction of models from the interactions of components through the discussion of spatial and temporal structures and substructures. Such interactions can be seen in many different forms like cooperation, communication, reproduction, competition, consumption, exploitation and so on. While individual behavior is often complex, it is not very clear if the emergent collective behavior of many individuals is com- plex. For this reason, the behavior of complex individuals at interaction with their environment becomes a subject of investigation. Recent developments in artificial neural networks, in modern scientific view, have provided the robustness of millions of synaptic weights’ optimization with observations operating powerfully for many different phenomena. Correspondingly, the related models make use of local computations to interpolate task-related mani- folds within high-dimensional parameter spaces rather than learning simple rules or world representations. Anal- ogous to evolutionary processes, models which are over- parameterized can be parsimonious with their provision of robust, applicable and versatile solutions to learn a diverse set of functions and reach optimized outcomes. These models and outlook the models are derived from establish links to consider unpredictability and more importantly, pose a far-reaching challenge to many perspectives in different areas of science relying merely on theoretical as- sumptions. In this context, the construction of models related to life should also include a model of the environ- ment when behaviors of individuals are measured as reac- tion to an external stimulus. In addition, the link between the individual’s capabilities and the environment’s de- mands can provide a comprehensive description of the or- ganism. In sum, all these points demonstrate the critical role the environment plays in the complex dynamics of Multi-chaos, fractal and multi-fractional AI in different complex systems Chapter | 3 25
- 49. evolutionary change and that there is a close relationship between an organism’s complexity and the organism’s environment. It is important to establish a balance between providing rigorousness, pursuit of distinctness as well as clarity and not abstracting the complexity of the real world so a junction, rather than a disjunction, is required between scientific wisdom and reality. One challenge is to stimulate mathematical intellect without having a reference to the interests of science that are geared toward the real world. The mathematical structure, known as nonlinearity, in this regard, refers to the system’s autonomia which may not have a single variable but operates in a system of high dimensions where numerous species and resources are connected very closely, interacting by producing highly complex entanglements and/or cooperating at times. In ecosystems, nonlinearity signifies the coefficient of repro- ductive rate with time constant not being a priori at all but one that changes according to the conditions of the ecosystem in question like the function of population. Through nonlinearity, the ecosystem can either end up with stability or it may produce unpredictable or complex os- cillations [45]. As a system, a simple one provides one single path to one answer only, providing one solution and one way for sorting out the problem. In this sense, a complex system has different and multiple ways toward multiple answers; so based on the choices made, it is likely to encounter a sys- tem that changes according to those selections, which also refer to the adaptiveness of the complex systems. The more insight is developed, the answers may keep changing and more learning occurs. While searching for the use of appropriate mathematical model, the description and anal- ysis of a system relies on the way the system is perceived [46]. Thus, when complex systems are at stake, with large collection of components interacting locally with one another at smaller scales and having self-organizing to manifest global behaviors and structures at larger scales without external intervention from the environment, it is necessary to understand and/or predict the properties of such an intricate collection based on the “whole” knowl- edge pertaining to its constituents. All these elements require novel mathematical frameworks in ever-changing current landscape, which was put very well by Stephen Hawking as: “I think the next [21st] century will be the century of complexity.” Accordingly, the aim of complexity and nonlinear science is to gain global under- standing by taking into consideration the multiple inter- acting factors of systems, many branches of possible states, and high-dimensional manifolds while monitoring actuality regarded as diachrony, which is to say the historical and evolutionary path that has been through many different critical points on the manifold. Within such settings as described and outlined above, dynamic modeling as a process allows the extension of our knowledge about reality, with computer being the main means due to allowing the user to trace the evolution of the dynamic system, which is represented by the model by numerically integrating the equations. Therefore, the chief objectives of dynamic modeling are intended to describe the flow of the situation and make prediction on its future evolution. As already mentioned, the construction of the model is a process, which needs to consider the aspects that are regarded as secondary so that results for the assump- tions made are provided, which will help us do the final analysis by handling the implications of one’s own opin- ions for the correct and applicable interpretation of related phenomena [38]. Given that, dynamics of coevolving sys- tems also need to be taken into account carefully as the success of natural sciences has reflected in that organisms do not only evolve but also coevolve with the other or- ganisms in the environment. The exploration of the re- quirements of order and capacity to evolve needs networks that manifest parallel processes where selection is critical in the emergence of entities which coevolve with each other; and thus, it is important to comprehend the ways in which selection achieves systems that are capable of coevolving [166]. All these elements and intricate qualities reflect Darwin’s perception of nature, which is described as “There is grandeur in this view of life” with bewildering interconnections between laws and amazing organic beings in endless forms [47]. This vision encompasses multiple facets like intellectual, sentimental and aesthetic dimensions, which is a key to dig into the layers of complexity. 2.1 Data reliability and complexity Data complexity is a challenging issue and as regards managing and leveraging data, the problems arise since significant data may be scattered across different platforms, some being inaccessible, some incomplete, outdated, inaccurate, misleading or inconsistent. At times lack of detailed knowledge about the source system data may be at the core of the problem, and at other times building data integration processes may be haphazard due to the lack of knowledge and skills required. For all these reasons, it is accepted that data complexity has a huge impact; and hence, while making critical decisions, it becomes impor- tant to address such complex problems and barriers to data reliability, accuracy and quality so that time, efforts and resources can be used efficiently. As for data management, complexity and data protection in the digitalized era where cloud, hybrid, and other type of workloads have been some common scenarios, the important keys are stated to be performance, scalability and reliability. To tackle data 26 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 50. protection complexity, cut down on disruption, and miti- gate data loss risk, simple, automated and reliable solutions are required. Consequently, another issue other than data complexity, protection and management aspects, reliability comes as one further noteworthy challenge. Describing the ability of a component or system to function under speci- fied circumstances, reliability is tightly connected to availability, which refers to the temporal ability of a component or system to function at a certain interval of time or moment. Reliability and availability are two closely related terms in data complexity issues to deal with accurate predictive measurement, uncertainty and risks of failure. Complexity brings about unique challenges in big data analytics in which big data does not only refer to huge volumes of data but its “bigness” has also to do with their complicated structures. Volume, velocity and variety are the three common characteristics of big data, as described by Zhang [48]. The last feature mentioned, namely variety, is concerned with data complexity, which can refer to complex relationships, high dimensionality and many other complications in a given dataset. The challenges involved in dealing with complexity in big data concerning modeling and analysis also provide opportunities for the development and application of related statistical methods, which also applies for reliability applications. Through engineering systems that generate big data for reliability analysis, it becomes possible to deal with degradation measurements, time to failure and time to recurrence of events [49]. Reliability analyses are important for complicated struc- tures while focusing on the use of data [50]. Furthermore, big data also requires new processing means to possess insight discovery, more robust decision-making power and better process of optimization ability along with the ways of effectively excavating useful information as another challenge faced in the current present big data background [51]. In this background, data complexity measures can be classified into three preliminary categories, which are measures of separability of classes, measures of overlap of individual feature values and measures of geometry, to- pology and density of manifolds. Considering these factors are of central importance while tackling multi-class datasets [52]. When missing data is the case, this will be a situation which would degrade performance, and incorrect imputa- tion of missing values can further cause inaccurate pre- dictions [120]. Since missing data could have serious impact when not handled appropriately, analytical methods can be applied to fix the missingness aspect [53]. Incom- plete data or information could also have serious impacts on the subsequent data processing and reasoning, which could cause one to end up in wrong decision at critical times. Some reasons of missing data are as follows: the attribute of the data may not exist, data may be not avail- able temporarily or data loss or damage might be the case [54]. All these interrelated problems bring the utilization of data to the foreground in pursuit of novel and highly effi- cient solutions, which is sorted out by the proposing of new schemes based on algorithms in tandem with technological advancements. As a result of progress in imaging tech- nologies and integration of systems approaches, quantita- tive science is also going through a reform. Within this context, robust conclusions to be drawn from