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FOUNDATIONS OF
COMPLEX SYSTEMS
Nonlinear Dynamics, StatisticalPhysics, Information
and Prediction

This page intentionally left blank
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FOUNDATIONS OF
COMPLEX SYSTEMS
Nonlinear Dynamics, StatisticalPhysics, Information
and Prediction
Gregoire Nicolis
University o
f Brussels,Belgium
Catherine Nicolis
Royal MeteorologicalInstitute o
f Belgium, Belgium
vpWorld Scientific
NEW JERSEY - LONDON * SINGAPORE * BElJlNG SHANGHAI * HONG KONG * TAIPEI * CHENNAI

British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN-13 978-981-270-043-8
ISBN-10 981-270-043-9
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Printed in Singapore.
FOUNDATIONS OF COMPLEX SYSTEMS
Nonlinear Dynamics, Statistical Physics, Information and Prediction

To Helen, Stamatis and little Katy
v

Preface
Complexity became a major scientific field in its own right as recently as 15
years ago, and since then it has modified considerably the scientific land-
scape through thousands of high-impact publications as well as through the
creation of specialized journals, Institutes, learned societies and University
chairs devoted specifically to it. It constitutes today a paradigm for ap-
proaching a large body of phenomena of concern at the crossroads of physical,
engineering, environmental, life and human sciences from a unifying point of
view.
Nonlinear science and statistical physics had been addressing for some
time phenomena of this kind: self-organization in nonequilibrium systems,
glassy materials, pattern formation, deterministic chaos are landmarks, wit-
nessing the success they have achieved in explaining how unexpected struc-
tures and events can be generated from the laws of nature in systems involv-
ing interacting subunits when appropriate conditions are satisfied - an issue
closely related to the problematics of complexity. And yet, on the one side,
for quite some time these attempts were progressing in rather disconnected
ways following their own momentum and success; and on the other side,
they were remaining confined to a large extent within a community of strong
background in physical and mathematical science, and did not incorporate
to a sufficient degree insights from the practitioner confronted with naturally
occurring systems where issues eliciting the idea of complexity show up in
a most pressing way. Last but not least, there was a lack of insight and of
illustrative power of just what are the minimal ingredients for observing the
sort of behaviors that would qualify as “complex”.
A first breakthrough that contributed significantly to the birth of com-
plexity research occurred in the late 1980’s - early 1990’s. It arose from the
cross-fertilization of ideas and tools from nonlinear science, statistical physics
and numerical simulation, the latter being a direct offspring of the increasing
availability of computers. By bringing chaos and irreversibility together it
showed that deterministic and probabilistic views, causality and chance, sta-
bility and evolution were different facets of a same reality when addressing
vii

viii Preface
certain classes of systems. It also provided insights on the relative roles of the
number of elements involved in the process and the nature of the underlying
dynamics. Paul Anderson’s well-known aphorism, “more is different”, that
contributed to the awareness of the scientific community on the relevance of
complexity, is here complemented in a most interesting way.
The second breakthrough presiding in the birth of complexity coincides
with the increasing input of fields outside the strict realm of physical science.
The intrusion of concepts that were till then not part of the vocabulary of fun-
damental science forced a reassessment of ideas and practices. Predictability,
in connection with the increasing concern about the evolution of the atmo-
sphere, climate and financial activities; algorithms, information, symbols,
networks, optimization in connection with life sciences, theoretical informat-
ics, computer science, engineering and management; adaptive behavior and
cognitive processes in connection with brain research, ethology and social
sciences are some characteristic examples.
Finally, time going on, it became clear that generic aspects of the complex
behaviors observed across a wide spectrum of fields could be captured by
minimal models governed by simple local rules. Some of them gave rise in
their computer implementation to attractive visualizations and deep insights,
from Monte Carlo simulations to cellular automata and multi-agent systems.
These developments provided the tools and paved the way to an under-
standing, both qualitative and quantitative, of the complex systems encoun-
tered in nature, technology and everyday experience. In parallel, natural
complexity acted as a source of inspiration generating progress at the funda-
mental level. Spontaneously, in a very short time interval complexity became
in this way a natural reference point for all sorts of communities and prob-
lems. Inevitably, in parallel with the substantial progress achieved ambiguous
statements and claims were also formulated related in one way or the other
to the diversity of backgrounds of the actors involved and their perceptions
as to the relative roles of hard facts, mechanisms, analogies and metaphors.
As a result complexity research is today both one of the most active and
fastest growing fields of science and a forum for the exchange of sometimes
conflicting ideas and views cutting across scientific disciplines.
In this book the foundations of complex systems are outlined. The vision
conveyed is that of complexity as a part of fundamental science, in which
the insights provided by its cross-fertilization with other disciplines are in-
corporated. What is more, we argue that by virtue of this unique blending
complexity ranks among the most relevant parts of fundamental science as it
addresses phenomena that unfold on our own scale, phenomena in the course
of which the object and the observer are co-evolving. A unifying presentation
of the concepts and tools needed to analyze, to model and to predict com-

Preface ix
plex systems is laid down and links between key concepts such as emergence,
irreversibility, evolution, randomness and information are established in the
light of the complexity paradigm. Furthermore, the interdisciplinary dimen-
sion of complexity research is brought out through representative examples.
Throughout the presentation emphasis is placed on the need for a multi-
level approach to complex systems integrating deterministic and probabilis-
tic views, structure and dynamics, microscopic, mesoscopic and macroscopic
level descriptions.
The book is addressed primarily to graduate level students and to re-
searchers in physics, mathematics and computer science, engineering, envi-
ronmental and life sciences, economics and sociology. It can constitute the
material of a graduate-level course and we also hope that, outside the aca-
demic community, professionals interested in interdisciplinary issues will find
some interest in its reading. The choice of material, the style and the cov-
erage of the items reflect our concern to do justice to the multiple facets
of complexity. There can be no “soft” approach to complexity: observing,
monitoring, analyzing, modeling, predicting and controlling complex systems
can only be achieved through the time-honored approach provided by “hard”
science. The novelty brought by complex systems is that in this endeavor the
goals are reassessed and the ways to achieve them are reinvented in a most
unexpected way as compared to classical approaches.
Chapter 1 provides an overview of the principal manifestations of com-
plexity. Unifying concepts such as instability, sensitivity, bifurcation, emer-
gence, self-organization, chaos, predictability, evolution and selection are
sorted out in view of later developments and the need for a bottom-up ap-
proach to complexity is emphasized. In Chapter 2 the basis of a deterministic
approach to the principal behaviors characteristic of the phenomenology of
complex systems at different levels of description is provided, using the for-
malism of nonlinear dynamical systems. The fundamental mechanism under-
lying emergence is identified. At the same time the limitations of a universal
description of complex systems within the framework of a deterministic ap-
proach are revealed and the “open future” character of their evolution is
highlighted. Some prototypical ways to model complexity in physical science
and beyond are also discussed, with emphasis on the role of the coupling
between constituting elements. In Chapter 3 an analysis incorporating the
probabilistic dimension of complex systems is carried out. It leads to some
novel ways to characterize complex systems, allows one to recover universal
trends in their evolution and brings out the limitations of the determinis-
tic description. These developments provide the background for different
ways to simulate complex systems and for understanding the relative roles
of dynamics and structure in their behavior. The probabilistic approach to

x Preface
complexity is further amplified in Chapter 4 by the incorporation of the con-
cepts of symbolic dynamics and information. A set of entropy-like quantities
is introduced and their connection with their thermodynamic counterparts is
discussed. The selection rules presiding the formation of complex structures
are also studied in terms of these quantities and the nature of the underlying
dynamics. The stage is thus set for the analysis of the algorithmic aspects of
complex systems and for the comparison between algorithmic complexity as
defined in theoretical computer science and natural complexity.
Building on the background provided by Chapters 1 to 4, Chapter 5 ad-
dresses “operational” aspects of complexity, such as monitoring and data
analysis approaches targeted specifically to complex systems. Special em-
phasis is placed on the mechanisms underlying the propagation of prediction
errors and the existence of a limited predictability horizon. The chapter ends
with a discussion of recurrence and extreme events, two prediction-oriented
topics of increasing concern. Finally, in Chapter 6 complexity is shown “in
action” on a number of selected topics. The choices made in this selection out
of the enormous number of possibilities reflect our general vision of complex-
ity as part of fundamental science but also, inevitably, our personal interests
and biases. We hope that this coverage illustrates adequately the relevance
and range of applicability of the ideas and tools outlined in the book. The
chapter ends with a section devoted to the epistemological aspects of com-
plex systems. Having no particular background in epistemology we realize
that this is a risky enterprise, but we feel that it cannot be dispensed with
in a book devoted to complexity. The presentation of the topics of this final
section is that of the practitioner of physical science, and contains only few
elements of specialized jargon in a topic that could by itself give rise to an
entire monograph.
In preparing this book we have benefitted from discussions with, com-
ments and help in the preparation of figures by Y. Almirantis, V. Basios, A.
Garcia Cantu, P. Gaspard, M. Malek Mansour, J. S. Nicolis, S. C. Nicolis,
A. Provata, R. Thomas and S. Vannitsem. S. Wellens assumed the hard task
of typing the first two versions of the manuscript.
Our research in the subject areas covered in this book is sponsored by
The University of Brussels, the Royal Meteorological Institute of Belgium,
the Science Policy Office of the Belgian Federal Government, the European
Space Agency and the European Commission. Their interest and support
are gratefully acknowledged.
G. Nicolis, C. Nicolis
Brussels, February 2007