quantitative data require a measure of their variability in complex sys- tems with experiments being carried out under intricate and measured complex processes. When one is exploring a complex system, it is important not to discard outlier data points, as those may be relevant as clustered measurements. In other words, it is important not to fall into the trap of ignoring data that do not match the related hypothesis tested, since the subject of interest may not be simple or straightforward that would provide only black or white answers. It is recommended not to turn the hypothesis driven research into a hypothesis forced one [55]. Reliability and robustness of data and its complexity in collection, storage, sharing and utilization of data in different areas also bring about another highly controversial and challenging point which are the ethical considerations. In that regard, it is important to be cognizant of different dimensions in various fields, particularly in health care. Some of the points associated with ethics are informed consent providing the person in question why data are collected, from whom and how, its storage way, length of keeping, and who will have access to it, all of which have legal dimensions. Another important aspect related to ethics is data transparency which suggests openness, communi- cation and accountability. It also refers to the control flow of the data in machine learning algorithm [56], while in medicine, transparency helps the enabling of evidence- based decisions that is critical to foster trust among the related parties [57]. Used in science, engineering, business and social sciences, transparency is practiced in many different systems ranging from communities to adminis- trations, companies to organizations. As a related compo- nent, accountability, namely tracking where data come from, can be taken as an operational issue since different queries are at stake like how were the data produced, who produced the data, when were the data extract produced and what data were produced [58]. Along with these di- mensions, one argument, in terms of big data, states that big data algorithm designers must make data sources and pro- files public to improve transparency and avoid bias [59]. Accountability also includes the related systems, laws, and regulations, which point to the complexity of the issue, in other words, even one single dimension of ethics provides complexity on its own operating within complex systems that have interacting and variable elements. Clear communication and data sharing are other important ethical Multi-chaos, fractal and multi-fractional AI in different complex systems Chapter | 3 27
- 51. aspects, since they are important to have clear processes to share data, which is as important as having clear processes to collect data. This becomes more critical when individual data are under consideration, which indicate privacy and sensitivity in widespread data such as mental health, sui- cide, patient details, biological information like DNA, personal financial data and so on. Furthermore, data privacy refers to the relationship between collection and dissemi- nation of data, which may be sensitive that would not be apt to reveal for the sake of trust. Also known as information privacy, data privacy is one branch of data security that deals with the proper handling of data, including factors like consent, notice as well as regulatory obligations. Practically speaking, data privacy concerns focus on whether or not data will be shared with third parties, if it is to be shared, then how it will be shared, how data are le- gally collected and stored, and finally regulatory re- strictions are to be observed [60]. Controlling access to personal information is viewed to be a significant aspect of maintaining privacy, and if information is accessed or revealed against the wish of the person, this situation has the potential to impact the well-being and breaching of rights, which include the respect dimension too [61]. Confidentiality refers to keeping data like medical records or service records confidential, while anonymous data refer to information that cannot be traced back to a particular person. Data linkability is also a related concept, which refers to the ability for anonymized data to remain linkable so that the data value will not be diminished; whereas data composability points to the privacy guarantees which could be provided when data from multiple sources (to which the same or different privacy models have been applied) are integrated to one single fused data-rich source [62]. Confidentiality and anonymity as well as bias include the ethical considerations linked with the collection and use of data in nonclinical areas with the relationship between patients as a group and organizations [63]. In business settings, online data collection has its ethical dimensions from the customers’ side, since they need to know that the data they are using are reliable and sourced ethically. On the other hand, companies that provide online data collec- tion technology have to commit themselves to trans- parency. Therefore, regulators have started to focus on the background of data collection process, which seems to be among the other future challenges related to this issue. More formally, concerning governance and law, legal in- struments protect privacy in different ways through acts and data protection laws and limitations [61]. In health and research characterized by evolving land- scape of big health data, the use, sharing and reusing of big data have become an important feature, and with the advent of technological developments, particularly artificial intel- ligence, the use of big data has become far-reaching. Although ethics of big data has become a topic of extensive research, guidance and ethical decision-making frameworks are required further, which will provide insight into the values and how decisions will be made in an increasingly complex environment of health and research. Ethical decision-making frameworks, in this regard, provide assistance to identify the related issues in different settings and help think through in a methodical manner while at the same time these frameworks integrate a deliberative compensating process to manage tensions between con- flicting values step by step and systematically. The values concerning the ethical issues point to being practical, explicit, of equal weight, flexible, problem-driven, rather than theory-driven so that real-world dilemmas can be addressed beyond theoretical positions and guide decision- making (see [62] for further details and the key ethical values defined). In the age of big data, it has become possible to generate more and more data from growing number of sources about health and biology. While access has become easier, challenges also arise in terms of ethics of data, when the relationship between privacy and public interest is considered. This is because data science and computing developments have imposed pressure on tradi- tional approaches to information governance like seeking consent or making data anonymous. For these reasons, data initiatives, governance and design thereof address key ethical principles so that good practice can be established [61]. One significant idea about consent is that it needs to be obtained prior to an intervention, which is to be based on a sound understanding of its implications and likely out- comes consequences. Yet, in the modern data world, it is disputable to what extent this is practicable or relevant [64]. Consent becomes a critical concern and an ethical requirement while carrying out research with human par- ticipants who are expected to comprehend the presented information like the aim of the research, predictable risks, likely outcomes of any decision to be made based on the information [65]. In addition, there is a surging interest toward aggregating biomedical and patient data into big datasets intended for research purposes and also public benefits, which causes new ethical issues to emerge with regard to social justice, human rights and trust that concern many parties including but not limited to researchers, pol- icy makers, regulators and healthcare professionals [66]. Moreover, responsible data sharing in health research needs to be addressed as well since it should have principles and norms when large-scale of dataset linkage in clinical set- tings is at stake considering treatment evaluation, disease etiology, its facilitation, diagnostic purposes, to name just a few [67]. Consequently, both ethical concepts such as privacy, informed consent and confidentiality as well as new ones are likely to arise in the evolving landscape of big health data, which shows that ethical issues also show evolution due its dynamic features, all pointing to the transient and emerging challenges of modern science. 28 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 52. Considering the requirements of modern science, which also necessitates comprehensive and fruitful view of ethics, it should be noted that the dimensions of ethics need to be integrated into a complex theory that shows the interplay of tensions and finding a balance so that none will obscure the practice of ethics [68,69]. The far-reaching effects of data technologies and data science due to their intrinsic complexity are evident in all dimensions of modern life, which obligates those dealing with data to engage with the ethical issues. Even though data are available, powerful tools to extract information and economical storage ca- pacity seem to be the advantageous aspects, such advanced technologies also come with their challenges like their potential misuse [64]. As is the case with everything else, data are also evolving with time with some datasets accu- mulating over time, changing dramatically as a result of the spatial and temporal conditions. For these reasons, various properties of modern data and the use thereof necessitate taking care of changing ethical issues some of which are interconnectedness, real-time decision-making as data arrive, lack of time, space and social context restriction on the scope of data as well as use for unexpected reasons and revealing unexpected information [64]. In this landscape, data ethics has emerged accordingly as a new branch of ethics that explores and evaluates the moral problems concerning data, which include but are not limited to generation, recording, processing, publishing, sharing and use, along with algorithms, machine learning and robots all related to corresponding practices with responsible acting, innovation and programming. To come up with morally acceptable solutions, right conducts and values are to be observed. Consequently, the conditions require the shift from being information-centric to being data-centric, which also indicate the complexity of ethical challenges brought by complex data science considering the collection and analysis of big datasets in many different fields like biomedical research, social sciences, profiling, open data, data philanthropy and so forth [70]. With the accumulating and increasing quantity as well as variety, data with different types and nature pose serious challenges associated with the complexity of the matters