Contents
Preface vii
1 The phenomenology of complex systems 1
1.1 Complexity, a new paradigm . . . . . . . . . . . . . . . . . . . 1
1.2 Signatures of complexity . . . . . . . . . . . . . . . . . . . . . 3
1.3 Onset of complexity . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Four case studies . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Rayleigh-Bénard convection . . . . . . . . . . . . . . . 8
1.4.2 Atmospheric and climatic variability . . . . . . . . . . 11
1.4.3 Collective problem solving: food recruitment in ants . . 15
1.4.4 Human systems . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Deterministic view 25
2.1 Dynamical systems, phase space, stability . . . . . . . . . . . 25
2.1.1 Conservative systems . . . . . . . . . . . . . . . . . . . 27
2.1.2 Dissipative systems . . . . . . . . . . . . . . . . . . . . 27
2.2 Levels of description . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 The microscopic level . . . . . . . . . . . . . . . . . . . 34
2.2.2 The macroscopic level . . . . . . . . . . . . . . . . . . 36
2.2.3 Thermodynamic formulation . . . . . . . . . . . . . . . 38
2.3 Bifurcations, normal forms, emergence . . . . . . . . . . . . . 41
2.4 Universality, structural stability . . . . . . . . . . . . . . . . . 46
2.5 Deterministic chaos . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Aspects of coupling-induced complexity . . . . . . . . . . . . . 53
2.7 Modeling complexity beyond physical science . . . . . . . . . . 59
3 The probabilistic dimension of complex systems 64
3.1 Need for a probabilistic approach . . . . . . . . . . . . . . . . 64
3.2 Probability distributions and their evolution laws . . . . . . . 65
3.3 The retrieval of universality . . . . . . . . . . . . . . . . . . . 72
xi

xii Contents
3.4 The transition to complexity in probability space . . . . . . . 77
3.5 The limits of validity of the macroscopic description . . . . . . 82
3.5.1 Closing the moment equations in the mesoscopic
description . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5.2 Transitions between states . . . . . . . . . . . . . . . . 84
3.5.3 Average values versus fluctuations in
deterministic chaos . . . . . . . . . . . . . . . . . . . . 88
3.6 Simulating complex systems . . . . . . . . . . . . . . . . . . . 90
3.6.1 Monte Carlo simulation . . . . . . . . . . . . . . . . . 91
3.6.2 Microscopic simulations . . . . . . . . . . . . . . . . . 92
3.6.3 Cellular automata . . . . . . . . . . . . . . . . . . . . . 94
3.6.4 Agents, players and games . . . . . . . . . . . . . . . . 95
3.7 Disorder-generated complexity . . . . . . . . . . . . . . . . . . 96
4 Information, entropy and selection 101
4.1 Complexity and information . . . . . . . . . . . . . . . . . . . 101
4.2 The information entropy of a history . . . . . . . . . . . . . . 104
4.3 Scaling rules and selection . . . . . . . . . . . . . . . . . . . . 106
4.4 Time-dependent properties of information.
Information entropy and thermodynamic entropy . . . . . . . 115
4.5 Dynamical and statistical properties of time histories.
Large deviations, fluctuation theorems . . . . . . . . . . . . . 117
4.6 Further information measures. Dimensions and Lyapunov
exponents revisited . . . . . . . . . . . . . . . . . . . . . . . . 120
4.7 Physical complexity, algorithmic complexity,
and computation . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.8 Summing up: towards a thermodynamics of
complex systems . . . . . . . . . . . . . . . . . . . . . . . . . 128
5 Communicating with a complex system: monitoring,
analysis and prediction 131
5.1 Nature of the problem . . . . . . . . . . . . . . . . . . . . . . 131
5.2 Classical approaches and their limitations . . . . . . . . . . . . 131
5.2.1 Exploratory data analysis . . . . . . . . . . . . . . . . 132
5.2.2 Time series analysis and statistical forecasting . . . . . 135
5.2.3 Sampling in time and in space . . . . . . . . . . . . . . 138
5.3 Nonlinear data analysis . . . . . . . . . . . . . . . . . . . . . . 139
5.3.1 Dynamical reconstruction . . . . . . . . . . . . . . . . 139
5.3.2 Symbolic dynamics from time series . . . . . . . . . . . 143
5.3.3 Nonlinear prediction . . . . . . . . . . . . . . . . . . . 148
5.4 The monitoring of complex fields . . . . . . . . . . . . . . . . 151

Contents xiii
5.4.1 Optimizing an observational network . . . . . . . . . . 153
5.4.2 Data assimilation . . . . . . . . . . . . . . . . . . . . . 157
5.5 The predictability horizon and the limits of modeling . . . . . 159
5.5.1 The dynamics of growth of initial errors . . . . . . . . 160
5.5.2 The dynamics of model errors . . . . . . . . . . . . . . 164
5.5.3 Can prediction errors be controlled? . . . . . . . . . . . 170
5.6 Recurrence as a predictor . . . . . . . . . . . . . . . . . . . . 171
5.6.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . 172
5.6.2 Recurrence time statistics and dynamical
complexity . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.7 Extreme events . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.7.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . 180
5.7.2 Statistical theory of extremes . . . . . . . . . . . . . . 182
5.7.3 Signatures of a deterministic dynamics in
extreme events . . . . . . . . . . . . . . . . . . . . . . 185
5.7.4 Statistical and dynamical aspects of the Hurst
phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 191
6 Selected topics 195
6.1 The arrow of time . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.1.1 The Maxwell-Boltzmann revolution, kinetic theory,
Boltzmann’s equation . . . . . . . . . . . . . . . . . . . 196
6.1.2 First resolution of the paradoxes: Markov processes,
master equation . . . . . . . . . . . . . . . . . . . . . . 200
6.1.3 Generalized kinetic theories . . . . . . . . . . . . . . . 202
6.1.4 Microscopic chaos and nonequilibrium statistical
mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.2 Thriving on fluctuations: the challenge of being small . . . . . 208
6.2.1 Fluctuation dynamics in nonequilibrium steady
states revisited . . . . . . . . . . . . . . . . . . . . . . 210
6.2.2 The peculiar energetics of irreversible paths
joining equilibrium states . . . . . . . . . . . . . . . . . 211
6.2.3 Transport in a fluctuating environment far from
equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 214
6.3 Atmospheric dynamics . . . . . . . . . . . . . . . . . . . . . . 217
6.3.1 Low order models . . . . . . . . . . . . . . . . . . . . . 218
6.3.2 More detailed models . . . . . . . . . . . . . . . . . . . 222
6.3.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . 223
6.3.4 Modeling and predicting with probabilities . . . . . . . 224
6.4 Climate dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.4.1 Low order climate models . . . . . . . . . . . . . . . . 227