and considerations revolving around ethics during pro- cesses concerning critical decision-making, handling complexity, collecting or sharing data. Accordingly, different types of data that cause different instances of complexity and challenges can be briefly categorized as follows [71]. Data downloaded from databases that are available to public: These may be country-level data or international as well. They are exposed to updates all the time, thus, version and date of retrieval need to be specified. The typical relational databases have restricted ability to manage the heterogeneous nature of complex and modern data; therefore, high complexity of data in those databases could cause bottleneck in conventional information sys- tems concerning the efficient and reliable retrieval of in- formation [72]. (1) Data collected manually and coded accordingly: They could fall into the category of archival data and original survey data. Effort, time, and financial resources contribute to the complexity of such data, since select- ing and entering them are demanding and arduous. (2) Data utilized under license coming from a commercial data provider: Commercial databases are used exten- sively in many studies focusing on economics, finance, and strategic works. Copyright becomes a legal issue, since it is held by a private organization, which ensures data access through subscription. When data are shared with others, there would be cases of copyright viola- tions; and providing public access to the original data is not viable on legal terms. All these dimensions add to the complexity of the problems. (3) Data available on a remote computer: To preserve the confidentiality of data, some data providers allow re- searchers to analyze data from the computers of the or- ganization or researchers submit their codes to the staff of the organization so that they can analyze the data for them. Thus, researchers are in a position of being un- able to share raw data, since they do not have access to the data. (4) Data which are generated by laboratory experiments: These may concern experimental design-based studies, which have grown in recent times. New standards come into existence, indicating the dynamic changing environment of data as well. (5) Data which are generated by qualitative research: Such research can be observations and interviews which cause certain problems and conflicts regarding transparency and ethical obligations among scholars doing the research [73]. In contrast with quantitative research that uses secondary data, qualitative area brings the confidentiality notion in ethics into fore- ground. Direct interaction is at stake during interviews, which is a qualitative research method; hence, a partic- ipant’s confidence becomes integral to the reliability of the data [71]. In addition, regarding the secondary analysis of qualitative data, culture of data archiving is important to be cultivated in qualitative research, and since data archiving would include someone’s personal views, the best practice could be to practice anonymization during the initial transcription. Thus, a record or log of aggregations, replacements or re- movals need to be made and stored from the anony- mized data files separately. It should be noted that further ethical concerns may arise during the rein- terpretation of qualitative data at a later time period [74]. Multi-chaos, fractal and multi-fractional AI in different complex systems Chapter | 3 29
- 53. (6) Industrial data: The realization of data collection ar- chitectures in Industry 4.0 is linked with vast imple- mentation efforts as a result of the heteregeneous structure of systems, interfaces, protocols and other disciplines involved in the projects. Digitization of processes has increased due to industrial automation transforms and available data in the production has also been increasing in amount, which makes it essen- tial to leverage data to adjust to production plants and machine parameters so that efficient and flexible pro- duction can be ensured [75]. Handling of data also requires the accurate categoriza- tion and comprehension thereof; and in that sense, data can be categorized as numeric or nonnumeric with qualitative data belonging to nonnumeric category, which represents certain descriptive features, and they are referred to as categorical data. Quantitative data, on the other hand, are numeric, being further clustered as discrete or continuous [76]. Complex data types are divided into two descriptive classes, which are discrete and continuous. While the former represents the features that exist independently that have definable boundaries and a finite number of possible values, being any of the three geometries, that is to say polylines, polygons and points, continuous data do not have clear or definable boundaries. Thus, continuous data can be remembered as data with no defined boundaries that cover a scale of values and its opposite counterpart, namely discrete data refers to data that have defined and clear boundaries, which can be described by values that can be eliminated from a system without disrupting the whole system. In addition, while continuous data constitutes the rest of nu- merical data, usually associated with some kind of physical measurement; discrete data often occurs in an instance in which there is certain number of values or whole numbers are counted. Another difference is that discrete data can be represented by integer or whole numbers, whereas contin- uous data are usually regarded as exact and integer with infinite possibilities, represented by real numbers [77]. The complexity of the systems and data utilized, shared, handled and managed should involve optimal strategies that seek the understanding of the interactions relevant to all the different parts of the system and in this setting, data collection should be geared toward the assessment of re- lationships within a system so that the analyses can inte- grate notions and ideas taken from systems thinking approach and also complexity science [78]. Systems theory is important in this context, since it is the interdisciplinary study of systems with interrelated parts that can be either manmade or natural, with each system being bounded by time and space impacted by the related environment, defined by its purpose and structure; eventually expressed by its functioning. Since the change in one part of the system can affect the whole system, predicting the changes in patterns of behavior becomes a must therein. Systems thinking helps as an approach with regard to problem solving, which intends to strike a balance between holistic thinking and reductionist thinking. The related purposes of this mode of thinking are significant. It enables one to define a system in this way and apply the definition to a variety of complex systems. It also helps one recognize the features of systems that make them complex ones, while analyzing a system means identifying problems and formulating it in systemic way. Finally, it allows us to apply advanced systems thinking so that we can seek so- lutions to disorganized management faced in real life. For systems that are in continuous adaptation and learning, the goals with respect to systems theory become to model a system’s limitations, dynamics, and conditions while elucidating the related principles. These ideas are closely linked to evolution, which can only take place when something generates spontaneous heterogeneity. In critical decision-making and handling of the complex processes, the situation becomes more complicated when its dynamics are taken into consideration in conjunction with the internal structure as well as the environment. Accordingly, the science of complex systems provides us with the required conceptual and methodological equipment to deal with emergence, self-organization, learning, adaptation, path- dependency, diversity, transformation and evolution through which microlevel properties lead to macrolevel behaviors, which make it necessary to explain dynamic behaviors and novel emerging structure in time. In this regard, systems thinking enables one to choose and make use of abstractions to understand those dynamics that un- derline the individuals’ and elements’ behaviors. Measure of complexity, being the amount of information required to describe the behavior of a complex system, complex sys- tems thinking with its appealing features characterize be- haviors, allowing the model development possibility that is capable of capturing richness and diversity of human ex- istence. Thus, a framework is set that includes a variety of approaches, enabling us to tackle the notion that a system’s component parts can optimally be understood within the context of relationships with one another as well as with other systems, rather than viewing them in isolation and with reductionist approach [79]. 2.2 Chaos thinking, processes and complexity Chaos and its study along with the advances in scientific realm are important roots of modern study of complex systems, which display nonlinear, dynamic, open qualities and interconnection with the environment made up of many components which interact, and new unanticipated patterns emerge. Chaos, in this context, can be said to have more or less strict definitions portraying a nonlinear world, and it deals with deterministic systems with trajectories diverging 30 Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
- 54. exponentially in time, which is also among the properties of behaviors in complex systems. Models of chaos generally provide the description of the dynamics of one or few variables that are real and denoted by a decimal number, and with the utilization of these models, the characteristic makes the behaviors of their dynamics found and known. In contrast, it is not always necessary that complex systems possess a representation of this sort of form or such be- haviors due to having different degrees of freedom with many elements not being entirely but partially independent. Complexity becomes further complicated when aspects of system-environment interaction are modeled by chaos. Another different aspect is that while complex systems’ study deals with both the dynamics and structure of the structure, chaos, referring to randomness and disorder, ad- dresses a few parameters and the dynamics of their values. Randomness and disorder also belong to the part of com- plex systems’ study and the concept of a chaotic environ- ment can be replaced with complex environment, since complex refers to both randomness of disorder and also deterministic chaos [43,80]. Originated in mathematics and physical sciences and widely used in humanities and social sciences, complexity theory as a transdisciplinary systems theory deals with change basically speaking [81]. In terms of evolution, as Kauffman suggests, the fate of all complex adapting systems, ranging from single cells to economies, is to evolve into a natural state that is between order and chaos, which is considered to be a compromise between structure and unexpectedness [82]. In industrial, technical or organizational issues, a pattern inherent therein is also subtly poised between the pendulum order and chaos [83,84]. Einstein put it very well that imagination