xiv Contents
6.4.2 Predictability of meteorological versus climatic fields . 230
6.4.3 Climatic change . . . . . . . . . . . . . . . . . . . . . . 233
6.5 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
6.5.1 Geometric and statistical properties of networks . . . . 236
6.5.2 Dynamical origin of networks . . . . . . . . . . . . . . 239
6.5.3 Dynamics on networks . . . . . . . . . . . . . . . . . . 244
6.6 Perspectives on biological complexity . . . . . . . . . . . . . . 247
6.6.1 Nonlinear dynamics and self-organization at the
biochemical, cellular and organismic level . . . . . . . . 249
6.6.2 Biological superstructures . . . . . . . . . . . . . . . . 251
6.6.3 Biological networks . . . . . . . . . . . . . . . . . . . . 253
6.6.4 Complexity and the genome organization . . . . . . . . 260
6.6.5 Molecular evolution . . . . . . . . . . . . . . . . . . . . 263
6.7 Equilibrium versus nonequilibrium in complexity and
self-organization . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.7.1 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . 268
6.7.2 Stabilization of nanoscale patterns . . . . . . . . . . . 272
6.7.3 Supramolecular chemistry . . . . . . . . . . . . . . . . 274
6.8 Epistemological insights from complex systems . . . . . . . . . 276
6.8.1 Complexity, causality and chance . . . . . . . . . . . . 277
6.8.2 Complexity and historicity . . . . . . . . . . . . . . . . 279
6.8.3 Complexity and reductionism . . . . . . . . . . . . . . 283
6.8.4 Facts, analogies and metaphors . . . . . . . . . . . . . 285
Color plates 287
Suggestions for further reading 291
Index 321

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The whole is more than the sum
of its parts
Aristotle Metaphysica 1045a
Chapter 1
The phenomenology of complex
systems
1.1 Complexity, a new paradigm
Complexity is part of our ordinary vocabulary. It has been used in everyday
life and in quite different contexts for a long time and suddenly, as recently
as 15 years ago it became a major field of interdisciplinary research that
has since then modified considerably the scientific landscape. What is in
the general idea of complexity that was missing in our collective knowledge
-one might even say, in our collective consciousness- which, once recognized,
conferred to it its present prominent status? What makes us designate certain
systems as “complex” distinguishing them from others that we would not
hesitate to call “simple”, and to what extent could such a distinction be
the starting point of a new approach to a large body of phenomena at the
crossroads of physical, engineering, environmental, life and human sciences?
For the public and for the vast majority of scientists themselves science is
usually viewed as an algorithm for predicting, with a theoretically unlimited
precision, the future course of natural objects on the basis of their present
state. Isaac Newton, founder of modern physics, showed more than three
centuries ago how with the help of a few theoretical concepts like the law of
universal gravitation, whose statement can be condensed in a few lines, one
can generate data sets as long as desired allowing one to interpret the essence
of the motion of celestial bodies and predict accurately, among others, an
eclipse of the sun or of the moon thousands of years in advance. The impact
of this historical achievement was such that, since then, scientific thinking
has been dominated by the Newtonian paradigm whereby the world is re-
ducible to a few fundamental elements animated by a regular, reproducible
1

2 Foundations of Complex Systems
and hence predictable behavior: a world that could in this sense be qualified
as fundamentally simple.
During the three-century reign of the Newtonian paradigm science reached
a unique status thanks mainly to its successes in the exploration of the very
small and the very large: the atomic, nuclear and subnuclear constitution
of matter on the one side; and cosmology on the other. On the other hand
man’s intuition and everyday experience are essentially concerned with the
intermediate range of phenomena involving objects constituted by a large
number of interacting subunits and unfolding on his own, macroscopic, space
and time scales. Here one cannot avoid the feeling that in addition to regular
and reproducible phenomena there exist other that are, manifestly, much less
so. It is perfectly possible as we just recalled to predict an eclipse of the sun
or of the moon thousands of years in advance but we are incapable of pre-
dicting the weather over the region we are concerned more than a few days
in advance or the electrical activity in the cortex of a subject a few minutes
after he started performing a mental task, to say nothing about next day’s
Dow Jones index or the state of the planet earth 50 years from now. Yet the
movement of the atmosphere and the oceans that governs the weather and
the climate, the biochemical reactions and the transport phenomena that
govern the functioning of the human body and underlie, after all, human
behavior itself, obey to the same dispassionate laws of nature as planetary
motion.
It is a measure of the fascination that the Newtonian paradigm exerted
on scientific thought that despite such indisputable facts, which elicit to the
observer the idea of “complexity”, the conviction prevailed until recently that
the irregularity and unpredictability of the vast majority of phenomena on
our scale are not authentic: they are to be regarded as temporary drawbacks
reflecting incomplete information on the system at hand, in connection with
the presence of a large number of variables and parameters that the observer
is in the practical impossibility to manage and that mask some fundamental
underlying regularities.
If evidence on complexity were limited to the intricate, large scale systems
of the kind mentioned above one would have no way to refute such an asser-
tion and fundamental science would thus have nothing to say on complexity.
But over the years evidence has accumulated that quite ordinary systems
that one would tend to qualify as “simple”, obeying to laws known to their
least detail, in the laboratory, under strictly controlled conditions, generate
unexpected behaviors similar to the phenomenology of complexity as we en-
counter it in nature and in everyday experience: Complexity is not a mere
metaphor or a nice way to put certain intriguing things, it is a phenomenon
that is deeply rooted into the laws of nature, where systems involving large

The Phenomenology of Complex Systems 3
numbers of interacting subunits are ubiquitous.
This realization opens the way to a systematic search of the physical
and mathematical laws governing complex systems. The enterprise was
crowned with success thanks to a multilevel approach that led to the de-
velopment of highly original methodologies and to the unexpected cross-
fertilizations and blendings of ideas and tools from nonlinear science, sta-
tistical mechanics and thermodynamics, probability theory and numerical
simulation. Thanks to the progress accomplished complexity is emerging as
the new, post-Newtonian paradigm for a fresh look at problems of current
concern. On the one side one is now in the position to gain new understand-
ing, both qualitative and quantitative, of the complex systems encountered
in nature and in everyday experience based on advanced modeling, analysis
and monitoring strategies. Conversely, by raising issues and by introducing
concepts beyond the traditional realm of physical science, natural complexity
acts as a source of inspiration for further progress at the fundamental level.
It is this sort of interplay that confers to research in complexity its unique,
highly interdisciplinary character.
The objective of this chapter is to compile some representative facts il-
lustrating the phenomenology associated with complex systems. The subse-
quent chapters will be devoted to the concepts and methods underlying the
paradigm shift brought by complexity and to showing their applicability on
selected case studies.
1.2 Signatures of complexity
The basic thesis of this book is that a system perceived as complex induces a
characteristic phenomenology the principal signature of which is multiplicity.
Contrary to elementary physical phenomena like the free fall of an object
under the effect of gravity where a well-defined, single action follows an initial
cause at any time, several outcomes appear to be possible. As a result the
system is endowed with the capacity to choose between them, and hence
to explore and to adapt or, more generally, to evolve. This process can be
manifested in the form of two different expressions.
• The emergence, within a system composed of many units, of global
traits encompassing the system as a whole that can in no way be
reduced to the properties of the constituent parts and can on these
grounds be qualified as “unexpected”. By its non-reductionist charac-
ter emergence has to do with the creation and maintenance of hierar-
chical structures in which the disorder and randomness that inevitably

exist at the local level are controlled, resulting in states of order and
long range coherence. We refer to this process as self-organization. A
classical example of this behavior is provided by the communication
and control networks in living matter, from the subcellular to the or-
ganismic level.
• The intertwining, within the same phenomenon, of large scale regu-
larities and of elements of “surprise” in the form of seemingly erratic
evolutionary events. Through this coexistence of order and disorder
the observer is bound to conclude that the process gets at times out
of control, and this in turn raises the question of the very possibility
of its long-term prediction. Classical examples are provided by the
all-familiar difficulty to issue satisfactory weather forecasts beyond a
horizon of a few days as well as by the even more dramatic extreme
geological or environmental phenomena such as earthquakes or floods.
If the effects generated by some underlying causes were related to these
causes by a simple proportionality -more technically, by linear relationships-
there would be no place for multiplicity. Nonlinearity is thus a necessary con-
dition for complexity, and in this respect nonlinear science provides a natural
setting for a systematic description of the above properties and for sorting
out generic evolutionary scenarios. As we see later nonlinearity is ubiquitous
in nature on all levels of observation. In macroscopic scale phenomena it is
intimately related to the presence of feedbacks, whereby the occurrence of a
process affects (positively or negatively) the way it (or some other coexisting
process) will further develop in time. Feedbacks are responsible for the onset
of cooperativity, as illustrated in the examples of Sec. 1.4.
In the context of our study a most important question to address con-
cerns the transitions between states, since the question of complexity would
simply not arise in a system that remains trapped in a single state for ever.
To understand how such transitions can happen one introduces the concept
of control parameter, describing the different ways a system is coupled to its
environment and affected by it. A simple example is provided by a ther-
mostated cell containing chemically active species where, depending on the
environmental temperature, the chemical reactions will occur at different
rates. Another interesting class of control parameters are those associated to
a constraint keeping the system away of a state of equilibrium of some sort.
The most clearcut situation is that of the state of thermodynamic equilib-
rium which, in the absence of phase transitions, is known to be unique and
lack any form of dynamical activity on a large scale. One may then choose
this state as a reference, switch on constraints driving the system out of equi-
librium for instance in the form of temperature or concentration differences