has no boundaries, which also points to the fact that complexity theory definitely appeals to the imagination. Scientific theories are based on the premise that they are open to refutation by experimental studies, whereas math- ematical models remain to be noncontradictory relying on the mathematical logic. Scientists attempt to understand if events from the real world conform to the given mathe- matical model or not; and for this, they follow three steps to construct the model with the first step being the observation of the phenomenon, afterward, conversion into equations, and then solving of those equations. Being a mathematical theory and still in development, chaos theory allows one to describe a series of phenomena from dynamics that con- cerns the impact of forces on the objects’ motion, and in that regard, the archetype of all theories related to dynamics belongs to Newton, with regard to celestial motions. The quote by Oliver Sacks summarizes the order of mathe- matics in chaos: “I liked numbers because they were solid, invariant; they stood unmoved in a chaotic world” [85]. Developed by a group of scientists who came from different backgrounds like mathematics, physics and chemistry working on the complex physical phenomena’s dynamics, chaos theory which has some significant cornerstone concepts also points to the roots of modern science. Causality principle maybe the most fundamental principle that refers to the premise that each effect has the antecedent immediate cause. This principle, being non- refutable is derived from Descartes (1641, Third Medita- tion) and yet, experience does not confirm it. The consolidation and simplification of causality principle was achieved by Newton (1687) who maintained that initial conditions and law of motion needed to be viewed sepa- rately from each other, which had the calculations parallel to the laws of Kepler. Determinism, based on causality, is another principle which philosophically proposes that each event is determined physically by a chain of previous oc- currences and this chain is unbroken in nature. Laplace was the figure who evidently stated the concept of universal determinism following d’Holbach. It would be apt to note that the birth of the chaos theory is related to the work of Laplace, which posited that past and future of the solar system could be calculated with precision and relied on the capacity to be cognizant of the initial conditions of the system (for further timeline-related details see [86]). Chaos is stated to have three features which are the primary ones; these are unpredictability, boundedness, and sensitivity to initial conditions as put forth by Kaplan and Glass [87]. Unpredictability refers to the fact that a sequence of numbers generated from a chaotic function does not repeat itself; while boundedness refers that all points remain within certain boundaries for the motion’s unpredictability [88]. Since chaos is assumed to play different functional roles in living systems, the principles and methods to detect chaos should be present in the toolkit of the scientist. Different chaotic functions yield time series that may change in terms of complexity whose meaning relies on the nature of the system and its underlying theory. Lower complexity could hint rigidity in the system, whereas a higher level of complexity shows a greater adaptability as per the evolution [88]. Complexity theory, as a related component for complex systems, in the realm of modern science has no precise boundaries with no exact definition actually. Portrayed in two different ways, as general systems theory (GST), the grand theory and complex adaptive systems (CAS) theory is the umbrella term. GST includes chaos theory, adaptation and nonlinear dynamics, with some delineation with complexity theory. The second portrayal makes a distinc- tion between GST and complexity theory, placing systems theory under GST and CAS under complexity theory. Complexity science was inspired by the original systems sciences including information theory, cybernetics and GST. CAS includes pillars such as history, nonlinearity, emergence, adaptability, self-organization, path de- pendency, irreducibility, balance between order and chaos [89]. Nonlinear dynamics, chaos theory, and adaptation/ Multi-chaos, fractal and multi-fractional AI in different complex systems Chapter | 3 31
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- 56. Consider thou, O man, what these places to thee showed ✿ And be upon thy guard ere thou travel the same road: And prepare thee good provision some day may serve thy turn ✿ For each dweller in the house needs must yede wi’ those who yode Consider how this people their palaces adorned ✿ And in dust have been pledged for the seed of acts they sowed: They built but their building availed them not, and hoards ✿ Nor saved their lives nor day of Destiny forslowed: How often did they hope for what things were undecreed. ✿ And passed unto their tombs before Hope the bounty showed: And from high and awful state all a-sudden they were sent ✿ To the straitness of the grave and oh! base is their abode: Then came to them a Crier after burial and cried, ✿ What booted thrones or crowns or the gold to you bestowed: Where now are gone the faces hid by curtain and by veil, ✿ Whose charms were told in proverbs, those beauties à-la-mode? The tombs aloud reply to the questioners and cry, ✿ “Death’s canker and decay those rosy cheeks corrode!” Long time they ate and drank, but their joyaunce had a term; ✿ And the eater eke was eaten, and was eaten by the worm. When the Emir read this, he wept, till he was like to swoon away, ——And Shahrazad perceived the dawn of day and ceased saying her permitted say. Now when it was the Five Hundred and Seventy-fifth Night, She said, It hath reached me, O auspicious King, that the Emir wept till he was like to swoon away, and bade write down the verses, after which he passed on into the inner palace and came to a vast hall, at each of whose four corners stood a pavilion lofty and spacious, washed with gold and silver and painted in various colours. In the heart of the hall was a great jetting-fountain of alabaster, surmounted by a canopy of brocade, and in each pavilion was a sitting-place and each place had its richly-wrought fountain and tank
- 57. paved with marble and streams flowing in channels along the floor and meeting in a great and grand cistern of many-coloured marbles. Quoth the Emir to the Shaykh Abd al-Samad, “Come, let us visit yonder pavilion!” So they entered the first and found it full of gold and silver and pearls and jacinths and other precious stones and metals, besides chests filled with brocades, red and yellow and white. Then they repaired to the second pavilion, and, opening a closet there, found it full of arms and armour, such as gilded helmets and Davidean[140] hauberks and Hindi swords and Arabian spears and Chorasmian[141] maces and other gear of fight and fray. Thence they passed to the third pavilion, wherein they saw closets padlocked and covered with curtains wrought with all manner of embroidery. They opened one of these and found it full of weapons curiously adorned with open work and with gold and silver damascene and jewels. Then they entered the fourth pavilion, and opening one of the closets there, beheld in it great store of eating and drinking vessels of gold and silver, with platters of crystal and goblets set with fine pearls and cups of carnelian and so forth. So they all fell to taking that which suited their tastes and each of the soldiers carried off what he could. When they left the pavilions, they saw in the midst of the palace a door of teak-wood marqueteried with ivory and ebony and plated with glittering gold, over which hung a silken curtain purfled with all manner of embroideries; and on this door were locks of white silver, that opened by artifice without a key. The Shaykh Abd al-Samad went valiantly up thereto and by the aid of his knowledge and skill opened the locks, whereupon the door admitted them into a corridor paved with marble and hung with veil-like[142] tapestries embroidered with figures of all manner beasts and birds, whose bodies were of red gold and white silver and their eyes of pearls and rubies, amazing all who looked upon them. Passing onwards they came to a saloon builded all of polished marble, inlaid with jewels, which seemed to the beholder as though the floor were flowing water[143] and whoso walked thereon slipped. The Emir bade the Shaykh strew somewhat upon it, that they might walk over it; which being done, they made shift to fare forwards till they came to a great domed pavilion of stone, gilded with red gold and crowned
- 58. with a cupola of alabaster, about which were set lattice-windows carved and jewelled with rods of emerald,[144] beyond the competence of any King. Under this dome was a canopy of brocade, reposing upon pillars of red gold and wrought with figures of birds whose feet were of smaragd, and beneath each bird was a network of fresh-hued pearls. The canopy was spread above a jetting fountain of ivory and carnelian, plated with glittering gold and thereby stood a couch set with pearls and rubies and other jewels and beside the couch a pillar of gold. On the capital of the column stood a bird fashioned of red rubies and holding in his bill a pearl which shone like a star; and on the couch lay a damsel, as she were the lucident sun, eyes never saw a fairer. She wore a tight-fitting body-robe of fine pearls, with a crown of red gold on her head, filleted with gems, and on her forehead were two great jewels, whose light was as the light of the sun. On her breast she wore a jewelled amulet, filled with musk and ambergris and worth the empire of the Cæsars; and around her neck hung a collar of rubies and great pearls, hollowed and filled with odoriferous musk. And it seemed as if she gazed on them to the right and to the left.——And Shahrazad perceived the dawn of day and ceased to say her permitted say. Now when it was the Five Hundred and Seventy-sixth Night, She said, It hath reached me, O auspicious King, that the damsel seemed to be gazing at the folk to the right and to the left. The Emir Musa marvelled at her exceeding beauty and was confounded at the blackness of her hair and the redness of her cheeks, which made the beholder deem her alive and not dead, and said to her, “Peace be with thee, O damsel!” But Talib ibn Sahl said to him, “Allah preserve thee, O Emir, verily this damsel is dead and there is no life in her; so how shall she return thy salam?”; adding, “Indeed, she is but a corpse embalmed with exceeding art; her eyes were taken out after
- 59. her death and quicksilver set under them, after which they were restored to their sockets. Wherefore they glisten and when the air moveth the lashes, she seemeth to wink and it appeareth to the beholder as though she looked at him, for all she is dead.” At this the Emir marvelled beyond measure and said, “Glory be to God who subjugateth His creatures to the dominion of Death!” Now the couch on which the damsel lay, had steps, and thereon stood two statues of Andalusian copper representing slaves, one white and the other black. The first held a mace of steel[145] and the second a sword of watered steel which dazzled the eye; and between them, on one of the steps of the couch, lay a golden tablet, whereon were written, in characters of white silver, the following words: “In the name of God, the Compassionating, the Compassionate! Praise be to Allah, the Creator of mankind; and He is the Lord of Lords, the Causer of Causes! In the name of Allah, the Never-beginning, the Everlasting, the Ordainer of Fate and Fortune! O son of Adam! what hath befooled thee in this long esperance? What hath unminded thee of the Death-day’s mischance? Knowest thou not that Death calleth for thee and hasteneth to seize upon the soul of thee? Be ready, therefore, for the way and provide thee for thy departure from the world; for, assuredly, thou shalt leave it without delay. Where is Adam, first of humanity? Where is Noah with his progeny? Where be the Kings of Hind and Irak-plain and they who over earth’s widest regions reign? Where do the Amalekites abide and the giants and tyrants of olden tide? Indeed, the dwelling-places are void of them and they have departed from kindred and home. Where be the Kings of Arab and Ajem? They are dead, all of them, and gone and are become rotten bones. Where be the lords so high in stead? They are all done dead. Where are Kora and Haman? Where is Shaddad son of Ad? Where be Canaan and Zu’l-Autád,[146] Lord of the Stakes? By Allah, the Reaper of lives hath reaped them and made void the lands of them. Did they provide them against the Day of Resurrection or make ready to answer the Lord of men? O thou, if thou know me not, I will acquaint thee with my name: I am Tadmurah,[147] daughter of the Kings of the Amalekites, of those who held dominion over the lands in equity and brought low the necks of humanity. I
- 60. possessed that which never King possessed and was righteous in my rule and did justice among my lieges; yea, I gave gifts and largesse and freed bondsmen and bondswomen. Thus lived I many years in all ease and delight of life, till Death knocked at my door and to me and to my folk befel calamities galore; and it was on this wise. There betided us seven successive years of drought, wherein no drop of rain fell on us from the skies and no green thing sprouted for us on the face of earth.[148] So we ate what was with us of victual, then we fell upon the cattle and devoured them, until nothing was left. Thereupon I let bring my treasures and meted them with measures and sent out trusty men to buy food. They circuited all the lands in quest thereof and left no city unsought, but found it not to be bought and returned to us with the treasure after a long absence; and gave us to know that they could not succeed in bartering fine pearls for poor wheat, bushel for bushel, weight for weight. So, when we despaired of succour, we displayed all our riches and things of price and, shutting the gates of the city and its strong places, resigned ourselves to the deme of our Lord and committed our case to our King. Then we all died,[149] as thou seest us, and left what we had builded and all we had hoarded. This, then, is our story, and after the substance naught abideth but the trace.” Then they looked at the foot of the tablet and read these couplets:—
- 61. O child of Adam, let not hope make mock and flyte at thee, ✿ From all thy hands have treasurèd, removèd thou shalt be; I see thou covetest the world and fleeting worldly charms, ✿ And races past and gone have done the same as thou I see. Lawful and lawless wealth they got; but all their hoarded store, ✿ Their term accomplished, naught delayed of Destiny’s decree. Armies they led and puissant men and gained them gold galore; ✿ Then left their wealth and palaces by Fate compelled to flee, To straitness of the grave-yard and humble bed of dust ✿ Whence, pledged for every word and deed, they never more win free: As a company of travellers had unloaded in the night ✿ At house that lacketh food nor is o’erfain of company: Whose owner saith, ‘O folk, there be no lodging here for you;’ ✿ So packed they who had erst unpacked and farèd hurriedly: Misliking much the march, nor the journey nor the halt ✿ Had aught of pleasant chances or had aught of goodly gree. Then prepare thou good provision for to-morrow’s journey stored, ✿ Naught but righteous honest life shall avail thee with the Lord! And the Emir Musa wept as he read, “By Allah, the fear of the Lord is the best of all property, the pillar of certainty and the sole sure stay. Verily, Death is the truth manifest and the sure behest, and therein, O thou, is the goal and return-place evident. Take warning, therefore, by those who to the dust did wend and hastened on the way of the predestined end. Seest thou not that hoary hairs summon thee to the tomb and that the whiteness of thy locks maketh moan of thy doom? Wherefore be thou on the wake ready for thy departure and thine account to make. O son of Adam, what hath hardened thy heart in mode abhorred? What hath seduced thee from the service of thy Lord? Where be the peoples of old time? They are a warning to whoso will be warned! Where be the Kings of Al-Sín and the lords of majestic mien? Where is Shaddad bin Ad and whatso he built and he stablished? Where is Nimrod who revolted against Allah and defied Him? Where is Pharaoh who rebelled against God and denied Him? Death followed hard upon the trail of them all, and laid them low sparing neither great nor small, male nor female; and the Reaper of Mankind cut them off, yea, by Him who maketh night to return upon day! Know, O thou who comest to this
- 62. place, that she whom thou seest here was not deluded by the world and its frail delights, for it is faithless, perfidious, a house of ruin, vain and treacherous; and salutary to the creature is the remembrance of his sins; wherefore she feared her Lord and made fair her dealings and provided herself with provaunt against the appointed marching-day. Whoso cometh to our city and Allah vouchsafeth him competence to enter it, let him take of the treasure all he can, but touch not aught that is on my body, for it is the covering of my shame[150] and the outfit for the last journey; wherefore let him fear Allah and despoil naught thereof; else will he destroy his own self. This have I set forth to him for a warning from me and a solemn trust to be; wherewith, peace be with ye and I pray Allah to keep you from sickness and calamity.”——And Shahrazad perceived the dawn of day and ceased saying her permitted say. Now when it was the Five Hundred and Seventy-seventh Night, She said, it hath reached me, O auspicious King, that when the Emir Musa read this, he wept with exceeding weeping till he swooned away and presently coming to himself, wrote down all he had seen and was admonished by all he had witnessed. Then he said to his men, “Fetch the camels and load them with these treasures and vases and jewels.” “O Emir,” asked Talib, “shall we leave our damsel with what is upon her, things which have no equal and whose like is not to be found and more perfect than aught else thou takest; nor couldst thou find a goodlier offering wherewithal to propitiate the favour of the Commander of the Faithful?” But Musa answered, “O man, heardest thou not what the Lady saith on this tablet? More by token that she giveth it in trust to us who are no traitors.” “And shall we,” rejoined the Wazir Talib, “because of these words, leave all these riches and jewels, seeing that she is dead? What should she do with these that are the adornments of the world and the
- 63. ornament of the worldling, seeing that one garment of cotton would suffice for her covering? We have more right to them than she.” So saying he mounted the steps of the couch between the pillars, but when he came within reach of the two slaves, lo! the mace-bearer smote him on the back and the other struck him with the sword he held in his hand and lopped off his head, and he dropped down dead. Quoth the Emir, “Allah have no mercy on thy resting-place! Indeed there was enough in these treasures; and greed of gain assuredly degradeth a man.” Then he bade admit the troops; so they entered and loaded the camels with those treasures and precious ores; after which they went forth and the Emir commanded them to shut the gate as before. They fared on along the sea-shore a whole month, till they came in sight of a high mountain overlooking the sea and full of caves, wherein dwelt a tribe of blacks, clad in hides, with burnooses also of hide and speaking an unknown tongue. When they saw the troops they were startled like shying steeds and fled into the caverns, whilst their women and children stood at the cave- doors, looking on the strangers. “O Shaykh Abd al-Samad,” asked the Emir, “what are these folk?” and he answered, “They are those whom we seek for the Commander of the Faithful.” So they dismounted and setting down their loads, pitched their tents; whereupon, almost before they had done, down came the King of the blacks from the mountain and drew near the camp. Now he understood the Arabic tongue; so, when he came to the Emir he saluted him with the salam and Musa returned his greeting and entreated him with honour. Then quoth he to the Emir, “Are ye men or Jinn?” “Well, we are men,” quoth Musa; “but doubtless ye are Jinn, to judge by your dwelling apart in this mountain which is cut off from mankind, and by your inordinate bulk.” “Nay,” rejoined the black; “we also are children of Adam, of the lineage of Ham, son of Noah (with whom be peace!), and this sea is known as Al-Karkar.” Asked Musa, “O King, what is your religion and what worship ye?”; and he answered, saying, “We worship the God of the heavens and our religion is that of Mohammed, whom Allah bless and preserve!” “And how came ye by the knowledge of this,” questioned the Emir, “seeing that no prophet was inspired to visit this country?” “Know,
- 64. Emir,” replied the King, “that there appeared to us whilere from out the sea a man, from whom issued a light that illumined the horizons and he cried out, in a voice which was heard of men far and near, saying:—O children of Ham, reverence to Him who seeth and is not seen and say ye, There is no god but the God, and Mohammed is the messenger of God! And he added:—I am Abu al-Abbás al-Khizr. Before this we were wont to worship one another, but he summoned us to the service of the Lord of all creatures; and he taught us to repeat these words, There is no god save the God alone, who hath for partner none, and His is the kingdom and His is the praise. He giveth life and death and He over all things is Almighty. Nor do we draw near unto Allah (be He exalted and extolled!) except with these words, for we know none other; but every eve before Friday[151] we see a light upon the face of earth and we hear a voice saying, Holy and glorious, Lord of the Angels and the Spirit! What He willeth is, and what He willeth not, is not. Every boon is of His grace and there is neither Majesty nor is there Might save in Allah, the Glorious, the Great!” “But ye,” quoth the King, “who and what are ye and what bringeth you to this land?” Quoth Musa, “We are officers of the Sovereign of Al-Islam, the Commander of the Faithful, Abd al-Malik bin Marwan, who hath heard tell of the lord Solomon, son of David (on whom be peace!) and of that which the Most High bestowed upon him of supreme dominion; how he held sway over Jinn and beast and bird and was wont when he was wroth with one of the Marids, to shut him in a cucurbite of brass and, stopping its mouth on him with lead, whereon he impressed his seal-ring, to cast him into the sea of Al-Karkar. Now we have heard tell that this sea is nigh your land; so the Commander of the Faithful hath sent us hither, to bring him some of these cucurbites, that he may look thereon and solace himself with their sight. Such, then, is our case and what we seek of thee, O King, and we desire that thou further us in the accomplishment of our errand commanded by the Commander of the Faithful.” “With love and gladness,” replied the black King, and carrying them to the guest-house, entreated them with the utmost honour and furnished them with all they needed, feeding them upon fish. They abode thus three days, when he bade