across the interface between the system and the external world, and see to
what extent the new states generated as a response to the constraint could
exhibit qualitatively new properties that are part of the phenomenology of
complexity. These questions, which are at the heart of complexity theory,
are discussed in the next section.
1.3 Onset of complexity
The principal conclusion of the studies of the response of a system to changes
of a control parameter is that the onset of complexity is not a smooth process.
Quite to the contrary, it is manifested by a cascade of transition phenomena
of an explosive nature to which is associated the universal model of bifurcation
and the related concepts of instability and chaos. These catastrophic events
are not foreseen in the fundamental laws of physics in which the dependence
on the parameters is perfectly smooth. To use a colloquial term, one might
say that they come as a “surprise”.
Figure 1.1 provides a qualitative representation of the foregoing. It de-
picts a typical evolution scenario in which, for each given value of a control
parameter λ, one records a certain characteristic property of the system as
provided, for instance, by the value of one of the variables X (temperature,
chemical concentration, population density, etc.) at a given point. For values
of λ less than a certain limit λc only one state can be realized. This state
possesses in addition to uniqueness the property of stability, in the sense that
the system is capable of damping or at least of keeping under control the in-
fluence of the external perturbations inflicted by the environment or of the
internal fluctuations generated continuously by the locally prevailing disor-
der, two actions to which a natural system is inevitably subjected. Clearly,
complexity has no place and no meaning under these conditions.
The situation changes radically beyond the critical value λc. One sees that
if continued, the unique state of the above picture would become unstable:
under the influence of external perturbations or of internal fluctuations the
system responds now as an amplifier, leaves the initial “reference” state and
is driven to one or as a rule to several new behaviors that merge to the
previous state for λ = λc but are differentiated from it for λ larger than λc.
This is the phenomenon of bifurcation: a phenomenon that becomes possible
thanks to the nonlinearity of the underlying evolution laws allowing for the
existence of multiple solutions (see Chapter 2 for quantitative details). To
understand its necessarily catastrophic character as anticipated earlier in this
section it is important to account for the following two important elements.
(a) An experimental measurement -the process through which we com-

X
λc
λ
(a)
(b1)
(a' )
(b2)
Fig. 1.1. A bifurcation diagram, describing the way a variable X characteriz-
ing the state of a system is affected by the variations of a control parameter
λ. Bifurcation takes place at a critical value λc beyond which the original
unique state (a) loses its stability, giving rise to two new branches of solutions
(b1) and (b2).
municate with a system- is necessarily subjected to finite precision. The
observation of a system for a given value of control parameter entails that
instead of the isolated point of the λ axis in Fig. 1.1 one deals in reality with
an “uncertainty ball” extending around this axis. The system of interest lies
somewhere inside this ball but we are unable to specify its exact position,
since for the observer all of its points represent one and the same state.
(b) Around and beyond the criticality λc we witness a selection between
the states available that will determine the particular state to which the sys-
tem will be directed (the two full lines surrounding the intermediate dotted
one -the unstable branch in Fig. 1.1- provide an example). Under the con-
ditions of Fig. 1.1 there is no element allowing the observer to determine
beforehand this state. Chance and fluctuations will be the ones to decide.
The system makes a series of attempts and eventually a particular fluctu-
ation takes over. By stabilizing this choice it becomes a historical object,
since its subsequent evolution will be conditioned by this critical choice. For
the observer, this pronounced sensitivity to the parameters will signal its in-
ability to predict the system’s evolution beyond λc since systems within the
uncertainty ball, to him identical in any respect, are differentiated and end
up in states whose distance is much larger than the limits of resolution of
the experimental measurement.

0.25
0.5
0.75
0 5 10 15 20 25
x
n
Fig. 1.2. Illustration of the phenomenon of sensitivity to the initial conditions
in a model system giving rise to deterministic chaos. Full and dashed lines
denote the trajectories (the set of successive values of the state variable X)
emanating from two initial conditions separated by a small difference =
10−3
.
We now have the basis of a mechanism of generation of complexity. In
reality this mechanism is the first step of a cascade of successive bifurca-
tions through which the multiplicity of behaviors may increase dramatically,
culminating in many cases in a state in which the system properties change
in time (and frequently in space as well) in a seemingly erratic fashion, not
any longer because of external disturbances or random fluctuations as be-
fore but, rather, as a result of deterministic laws of purely intrinsic origin.
The full line of Fig. 1.2 depicts a time series -a succession of values of a
relevant variable in time- corresponding to this state of deterministic chaos.
Its comparison with the dotted line reveals what is undoubtedly the most
spectacular property of deterministic chaos, the sensitivity to the initial con-
ditions: two systems whose initial states are separated by a small distance,
smaller than the precision of even the most advanced method of experimen-
tal measurement, systems that will therefore be regarded by the observer as
indistinguishable (see also point (a) above) will subsequently diverge in such
a way that the distance between their instantaneous states (averaged over
many possible initial states, see Chapters 2 and 3) will increase exponentially.
As soon as this distance will exceed the experimental resolution the systems
will cease to be indistinguishable for the observer. As a result, it will be
impossible to predict their future evolution beyond this temporal horizon.

We here have a second imperative reason forcing us to raise the question of
predictability of the phenomena underlying the behavior of complex systems.
All elements at our disposal from the research in nonlinear science and
chaos theory lead to the conclusion that one cannot anticipate the full list of
the number or the type of the evolutionary scenarios that may lead a system
to complex behavior. In addition to their limited predictability complex
systems are therefore confronting us with the fact that we seem to be stuck
with a mode of description of a limited universality. How to reconcile this
with the requirement that the very mission of science is to provide a universal
description of phenomena and to predict their course? The beauty of complex
systems lies to a great extent in that despite the above limitations this mission
can be fulfilled, but that its realization necessitates a radical reconsideration
of the concepts of universality and prediction. We defer a fuller discussion of
this important issue to Chapters 2 and 3.
1.4 Four case studies
1.4.1 Rayleigh-Bénard convection
Consider a shallow layer of a fluid limited by two horizontal plates brought to
identical temperatures. As prescribed by the second law of thermodynamics,
left to itself the fluid will tend rapidly to a state where all its parts along
the horizontal are macroscopically identical and where there is neither bulk
motion nor internal differentiation of temperatures: T = T1 = T2, T2 and T1
being respectively the temperatures of the lower and upper plate. This is the
state we referred to in Sec. 1.2 as the state of thermodynamic equilibrium.
Imagine now that the fluid is heated from below. By communicating to
it in this way energy in the form of heat one removes it from the state of
equilibrium, since the system is now submitted to a constraint ∆T = T2 −
T1 0, playing in this context the role of the control parameter introduced
in Sec. 1.2. As long at ∆T remains small the flux of energy traversing
the system will merely switch on a process of heat conduction, in which
temperature varies essentially linearly between the hot (lower) zone and the
cold (upper) one. This state is maintained thanks to a certain amount of
energy that remains trapped within the system -one speaks of dissipation-
but one can in no way speak here of complexity and emergence, since the
state is unique and the differentiation observed is dictated entirely by the
way the constraint has been applied: the behavior is as “simple” as the one
in the state of equilibrium.
If one removes now the system progressively from equilibrium, by increas-