- 65. his divers fetch from out the sea some of the vessels of Solomon. So they dived and brought up twelve cucurbites, whereat the Emir and the Shaykh and all the company rejoiced in the accomplishment of the Caliph’s need. Then Musa gave the King of the blacks many and great gifts; and he, in turn, made him a present of the wonders of the deep, being fishes in human form,[152] saying “Your entertainment these three days hath been of the meat of these fish.” Quoth the Emir, “Needs must we carry some of these to the Caliph, for the sight of them will please him more than the cucurbites of Solomon.” Then they took leave of the black King and, setting out on their homeward journey, travelled till they came to Damascus, where Musa went in to the Commander of the Faithful and told him all that he had sighted and heard of verses and legends and instances, together with the manner of the death of Talib bin Sahl; and the Caliph said, “Would I had been with you, that I might have seen what you saw!” Then he took the brazen vessels and opened them, cucurbite after cucurbite, whereupon the devils came forth of them, saying, “We repent, O Prophet of Allah! Never again will we return to the like of this thing; no never!” And the Caliph marvelled at this. As for the daughters of the deep presented to them by the black King, they made them cisterns of planks, full of water, and laid them therein; but they died of the great heat. Then the Caliph sent for the spoils of the Brazen City and divided them among the Faithful,—— And Shahrazad perceived the dawn of day and ceased to say her permitted say. Now when it was the Five Hundred and Seventy-eighth Night, She said, It hath reached me, O auspicious King, that the Caliph marvelled much at the cucurbites and their contents; then he sent for the spoils and divided them among the Faithful, saying, “Never gave Allah unto any the like of that which he bestowed upon Solomon David-son!” Thereupon the Emir Musa sought leave of him
- 66. to appoint his son Governor of the Province in his stead, that he might betake himself to the Holy City of Jerusalem, there to worship Allah. So the Commander of the Faithful invested his son Harun with the government and Musa repaired to the Glorious and Holy City, where he died. This, then, is all that hath come down to us of the story of the City of Brass, and God is All-knowing!——Now (continued Shahrazad) I have another tale to tell anent the 104. This is a true “City of Brass.” (Nuhás asfar = yellow copper), as we learn in Night dcclxxii. It is situated in the “Maghrib” (Mauritania), the region of magic and mystery; and the idea was probably suggested by the grand Roman ruins which rise abruptly from what has become a sandy waste. Compare with this tale “The City of Brass” (Night cclxxii). In Egypt Nuhás is vulg. pronounced Nihás. 105. The Bresl. Edit. adds that the seal-ring was of stamped stone and iron, copper and lead. I have borrowed copiously from its vol. vi. pp. 343, et seq. 106. As this was a well-known pre-Islamitic bard, his appearance here is decidedly anachronistic, probably by intention. 107. The first Moslem conqueror of Spain whose lieutenant, Tárik, the gallant and unfortunate, named Gibraltar (Jabal al-Tarik). 108. The colours of the Banú Umayyah (Ommiade) Caliphs were white; of the Banú Abbás (Abbasides) black, and of the Fatimites green. Carrying the royal flag denoted the generalissimo or plenipotentiary. 109. i.e. Old Cairo, or Fustat: the present Cairo was then a Coptic village founded on an old Egyptian settlement called Lui-Tkeshroma, to which belonged the tanks on the hill and the great well, Bir Yusuf, absurdly attributed to Joseph the Patriarch. Lui is evidently the origin of Levi and means a high priest (Brugsh ii. 130) and his son’s name was Roma.
- 67. 110. I cannot but suspect that this is a clerical error for “Al-Samanhúdi,” a native of Samanhúd (Wilkinson’s “Semenood”) in the Delta on the Damietta branch, the old Sebennytus (in Coptic Jem-nuti = Jem the God), a town which has produced many distinguished men in Moslem times. But there is also a Samhúd lying a few miles down stream from Denderah and, as its mounds prove, it is an ancient site. 111. Egypt had not then been conquered from the Christians. 112. Arab. “Kízán fukká’a,” i.e. thin and slightly porous earthenware jars used for Fukká’a, a fermented drink, made of barley or raisins. 113. I retain this venerable blunder: the right form is Samúm, from Samm, the poison-wind. 114. i.e. for worship and to prepare for futurity. 115. The camel carries the Badawi’s corpse to the cemetery which is often distant: hence to dream of a camel is an omen of death. 116. Koran xxiv. 39. The word “Saráb” (mirage) is found in Isaiah (xxxv. 7) where the passage should be rendered “And the mirage (sharab) shall become a lake” (not, “and the parched ground shall become a pool”). The Hindus prettily call it “Mrigatrishná” = the thirst of the deer. 117. A name of Allah. 118. Arab. “Kintár” = a hundredweight (i.e. 100 lbs.), about 98¾ lbs. avoir. Hence the French quintal and its congeners (Littré). 119. i.e. “from Shám” (Syria) to (the land of) Adnan, ancestor of the Naturalized Arabs that is, to Arabia.
- 68. 120. Koran lii. 21. “Every man is given in pledge for that which he shall have wrought.” 121. There is a constant clerical confusion in the texts between “Arar” (Juniperus Oxycedrus used by the Greeks for the images of their gods) and “Marmar” marble or alabaster, in the Talmud “Marmora” = marble, evidently from μάρμαρος = brilliant, the brilliant stone. 122. These Ifritical names are chosen for their bizarrerie. “Al-Dáhish” = the Amazed; and “Al-A’amash” = one with weak eyes always watering. 123. The Arabs have no word for million; so Messer Marco Miglione could not have learned it from them. On the other hand the Hindus have more quadrillions than modern Europe. 124. This formula, according to Moslems, would begin with the beginning “There is no iláh but Allah and Adam is the Apostle (rasúl = one sent, a messenger; not nabí = prophet) of Allah.” And so on with Noah, Moses, David (not Solomon as a rule) and Jesus to Mohammed. 125. This son of Barachia has been noticed before. The text embroiders the Koranic chapter No. xxvii. 126. The Bresl. Edit. (vi. 371) reads “Samm-hu” = his poison, prob. a clerical error for “Sahmhu” = his shaft. It was a duel with the “Shiháb” or falling stars, the meteors which are popularly supposed, I have said, to be the arrows shot by the angels against devils and evil spirits when they approach too near Heaven in order to overhear divine secrets. 127. A fancy sea from the Lat. “Carcer” (?). 128. Andalusian = Spanish, the Vandal-land, a term accepted by the Moslem invader.
- 69. 129. This fine description will remind the traveller of the old Haurani towns deserted since the sixth century, which a silly writer miscalled the “Giant Cities of Bashan.” I have never seen anything weirder than a moonlight night in one of these strong places whose masonry is perfect as when first built, the snowy light pouring on the jet-black basalt and the breeze sighing and the jackal wailing in the desert around. 130. “Zanj,” I have said, is the Arab. form of the Persian “Zang-bar” (= Black- land), our Zanzibar. Those who would know more of the etymology will consult my “Zanzibar,” etc., chapt. i. 131. Arab. “Tanjah” = Strabo Τίγγις (derivation uncertain), Tingitania, Tangiers. But why the terminal s? 132. Or Amidah, by the Turks called “Kara (black) Amid” from the colour of the stones; and the Arabs “Diyar-bakr” (Diarbekir), a name which they also give to the whole province—Mesopotamia. 133. Mayyáfárikín, an episcopal city in Diyar-bakr: the natives are called Fárikí; hence the abbreviation in the text. 134. Arab. “Ayát al-Naját,” certain Koranic verses which act as talismans, such as, “And wherefore should we not put our trust in Allah?” (xiv. 15); “Say thou, ‘Naught shall befal us save what Allah hath decreed for us.’” (ix. 51), and sundry others. 135. These were the “Brides of the Treasure,” alluded to in the story of Hasan of Bassorah and elsewhere. 136. Arab. “Ishárah,” which may also mean beckoning. Easterns reverse our process: we wave band or finger towards ourselves; they towards the object; and our fashion represents to them, Go away! 137. i.e. musing a long time and a longsome.
- 70. 138. Arab. “Dihlíz” from the Persian. This is the long dark passage which leads to the inner or main gate of an Eastern city, and which is built up before a siege. It is usually furnished with Mastabah-benches of wood and masonry, and forms a favourite lounge in hot weather. Hence Lot and Moses sat and stood in the gate, and here man speaks with his enemies. 139. The names of colours are as loosely used by the Arabs as by the Classics of Europe; for instance, a light grey is called a “blue or a green horse.” Much nonsense has been written upon the colours in Homer by men who imagine that the semi-civilised determine tints as we do. They see them but they do not name them, having no occasion for the words. As I have noticed, however, the Arabs have a complete terminology for the varieties of horse- hues. In our day we have witnessed the birth of colours, named by the dozen, because required by women’s dress. 140. For David’s miracles of metallurgy see vol. i. 286. 141. Arab. “Khwárazm,” the land of the Chorasmioi, who are mentioned by Herodotus (iii. 93) and a host of classical geographers. They place it in Sogdiana (hod. Sughd) and it corresponds with the Khiva country. 142. Arab. “Burka’,” usually applied to a woman’s face-veil and hence to the covering of the Ka’abah, which is the “Bride of Meccah.” 143. Alluding to the trick played upon Bilkís by Solomon who had heard that her legs were hairy like those of an ass: he laid down a pavement of glass over flowing water in which fish were swimming and thus she raised her skirts as she approached him and he saw that the report was true. Hence, as I have said, the depilatory (Koran xxvii.). 144. I understand the curiously carved windows cut in arabesque-work of marble (India) or basalt (the Haurán) and provided with small panes of glass set in emeralds where tinfoil would be used by the vulgar. 145. Arab. “Bulád” from the Pers. “Pulád.” Hence the name of the famous Druze family “Jumblat,” a corruption of “Ján-pulád” = Life o’ Steel.