Fig. 1.3. Rayleigh-Bénard convection cells appearing in a liquid maintained
between a horizontal lower hot plate and an upper cold one, below a critical
value of the temperature difference ∆T (see Color Plates).
ing ∆T, one suddenly observes, for a critical value ∆Tc, the onset of bulk
motion in the layer. This motion is far from sharing the randomness of the
motion of the individual molecules: the fluid becomes structured and displays
a succession of cells along a direction transversal to that of the constraint, as
seen in Fig. 1.3. This is the regime of thermal, or Rayleigh-Bénard convec-
tion. Now one is entitled to speak of complexity and emergence, since the
spatial differentiation along a direction free from any constraint is the result
of processes of internal origin specific to the system, maintained by the flow
of energy communicated by the external world and hence by the dissipation.
We have thus witnessed a particular manifestation of emergence, in the form
of the birth of a dissipative structure. In a way, one is brought from a static,
geometric view of space, to one where space is modeled by the dynamical
processes switched on within the system. One can show that the state of rest
is stable below the threshold ∆Tc but loses its stability above it while still
remaining a solution -in the mathematical sense of the term- of the evolution
laws of the fluid. As for the state of thermal convection, it simply does not
exist below ∆Tc and inherits above it the stability of the state of rest. For
∆T = ∆Tc there is degeneracy in the sense that the two states merge. We
here have a concrete illustration of the generic phenomenon of bifurcation
introduced in Sec. 1.3, see Fig. 1.1. Similar phenomena are observed in
a wide range of laboratory scale systems, from fluid mechanics to chemical
kinetics, optics, electronics or materials science. In each case one encoun-

ters essentially the same phenomenology. The fact that this is taking place
under perfectly well controlled conditions allows one to sort out common fea-
tures and set up a quantitative theory, as we see in detail in the subsequent
chapters.
A remarkable property of the state of thermal convection is to possess a
characteristic space scale -the horizontal extent of a cell (Fig. 1.3) related,
in turn, to the depth of the layer. The appearance of such a scale reflects the
fact that the states generated by the bifurcation display broken symmetries.
The laws of fluid dynamics describing a fluid heated from below and con-
tained between two plates that extend indefinitely in the horizontal direction
remain invariant -or more plainly look identical- for all observers displaced
to one another along this direction (translational invariance). This invari-
ance property is shared by the state realized by the fluid below the threshold
∆Tc but breaks down above it, since a state composed of a succession of
Bénard cells displays an intrinsic differentiation between its different parts
that makes it less symmetrical than the laws that generated it. A differentia-
tion of this sort may become in many cases one of the prerequisites for further
complexification, in the sense that processes that would be impossible in an
undifferentiated medium may be switched on. In actual fact this is exactly
what is happening in the Rayleigh-Bénard and related problems. In addition
to the first bifurcation described above, as the constraint increases beyond
∆Tc the system undergoes a whole series of successive transitions. Several
scenarios have been discovered. If the horizontal extent of the cell is much
larger than the depth the successive transition thresholds are squeezed in a
small vicinity of ∆Tc. The convection cells are first maintained globally but
are subsequently becoming fuzzy and eventually a regime of turbulence sets
in, characterized by an erratic-looking variability of the fluid properties in
space (and indeed in time as well). In this regime of extreme spatio-temporal
chaos the motion is ordered only on a local level. The regime dominated by
a characteristic space scale has now been succeeded by a scale-free state in
which there is a whole spectrum of coexisting spatial modes, each associated
to a different space scale. Similar phenomena arise in the time domain, where
the first bifurcation may lead in certain types of systems to a strictly periodic
clock-like state which may subsequently lose its coherence and evolve to a
regime of deterministic chaos in which the initial periodicity is now part of
a continuous spectrum of coexisting time scales.
As we see throughout this book states possessing a characteristic scale
and scale-free states are described, respectively, by exponential laws and by
power laws. There is no reason to restrict the phenomenology of complexity
to the class of scale free states as certain authors suggest since, for one thing,
coherence in living matter is often reflected by the total or partial synchro-

nization of the activities of the individual cells to a dominant temporal or
spatial mode.
In concluding this subsection it is appropriate to stress that configurations
of matter as unexpected a priori as the Bénard cells, involving a number of
molecules (each in disordered motion !) of the order of the Avogadro number
N ≈ 1023
are born spontaneously, inevitably, at a modest energetic and
informational cost, provided that certain conditions related to the nature
of the system and the way it is embedded to its environment are fulfilled.
Stated differently the overall organization is not ensured by a centralized
planification and control but, rather, by the “actors” (here the individual
fluid parcels) present. We refer to this process as the bottom-up mechanism.
1.4.2 Atmospheric and climatic variability
Our natural environment plays a central role in this book, not only on the
grounds of its importance in man’s everyday activities but also because it
qualifies in any respect as what one intuitively means by complex system and
forces upon the observer the need to cope with the problem of prediction.
Contrary to the laboratory scale systems considered in the previous subsec-
tion we have no way to realize at will the successive transitions underlying
its evolution to complexity. The best one can expect is that a monitoring
in the perspective of the complex systems approach followed by appropriate
analysis and modeling techniques, will allow one to constitute the salient
features of the environment viewed as a dynamical system and to arrive at a
quantitative characterization of the principal quantities of interest.
To an observer caught in the middle of a hurricane, a flood or a long
drought the atmosphere appears as an irrational medium. Yet the atmo-
spheric and climatic variables are far from being distributed randomly. Our
environment is structured in both space and time, as witnessed by the strati-
fication of the atmospheric layers, the existence of global circulation patterns
such as the planetary waves, and the periodicities arising from the daily or
the annual cycle. But in spite of this global order one observes a pronounced
superimposed variability, reflected by marked deviations from perfect or even
approximate regularity.
An example of such a variability is provided by the daily evolution of
air temperature at a particular location (Fig. 1.4). One observes small
scale irregular fluctuations that are never reproduced in an identical fashion,
superimposed on the large scale regular seasonal cycle of solar radiation. A
second illustration of variability pertains to the much larger scale of global
climate. All elements at our disposal show indeed that the earth’s climate
has undergone spectacular changes in the past, like the succession of glacial-

-10
0
10
20
30
1998 2000 2002 2004
Temperature
Year
Fig. 1.4. Mean daily temperature at Uccle (Brussels) between January 1st,
1998 and December 31, 2006.
Time (103
yrs B.P.)
Ice
volume
0 200 400 600 800 1000
Fig. 1.5. Evolution of the global ice volume on earth during the last million
years as inferred from oxygen isotope data.

interglacial periods. Figure 1.5 represents the variation of the volume of
continental ice over the last million years as inferred from the evolution of
the composition of marine sediments in oxygen 16 and 18 isotopes. Again,
one is struck by the intermittent character of the evolution, as witnessed
by a marked aperiodic component masking to a great extent an average
time scale of 100 000 years that is sometimes qualified as the Quaternary
glaciation “cycle”. An unexpected corollary is that the earth’s climate can
switch between quite different modes over a short time in the geological scale,
of the order of a few thousand years.
Rainfall
departures
(×
10
mm)
3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
1910 1930 1950 1970 Year
Fig. 1.6. Yearly rainfall departures from the long time average value at Kaédi
(Mauritania) between 1904 and 1987.
Figure 1.6 depicts another example of climatic variability and regime
switching, on a scale that is intermediate between those in Figs 1.4 and 1.5.
It has to do with the time variation of the precipitation in western Sahel, and
signals the onset of a regime of drought in this region, a phenomenon known
to occur in several other areas of the globe. Again, one is struck by the
irregular character of the process. The new element as compared to Figs 1.4
and 1.5 is that in the language of statistics the signal is no longer stationary:
rather than succeeding each other without exhibiting a systematic trend, the
states are here undergoing an abrupt transition between a regime of a quasi-
normal and a weak rainfall that one can locate using traditional statistical
analysis around the mid-1960’s. It is likely that observations over a much
longer time scale will reestablish the stationarity of the process, in the sense
that the state of drought will sooner or later be succeeded by a quasi-normal
state which will subsequently switch again to a state of drought, and so forth.