- 71. 146. Pharaoh, so called in Koran (xxxviii. 11) because he tortured men by fastening them to four stakes driven into the ground. Sale translates “the contriver of the stakes” and adds, “Some understand the word figuratively, of the firm establishment of Pharaoh’s kingdom, because the Arabs fix their tents with stakes; but they may possibly intend that prince’s obstinacy and hardness of heart.” I may note that in “Tasawwuf,” or Moslem Gnosticism, Pharaoh represents, like Prometheus and Job, the typical creature who upholds his own dignity and rights in presence and despight of the Creator. Sáhib the Súfí declares that the secret of man’s soul (i.e. its emanation) was first revealed when Pharaoh declared himself god; and Al-Ghazálí sees in his claim the most noble aspiration to the divine, innate in the human spirit (Dabistan, vol. iii.). 147. In the Calc. Edit. “Tarmuz, son of the daughter,” etc. According to the Arabs, Tadmur (Palmyra) was built by Queen Tadmurah, daughter of Hassán bin Uzaynah. 148. It is only by some such drought that I can account for the survival of those marvellous Haurani cities in the great valley S. E. of Damascus. 149. So Moses described his own death and burial. 150. A man’s “aurat” (shame) extends from the navel (included) to his knees; a woman’s from the top of the head to the tips of her toes. I have before noticed the Hindostaní application of the word. 151. Arab. “Jum’ah” (= the assembly) so called because the General Resurrection will take place on that day and it witnessed the creation of Adam. Both these reasons are evidently after-thoughts; as the Jews received a divine order to keep Saturday, and the Christians, at their own sweet will, transferred the weekly rest-day to Sunday, wherefore the Moslem preferred Friday. Sabbatarianism, however, is unknown to Al-Islam and business is interrupted, by Koranic order (lxii. 9–10), only during congregational prayers in the Mosque. The most a Mohammedan does is not to work or travel till after public service. But the Moslem hardly wants a “day of rest;” whereas a Christian, especially in the desperately dull routine of daily life and toil, without a gleam of light to break the darkness of his civilised and most unhappy existence, distinctly requires it.
- 72. 152. Mankind, which sees itself everywhere and in everything, must create its own analogues in all the elements, air (Sylphs), fire (Jinns), water (Mermen and Mermaids) and earth (Kobolds). These merwomen were of course seals or manatees, as the wild women of Hanno were gorillas.
- 73. CRAFT AND MALICE OF WOMEN,[153] OR THE TALE OF THE KING, HIS SON, HIS CONCUBINE AND THE SEVEN WAZIRS. There was, in days of yore and in ages and times long gone before, a puissant King among the Kings of China, the crown of crowned heads, who ruled over many men of war and vassals with wisdom and justice, might and majesty; equitable to his Ryots, liberal to his lieges and dearly beloved by the hearts of his subjects. He was wealthy as he was powerful, but he had grown old without being blessed with a son, and this caused him sore affliction. He could only brood over the cutting off of his seed and the oblivion that would bury his name and the passing of his realm into the stranger’s hands. So he secluded himself in his palace, never going in and out or rising and taking rest till the lieges lost all tidings of him and were sore perplexed and began to talk about their King. Some said, “He’s dead”; others said, “No, he’s not but all resolved to find a ruler who could reign over them and carry out the customs of government.” At last, utterly despairing of male issue, he sought the intercession of the Prophet (whom Allah bless and keep!) with the Most High and implored Him, by the glory of His Prophets and Saints and Martyrs and others of the Faithful who were acceptable to Heaven that he would grant him a son, to be the coolth of his eyes and heir to the kingdom after him. Then he rose forthright and, withdrawing to his sitting-saloon, sent for his wife who was the daughter of his uncle. Now this Queen was of surpassing beauty and loveliness, the fairest of all his wives and the dearest to him as she was the nearest: and to boot a woman of excellent wit and passing judgement. She found the King dejected and sorrowful, tearful-eyed and heavy-hearted; so
- 74. she kissed ground between his hands and said, “O King, may my life ransom thy life! may Time never prove thy foe, nor the shifts of Fortune prevail over thee; may Allah grant thee every joy and ward off from thee all annoy! How is it I see thee brooding over thy case and tormented by the displeasures of memory?” He replied, “Thou wottest well that I am a man now shotten in years, who hath never been blessed with a son, a sight to cool his eyes; so I know that my kingdom shall pass away to the stranger in blood and my name and memory will be blotted out amongst men. ‘Tis this causeth me to grieve with excessive grief.” “Allah do away with thy sorrows,” quoth she: “long ere this day a thought struck me; and yearning for issue arose in my heart even as in thine. One night I dreamed a dream and a voice said to me:—The King thy husband pineth for progeny: if a daughter be vouchsafed to him, she will be the ruin of his realm; if a son, the youth will undergo much trouble and annoy but he will pass through it without loss of life. Such a son can be conceived by thee and thee only and the time of thy conception is when the moon conjoineth with Gemini! I woke from my dream, but after what I heard that voice declare I refrained from breeding and would not consent to bear children.” “There is no help for it but that I have a son, Inshallah,—God willing!” cried the King. Thereupon she soothed and consoled him till he forgot his sorrows and went forth amongst the lieges and sat, as of wont, upon his throne of estate. All rejoiced to see him once more and especially the Lords of his realm. Now when the conjunction of the moon and Gemini took place, the King knew his wife carnally and, by order of Allah Almighty she became pregnant. Presently she announced the glad tidings to her husband and led her usual life until her nine months of pregnancy were completed and she bare a male child whose face was as the rondure of the moon on its fourteenth night. The lieges of the realm congratulated one another thereanent and the King commanded an assembly of his Olema and philosophers, astrologers and horoscopists, whom he thus addressed, “I desire you to forecast the fortune of my son and to determine his ascendant[154] and whatever is shown by his nativity.” They replied “‘Tis well, in Allah’s name, let us do so!” and cast his nativity with all diligence. After ascertaining
- 75. his ascendant, they pronounced judgement in these words, “We see his lot favourable and his life viable and durable; save that a danger awaiteth his youth.” The father was sorely concerned at this saying, when they added “But, O King, he shall escape from it nor shall aught of injury accrue to him!” Hereupon the King cast aside all cark and care and robed the wizards and dismissed them with splendid honoraria; and he resigned himself to the will of Heaven and acknowledged that the decrees of Destiny may not be countervailed. He committed his boy to wet nurses and dry nurses, handmaids and eunuchs, leaving him to grow and fill out in the Harim till he reached the age of seven. Then he addressed letters to his Viceroys and Governors in every clime and by their means gathered together Olema and philosophers and doctors of law and religion, from all countries, to a number of three hundred and three score. He held an especial assembly for them and, when all were in presence, he bade them draw near him and be at their ease while he sent for the food- trays and all ate their sufficiency. And when the banquet ended and the wizards had taken seats in their several degrees, the King asked them, “Wot ye wherefore I have gathered ye together?”; whereto all answered, “We wot not, O King!” He continued, “It is my wish that you select from amongst you fifty men, and from these fifty ten, and from these ten one, that he may teach my son omnem rem scibilem; for whenas I see the youth perfect in all science, I will share my dignity with the Prince and make him partner with me in my possessions.” “Know, O King,” they replied, “that among us none is more learned or more excellent than Al-Sindibad,[155] hight the Sage, who woneth in thy capital under thy protection. If such be thy design, summon him and bid him do thy will.” The King acted upon their advice and the Sage, standing in the presence, expressed his loyal sentiments with his salutation, whereupon his Sovereign bade him draw nigh and thus raised his rank, saying, “I would have thee to know, O Sage, that I summoned this assembly of the learned and bade them choose me out a man to teach my son all knowledge; when they selected thee without dissenting thought or voice. If, then, thou feel capable of what they claimed for thee, come thou to the task and understand that a man’s son and heir is the very fruit of