A fundamental consequence of the aperiodicity of the atmospheric and
climate dynamics is the well-known difficulty to make reliable predictions.
Contrary to simple periodic or multiperiodic phenomena for which a long
term prediction is possible, predictions in meteorology are limited in time.
The most plausible (and currently admitted) explanation is based on the re-
alization that a small uncertainty in the initial conditions used in a prediction
scheme (usually referred as “error”) seems to be amplified in the course of
the evolution. Such uncertainties are inherent in the process of experimen-
tal measurement, as pointed out already in Sec. 1.3. A great deal of effort
is devoted in atmospheric sciences in the development of data assimilation
techniques aiming to reduce them as much as possible (cf. also Sec. 5.4), but
it is part of the laws of nature that they will never be fully eliminated. This
brings us to the picture drawn in connection with Fig. 1.2, suggesting that
the atmosphere displays sensitivity to the initial conditions because it is in
a state of deterministic chaos. This conjecture seems to be compatible both
with the analysis of the data available and with the modeling of atmospheric
dynamics. This aspect is discussed more amply in Chapters 5 and 6, but
one may already notice at this stage that much like experiment, modeling
is also limited in practice by a finite resolution (of the order of several kilo-
meters) and the concomitant omission of “subgrid” processes like e.g. local
turbulence. Furthermore, many of the parameters are not known to a great
precision. In addition to initial errors prediction must thus cope with model
errors, reflecting the fact that a model is only an approximate representa-
tion of nature. This raises the problem of sensitivity to the parameters and
brings us to the picture drawn in connection with Fig. 1.1. If the dynamics
were simple like in the part of Fig. 1.1 left to λc neither of these errors would
matter. But this is manifestly not the case. Initial and model errors can thus
be regarded as probes revealing the fundamental instability and complexity
underlying the atmosphere.
In all the preceding examples it was understood that the characteristic
parameters of the atmosphere remained fixed. Over the last years there
has been growing interest in the response of the weather and climate to
changing parameter values - for instance, as a result of anthropogenic effects.
In the representation of Fig. 1.1, the question would then be, whether the
underlying dynamical system would undergo transitions to new regimes and
if so, what would be the nature of the most plausible transition scenarios.
This raises a whole new series of problems, some of which will be taken up
in the sequel.
As pointed out earlier in this subsection, in certain environmental phe-
nomena the variability is so considerable that no underlying regularity seems
to be present. This property, especially pronounced in hydrology and in par-

ticular in the regime of river discharges, entails that the average and other
quantifiers featured in traditional statistics are irrelevant. An ingenious way
to handle such records, suggested some time ago by Harold Hurst, is to
monitor the way the distance R between the largest and smallest value in a
certain time window τ -usually referred to as the range- varies with τ. Ac-
tually, to deal with a dimensionless quantity one usually reduces R by the
standard deviation C around the mean measured over the same interval. A
most surprising result is that in a wide spectrum of environmental records
R/C varies with τ as a power law of the form τH
, where the Hurst expo-
nent H turns out to be close to 0.70. To put this in perspective, for records
generated by statistically independent processes with finite standard devia-
tion, H is bound to be 1/2 and for records where the variability is organized
around a characteristic time scale there would simply not be a power law at
all. Environmental dynamics provides therefore yet another example of the
coexistence of phenomena possessing a characteristic scale and of scale free
ones.
An interesting way to differentiate between these processes is to see how
the law is changing upon a transformation of the variable (here the window
τ). For an exponential law, switching from τ to λτ (which can be interpreted
as a change of scale in measuring τ) maintains the exponential form but
changes the exponent multiplying τ, which provides the characteristic scale
of the process, by a factor of λ. But in a power law the same transformation
keeps the exponent H invariant, producing merely a multiplicative factor.
We express this by qualifying this law as scale invariant. The distinction
breaks down for nonlinear transformations, for which a power law can become
exponential and vice versa.
As we see later deterministic chaos can be associated with variabilities
of either of the two kinds, depending on the mechanisms presiding in its
generation.
1.4.3 Collective problem solving: food recruitment in
ants
In the preceding examples the elements constituting the system of interest
were the traditional ones considered in physical sciences: molecules, volume
elements in a fluid or in a chemical reagent, and so forth. In this subsection
we are interested in situations where the actors involved are living organisms.
We will see that despite this radical change, the principal manifestations of
complexity will be surprisingly close to those identified earlier. Our discussion
will focus on social insects, in particular, the process of food searching in ants.

Ants, like bees, termites and other social insects represent an enormous
ecological success in biological evolution. They are known to be able to
accomplish successfully number of collective activities such as nest construc-
tion, recruitment, defense etc. Until recently the view prevailed that in such
highly non-trivial tasks individual insects behave as small, reliable automa-
tons executing a well established genetic program. Today this picture is
fading and replaced by one in which adaptability of individual behavior, col-
lective interactions and environmental stimuli play an important role. These
elements are at the origin of a two-scale process. One at the level of the in-
dividual, characterized by a pronounced probabilistic behavior, and another
at the level of the society as a whole, where for many species despite the in-
efficiency and unpredictability of the individuals, coherent patterns develop
at the scale of the entire colony.
Fig. 1.7. Schematic representation of recruitment: (a) discovery of the food
source by an individual; (b) return to the nest with pheromone laying; (c) the
pheromone trail stimulates additional individuals to visit the source, which
contribute to its reinforcement by further pheromone laying.
Let us see how these two elements conspire in the process of food search-
ing by ants. Consider first the case where a single food source (for instance
a saccharose solution) is placed close to the nest, as in Fig. 1.7 (here and in
the sequel laboratory experiments emulating naturally occurring situations
while allowing at the same time for detailed quantitative analyses are instru-

mental). A “scout” discovers the source in the course of a random walk.
After feeding on the source it returns to the nest and deposits along the way
a chemical signal known as trail pheromone, whose quantity is correlated to
the sugar concentration in the source. Subsequently a process of recruitment
begins in which two types of phenomena come into play:
- a first mechanism in which the scout-recruiter and/or the trail stimulate
individuals that were till then inactive to go out of the nest;
- and a second one where the trail guides the individuals so recruited to
the food source, entailing that as recruited individuals will sooner or later
become recruiters in their turn the process will be gradually amplified and a
substantial traffic will be established along the trail.
Consider now the more realistic situation where the colony disposes of
several food sources. A minimal configuration allowing one to study how it
then copes with the problem of choice to which it is confronted is depicted in
Fig. 1.8a: two equivalent paths leading from the nest to two simultaneously
present identical food sources. In a sufficiently numerous colony after a short
period of equal exploitation a bifurcation, in the precise sense of Fig. 1.1,
is then observed marking a preferential exploitation of one of the sources
relative to the other, to its exhaustion (Fig. 1.8b). Thereafter the second
source is fully colonized and its exploitation is intensified. When the colony
is offered two sources with different sugar concentrations and the richest
source is discovered before or at the same time as the poorer one, it is most
heavily exploited. But when it is discovered after the poorer one, it is only
weakly exploited. This establishes the primordial importance of the long-
range cooperativity induced by the presence of the trail.
It is tempting to conjecture that far from being a curiosity the above
phenomenon, which shares with the Rayleigh-Bénard instability the prop-
erty of spontaneous emergence of an a priori highly unexpected behavioral
pattern, is prototypical of a large class of systems, including socio-economic
phenomena in human populations (see also Sec. 1.4.4 below). The key point
lies in the realization that nature offers a bottom-up mechanism of organi-
zation that has no recourse to a central or hierarchical command process
as in traditional modes of organization. This mechanism leads to collective
decisions and to problem solving on the basis of (a) the local information
available to each “agent”; and (b) its implementation on global level without
the intervention of an information-clearing center. It opens the way to a
host of applications in the organization of distributed systems of interacting
agents as seen, for example, in communication networks, computer networks
and networks of mobile robots or static sensory devices. Such analogy-driven
considerations can stimulate new ideas in a completely different context by
serving as archetypes. They are important elements in the process of model

Fig. 1.8. (a) A typical experimental set up for the study of the process of
choice between two options. (b) Time evolution of the number of individuals
(here ants of the species Lasius niger) exploiting two equivalent (here 1 molar
saccharose rich) food sources offered simultaneously, in an experimental set
up of the type depicted in Fig. 1.8(a).