- 76. his vitals and core of his heart and liver. My desire of thee is thine instruction of him; and to happy issue Allah guideth!” The King then sent for his son and committed him to Al-Sindibad conditioning the Sage to finish his education in three years. He did accordingly but, at the end of that time, the young Prince had learned nothing, his mind being wholly occupied with play and disport; and when summoned and examined by his sire, behold, his knowledge was as nil. Thereupon the King turned his attention to the learned once more and bade them elect a tutor for his youth; so they asked, “And what hath his governor, Al-Sindibad, been doing?” and when the King answered, “He hath taught my son naught;” the Olema and philosophers and high officers summoned the instructor and said to him, “O Sage, what prevented thee from teaching the King’s son during this length of days?” “O wise men,” he replied, “the Prince’s mind is wholly occupied with disport and play; yet, an the King will make with me three conditions and keep to them, I will teach him in seven months what he would not learn (nor indeed could any other lesson him) within seven years.” “I hearken to thee,” quoth the King, “and I submit myself to thy conditions;” and quoth Al-Sindibad, “Hear from me, Sire, and bear in mind these three sayings, whereof the first is:—Do not to others what thou wouldest not they do unto thee;[156] and second:—Do naught hastily without consulting the experienced; and thirdly:—Where thou hast power show pity.[157] In teaching this lad I require no more of thee but to accept these three dictes and adhere thereto.” Cried the King, “Bear ye witness against me, O all ye here assembled, that I stand firm by these conditions!”; and caused a procès verbal to be drawn up with his personal security and the testimony of his courtiers. Thereupon the Sage, taking the Prince’s hand, led him to his place, and the King sent them all requisites of provaunt and kitchen-batteries, carpets and other furniture. Moreover the tutor bade build a house whose walls he lined with the whitest stucco painted over with ceruse,[158] and, lastly, he delineated thereon all the objects concerning which he proposed to lecture his pupil. When the place was duly furnished, he took the lad’s hand and installed him in the apartment which was amply furnished with belly-timber; and, after stablishing him therein,
- 77. went forth and fastened the door with seven padlocks. Nor did he visit the Prince save every third day when he lessoned him on the knowledge to be extracted from the wall-pictures and renewed his provision of meat and drink, after which he left him again to solitude. So whenever the youth was straitened in breast by the tedium and ennui of loneliness, he applied himself diligently to his object-lessons and mastered all the deductions therefrom. His governor seeing this turned his mind into other channel and taught him the inner meanings of the external objects; and in a little time the pupil mastered every requisite. Then the Sage took him from the house and taught him cavalarice and Jeríd play and archery. When the pupil had thoroughly mastered these arts, the tutor sent to the King informing him that the Prince was perfect and complete in all things required to figure favourably amongst his peers. Hereat the King rejoiced; and, summoning his Wazirs and Lords of estate to be present at the examination, commanded the Sage to send his son into the presence. Thereupon Al-Sindibad consulted his pupil’s horoscope and found it barred by an inauspicious conjunction which would last seven days; so, in sore affright for the youth’s life, he said, “Look into thy nativity-scheme.” The Prince did so and, recognising the potent, feared for himself and presently asked the Sage, saying, “What dost thou bid me do?” “I bid thee,” he answered, “remain silent and speak not a word during this se’nnight; even though thy sire slay thee with scourging. An thou pass safely through this period, thou shalt win to high rank and succeed to thy sire’s reign; but an things go otherwise then the behest is with Allah from the beginning to the end thereof.” Quoth the pupil, “Thou art in fault, O preceptor, and thou hast shown undue haste in sending that message to the King before looking into my horoscope. Hadst thou delayed till the week had passed all had been well.” Quoth the tutor, “O my son, what was to be was; and the sole defaulter therein was my delight in thy scholarship. But now be firm in thy resolve; rely upon Allah Almighty and determine not to utter a single word.” Thereupon the Prince fared for the presence and was met by the Wazirs who led him to his father. The King accosted him and addressed him but he answered not; and sought speech of him but
- 78. he spake not. Whereupon the courtiers were astounded and the monarch, sore concerned for his son, summoned Al-Sindibad. But the tutor so hid himself that none could hit upon his trace nor gain tidings of him; and folk said, “He was ashamed to appear before the King’s majesty and the courtiers.” Under these conditions the Sovereign heard some of those present saying, “Send the lad to the Serraglio where he will talk with the women and soon set aside this bashfulness;” and, approving their counsel, gave orders accordingly. So the Prince was led into the palace, which was compassed about by a running stream whose banks were planted with all manner of fruit-trees and sweet-smelling flowers. Moreover, in this palace were forty chambers and in every chamber ten slave-girls, each skilled in some instrument of music, so that whenever one of them played, the palace danced to her melodious strains. Here the Prince passed one night; but, on the following morning, the King’s favourite concubine happened to cast eyes upon his beauty and loveliness, his symmetrical stature, his brilliancy and his perfect grace, and love gat hold of her heart and she was ravished with his charms.[159] So she went up to him and threw herself upon him, but he made her no response; whereupon, being dazed by his beauty, she cried out to him and required him of himself and importuned him; then she again threw herself upon him and clasped him to her bosom kissing him and saying, “O King’s son, grant me thy favours and I will set thee in thy father’s stead; I will give him to drink of poison, so he may die and thou shalt enjoy his realm and wealth.” When the Prince heard these words, he was sore enraged against her and said to her by signs, “O accursed one, so it please Almighty Allah, I will assuredly requite thee this thy deed, whenas I can speak; for I will go forth to my father and will tell him, and he shall kill thee.” So signing, he arose in rage, and went out from her chamber; whereat she feared for herself. Thereupon she buffeted her face and rent her raiment and tare her hair and bared her head, then went in to the King and cast herself at his feet, weeping and wailing. When he saw her in this plight, he was sore concerned and asked her, “What aileth thee, O damsel? How is it with thy lord, my son? Is he not well?”; and she answered, “O King, this thy son, whom thy courtiers avouch
- 79. to be dumb, required me of myself and I repelled him, whereupon he did with me as thou seest and would have slain me; so I fled from him, nor will I ever return to him, nor to the palace again, no, never again!” When the King heard this, he was wroth with exceeding wrath and, calling his seven Wazirs, bade them put the Prince to death. However, they said one to other, “If we do the King’s Commandment, he will surely repent of having ordered his son’s death, for he is passing dear to him and this child came not to him save after despair; and he will round upon us and blame us, saying:—Why did ye not contrive to dissuade me from slaying him?” So they took counsel together, to turn him from his purpose, and the chief Wazir said, “I will warrant you from the King’s mischief this day.” Then he went in to the presence and prostrating himself craved leave to speak. The King gave him permission, and he said, “O King, though thou hadst a thousand sons, yet were it no light matter to thee to put one of them to death, on the report of a woman, be she true or be she false; and belike this is a lie and a trick of her against thy son; for indeed, O King, I have heard tell great plenty of stories of the malice, the craft and perfidy of women.” Quoth the King, “Tell me somewhat of that which hath come to thy knowledge thereof.” And the Wazir answered, saying:—Yes, there hath reached me, O King, a tale entituled
- 80. THE KING AND HIS WAZIR’S WIFE.[160] There was once a King of the Kings, a potent man and a proud, who was devoted to the love of women and one day being in the privacy of his palace, he espied a beautiful woman on the terrace-roof of her house and could not contain himself from falling consumedly in love with her.[161] He asked his folk to whom the house and the damsel belonged and they said, “This is the dwelling of the Wazir such an one and she is his wife.” So he called the Minister in question and despatched him on an errand to a distant part of the kingdom, where he was to collect information and to return; but, as soon as he obeyed and was gone, the King contrived by a trick to gain access to his house and his spouse. When the Wazir’s wife saw him, she knew him and springing up, kissed his hands and feet and welcomed him. Then she stood afar off, busying herself in his service, and said to him, “O our lord, what is the cause of thy gracious coming? Such an honour is not for the like of me.” Quoth he, “The cause of it is that love of thee and desire thee-wards have moved me to this.” Whereupon she kissed ground before him a second time and said, “By Allah, O our lord, indeed I am not worthy to be the handmaid of one of the King’s servants; whence then have I the great good fortune to be in such high honour and favour with thee?” Then the King put out his hand to her intending to enjoy her person, when she said, “This thing shall not escape us; but take patience, O my King, and abide with thy handmaid all this day, that she may make ready for thee somewhat to eat and drink.” So the King sat down on his Minister’s couch and she went in haste and brought him a book wherein he might read, whilst she made ready the food. He took the book and, beginning to read, found therein moral instances and exhortations, such as restrained him from adultery and broke his courage to commit sin and crime. After awhile, she returned and set before him some ninety dishes of different kinds and colours, and he ate a mouthful of each and found
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