building -an essential part of the research in complex systems- in situations
in which the evolution laws of the variables involved may not be known to
any comparable degree of detail as in physical systems.
1.4.4 Human systems
We now turn to a class of complexity related problems in which the actors
involved are human beings. Here the new element that comes into play is
the presence of such concepts as strategy, imitation, anticipation, risk assess-
ment, information, history, quite remote at first sight from the traditional
vocabulary of physical science. The expectation would be that thanks to the
rationality underlying these elements, the variability and unpredictability
should be considerably reduced. The data at our disposal show that this is
far from being the case. Human systems provide, in fact, one of the most au-
thentic prototypes of complexity. They also constitute a source of inspiration
for raising number of new issues, stimulating in turn fundamental research
in the area of complex systems.
A first class of instances pertains to cooperativity (imitation) driven socio-
cultural phenomena. They usually lead to bifurcations very similar to those
considered in the previous subsection in which the variability inherent in
the dynamics of the individuals is eventually controlled to yield an emergent
pattern arising through a sharp transition in the form of a bifurcation. The
propagation of rumors or of opinions is the most classical example in this
area, but in recent years some further unexpected possibilities have been
suggested, such as the genesis of a phonological system in a human society.
Ordinarily, the inherent capacity of humans to emit and recognize sounds and
to attribute them to objects is advanced as the most plausible mechanism
of this process. On the other hand, consider a population of N individuals
capable to emit M sounds to designate a given object. When two individuals
pronouncing sounds i and j meet, each one of them can convince, with cer-
tain probabilities, the other that his sound is more appropriate to designate
the object. This switches on a cooperativity in the process of competition
between the options available very similar to that between the two trails in
Fig. 1.8a, leading to the choice of one of them by the overwhelming part
of the population (being understood that N is large enough). This scenario
opens interesting perspectives, which need to be implemented by linguistic
analyses and real-time experiments.
Competition between different options is also expected to underlie the
origin of a variety of spatial patterns and organizational modes observed in
human systems. An example is provided by the formation and the evolution
of urban structures, as certain areas specialize in specific economic activities

and as residential differentiation produces neighborhoods differing in their
living conditions and access to jobs and services. In many cases this occurs
as a spontaneous process of endogenous origin. In addition to this evolu-
tionary scenario central planning may be present as well and provide a bias
in the individual decision making. It is, however, most unlikely that under
present conditions it will supersede the bottom-up mechanism operating in
complex systems: the chance of a modern Deinokrates or a modern Constan-
tine the Great designing from scratch an Alexandria or a Constantinople-like
structure are nowadays practically nil.
It is, perhaps, in the domain of economic and financial activities that the
specificity of the human system finds its most characteristic expression. In
addition to steps involving self-organization and emergence through bifur-
cation one witnesses here the entrance in force of the second fingerprint of
complexity, namely, the intertwining of order and disorder. This raises in
turn the problem of prediction in a most acute manner. The economics of
the stock market provides a striking example. On October 19, 1987 the Dow
Jones index of New York stock exchange dropped by 22.6%. This drop, the
highest registered ever in a single day, was preceded by three other substantial
ones on October 14, 15, 16. Impressive as they are, such violent phenomena
are far from being unique: financial history is full of stock market crises such
as the famous October 1929 one in which on two successive days the values
were depreciated cumulatively by 23.1%.
The first reaction that comes to mind when witnessing these events is
that of irrationality yet, much like in our discussion of subsection 1.4.2, the
evidence supports on the contrary the idea of perfectly rational attitudes be-
ing at work. Ideally, in a market a price should be established by estimating
the capacity of a company to make benefits which depends in turn on readily
available objective data such as its technological potential, its developmental
strategy, its current economic health and the quality of its staff. In reality,
observing the market one realizes that for a given investor these objective
criteria are in many instances superseded by observing the evolution of the
index in the past and, especially, by watching closely the attitude of the
other investors at the very moment of action. This may lead to strong co-
operative effects in which a price results in from an attitude adopted at a
certain time, and is subsequently affecting (e.g. reinforcing) this very atti-
tude (which was perhaps initially randomly generated). As a matter of fact
this largely endogenous mechanism seems to be operating not only during
major crises but also under “normal” conditions, as illustrated by Fig. 1.9
in which the “real” (full line) versus the “objective” (dashed line) value of a
certain product in the New York stock exchange is depicted for a period of
about 50 years. It may result in paradoxical effects such as the increase of

1930 1940 1950 1960 1970 1980
0
500
1000
1500
2000
year
ind
p
Fig. 1.9. Dow Jones industrial average p and a posteriori estimated rational
price p∗
of the New York stock market during the period 1928 to 1979. Raw
data have been detrended by dividing by the systematic growth factor.
a certain value merely because the investors anticipate at a certain moment
that this is indeed going to happen, though it has not happened yet! In this
logic the product that is supposed to guarantee this high value might even
be inferior to others, less well quoted ones. That such a priori unexpected
events actually occur with appreciable probability is reminiscent of the com-
ments made in subsections 1.4.1 and 1.4.3 in connection with the emergence
of Rayleigh-Bénard cells and pheromone trails. It suggests that key mani-
festations of economic activities are the result of constraints acting on the
system and activating intrinsic nonlinearities, as a result of which the con-
cept of economic equilibrium often becomes irrelevant. Of equal importance
is also the variability of the individual agents, reflected by the presence of
different goals and strategies amongst them (cf. also Sec. 3.7).
It is important to realize that the speculative character of the process
underlying Fig. 1.9 coexists with regular trends reflected by the generally
admitted existence of economic cycles. While the latter are manifested on
a rather long time scale, the behavior on a wide range covering short to
intermediate scales seems rather to share the features of a scale free process.
Again the situation looks similar in this respect to that encountered in the
previous subsections. An analysis of the range of variability normalized by

its standard deviation confirms this, with Hurst exponents H close to 0.5
for products most easily subject to speculation, and higher for products that
are less negotiable. As mentioned in connection with subsection 1.4.2 this
implies that the corresponding processes are, respectively, uncorrelated and
subjected to long range correlations.
An alternative view of financial fluctuations is provided by the construc-
tion of their histograms from the available data. Let Pt be the present price
of a given stock. The stock price return rt is defined as the change of the
logarithm of the stock price in a given time interval ∆t, rt = lnPt − lnPt−∆t.
The probability that a return is (in absolute value) larger than x is found
empirically to be a power law of the form
P(|rt| x) ≈ x−γt
(1.1)
with γt ≈ 3. This law which belongs to the family of probability distributions
known as Pareto distributions holds for about 80 stocks with ∆t ranging from
one minute to one month, for different time periods and for different sizes
of stocks. It may thus be qualified as “universal” in this precise sense. The
scale invariant (in ∆t and in size) behavior that it predicts in the above
range suggests that large deviations can occur with appreciable probability,
much more appreciable from what would be predicted by an exponential or
a Gaussian distribution. As a matter of fact such dramatic events as the
1929 and 1987 market crashes conform to this law. Surprisingly, Pareto’s
law seems also to describe the distribution of incomes of individuals in a
country, with an exponent that is now close to 1.5.
In an at first sight quite different context, power laws concomitant to self-
similarity and scale free behavior are also present whenever one attempts to
rank objects according to a certain criterion and counts how the frequency
of their occurrence depends on the rank. For instance, if the cities of a given
country are ranked by the integers 1, 2, 3,... according to the decreasing
order of population size, then according to an empirical discovery by George
Zipf the fraction of people living in the nth city varies roughly as
P(n) ≈ n−1
(1.2)
Zipf has found a similar law for the frequency of appearance of words in the
English prose, where P(n) represents now the relative frequency of the nth
most frequent word (“the”, “of”, “and” and “to” being the four successively
more used words in a ranking that extends to 10 000 or so).
Eq. (1.2) is parameter free, and on these grounds one might be tempted to
infer that it applies universally to all populations and to all languages. Benoı̂t
Mandelbrot has shown that this is not the case and proposed a two-parameter

extension of Zipf’s law accounting for the differences between subjects and
languages, in the form
P(n) ≈ (n + n0)−B
(1.3)
where n0 plays the role of a cutoff.
1.5 Summing up
The fundamental laws of nature governing the structure of the building blocks
of matter and their interactions are deterministic: a system whose state
is initially fully specified will follow a unique course. Yet throughout this
chapter we have been stressing multiplicity as the principal manifestation of
complexity; and have found it natural -and necessary- to switch continuously
on many occasions between the deterministic description of phenomena and
a probabilistic view.
Far from reflecting the danger of being caught in a contradiction already
at the very start of this book this opposition actually signals what is going to
become the leitmotiv of the chapters to come, namely, that when the funda-
mental laws of nature are implemented on complex systems the deterministic
and the probabilistic dimensions become two facets of the same reality: be-
cause of the limited predictability of complex systems in the sense of the
traditional description of phenomena one is forced to adopt an alternative
view, and the probabilistic description offers precisely the possibility to sort
out regularities of a new kind; but on the other side, far from being applied
in a heuristic manner in which observations are forced to fit certain a priori
laws imported from traditional statistics, the probabilistic description one
is dealing with here is intrinsic in the sense that it is generated by the un-
derlying dynamics. Depending on the scale of the phenomenon, a complex
system may have to develop mechanisms for controlling randomness in order
to sustain a global behavioral pattern thereby behaving deterministically or,
on the contrary, to thrive on randomness in order to acquire transiently the
variability and flexibility needed for its evolution between two such configu-
rations.
Similarly to the determinism versus randomness, the structure versus
dynamics dualism is also fading as our understanding of complex systems is
improving. Complex systems shape in many respects the geometry of the
space in which they are embedded, through the dynamical processes that
they generate. This intertwining can occur on the laboratory time scale as
in the Rayleigh-Bénard cells and the pheromone trails (1.4.1, 1.4.3); or on

the much longer scale of geological or biological evolution, as in e.g. the
composition of the earth’s atmosphere or the structure of biomolecules.
Complexity is the conjunction of several properties and, because of this,
no single formal definition doing justice to its multiple facets and manifesta-
tions can be proposed at this stage. In the subsequent chapters a multilevel
approach capable of accounting for these diverse, yet tightly intertwined el-
ements will be developed. The question of complexity definition(s) will be
taken up again in the end of Chapter 4.

Chapter 2
Deterministic view
2.1 Dynamical systems, phase space,
stability
Complexity finds its natural expression in the language of the theory of dy-
namical systems. Our starting point is to observe that the knowledge of
the instantaneous state of a system is tantamount to the determination of
a certain set of variables as a function of time: x1(t), ..., xn(t). The time
dependence of these variables will depend on the structure of the evolution
laws and, as stressed in Sec. 1.2, on the set of control parameters λ1, ..., λm
through which the system communicates with the environment. We qualify
this dependence as deterministic if it is of the form
xt = Ft
(x0, λ) (2.1)
Here xt is the state at time t ; x0 is the initial state, and Ft
is a smooth
function such that for each given x0 there exists only one xt. For compactness
we represented the state as a vector whose components are x1(t), ..., xn(t). Ft
is likewise a vector whose components F1(x1(0), ...xn(0); t, λ), ..., Fn(x1(0), ...
xn(0); t, λ) describe the time variation of the individual x0
s.
In many situations of interest the time t is a continuous (independent)
variable. There exists then, an operator f determining the rate of change of
xt in time :
Rate of change of xt in time = function of the xt and λ
or, more quantitatively
∂x
∂t
= f(x, λ) (2.2)
As stressed in Secs 1.2 and 1.3 in a complex system f depends on x in a
25

nonlinear fashion, a feature that reflects, in particular, the presence of coop-
erativity between its constituent elements.
An important class of complex systems are those in which the variables
xt depend only on time. This is not a trivial statement since in principle the
properties of a system are expected to depend on space as well, in which case
the xt’s define an infinite set (actually a continuum) of variables constituted
by their instantaneous values at each space point. Discounting this possibility
for the time being (cf. Sec. 2.2.2 for a full discussion), a very useful geometric
representation of the relations (2.1)-(2.2) is provided then by their embedding
onto the phase space. The phase space, which we denote by Γ, is an abstract
space spanned by coordinates which are the variables x1, ..., xn themselves.
An instantaneous state corresponds in this representation to a point Pt and
a time evolution between the initial state and that at time t to a curve γ, the
phase trajectory (Fig. 2.1). In a deterministic system (eq. (2.1)) the phase
trajectories emanating from different points will never intersect for any finite
time t, and will possess at any of their points a unique, well-defined tangent.
Fig. 2.1. Phase space trajectory γ of a dynamical system embedded in a
three-dimensional phase space Γ spanned by the variables x1, x2 and x3.
The set of the evolutionary processes governed by a given law f will be
provided by the set of the allowed phase trajectories, to which we refer as
phase portrait. There are two qualitatively different topologies describing

Deterministic View 27
these processes which define the two basic classes of dynamical systems en-
countered in theory and in practice, the conservative and the dissipative
systems.
In the discussion above it was understood that the control parameters λ
are time independent and that the system is not subjected to time-dependent
external forcings. Such autonomous dynamical systems constitute the core
of nonlinear dynamics. They serve as a reference for identifying the different
types of complex behaviors and for developing the appropriate methodologies.
Accordingly, in this chapter we will focus entirely on this class of systems.
Non-autonomous systems, subjected to random perturbations of intrinsic or
environmental origin will be considered in Chapters 3, 4 and onwards. The
case of time-dependent control parameters will be briefly discussed in Sec.
6.4.3.
2.1.1 Conservative systems
Consider a continuum of initial states, enclosed within a certain phase space
region ∆Γ0. As the evolution is switched on, each of these states will be
the point from which will emanate a phase trajectory. We collect the points
reached on these trajectories at time t and focus on the region ∆Γt that they
constitute. We define a conservative system by the property that ∆Γt will
keep the same volume as ∆Γ0 in the course of the evolution, |∆Γt| = |∆Γ0|
although it may end up having a quite different shape and location in Γ
compared to ∆Γ0. It can be shown that this property entails that the phase
trajectories are located on phase space regions which constitute a continuum,
the particular region enclosing a given trajectory being specified uniquely by
the initial conditions imposed on x1, ..., xn. We refer to these regions as
invariant sets.
A simple example of conservative dynamical system is the frictionless
pendulum. The corresponding phase space is two-dimensional and is spanned
by the particle’s position and instantaneous velocity. Each trajectory with
the exception of the equilibrium state on the downward vertical is an ellipse,
and there is a continuum of such ellipses depending on the total energy
(a combination of position and velocity variables) initially conferred to the
system.
2.1.2 Dissipative systems
Dissipative systems are defined by the property that the dynamics leads to
eventual contraction of the volume of an initial phase space region. As a
result the invariant sets containing the trajectories once the transients have

died out are now isolated objects in the phase space and their dimension is
strictly less than the dimension n of the full phase space. The most important
invariant sets for the applications are the attractors, to which tend all the
trajectories emanating from a region around the attractor time going on
(Fig. 2.2). The set of the trajectories converging to a given attractor is its
attraction basin. Attraction basins are separated by non-attracting invariant
sets which may have a quite intricate topology.
Fig. 2.2. Attraction basins in a 3-dimensional phase space separated by an
unstable fixed point possessing a 2-dimensional stable manifold and a one-
dimensional unstable one.
The simplest example of dissipative system is a one-variable system, for
which the attractors are necessarily isolated points. Once on such a point the
system will no longer evolve. Point attractors, also referred as fixed points,
are therefore models of steady-state solutions of the evolution equations.
A very important property providing a further characterization of the so-
lutions of eqs (2.1)-(2.2) and of the geometry of the phase space portrait is
stability, to which we referred already in qualitative terms in Sec. 1.3. Let
γs be a “reference” phase trajectory describing a particular long-time behav-
ior of the system at hand. This trajectory lies necessarily on an invariant
set like an attractor, or may itself constitute the attractor if it reduces to
e.g. a fixed point. Under the influence of the perturbations to which all real
world systems are inevitably subjected (see discussion in Secs 1.2 and 1.3)
the trajectory that will in fact be realized will be a displaced one, γ whose
instantaneous displacement from γs we denote by δxt (Fig. 2.3). The ques-
tion is, then, whether the system will be able to control the perturbations or,

Deterministic View 29
Fig. 2.3. Evolution of two states on the reference trajectory γs and on a per-
turbed one γ separated initially by a perturbation δx0, leading to a separation
δxt at time t.
on the contrary, it will be removed from γs as a result of their action. These
questions can be formulated more precisely by comparing the initial distance
|δx0| between γ and γs (where the bars indicate the length (measure) of the
vector δx0) and the instantaneous one |δxt| in the limit of long times. The
following situations may then arise:
(i) For each prescribed “level of tolerance”, for the magnitude of |δxt|,
it is impossible to find an initial vicinity of γs in which |δx0| is less than a
certain δ, such that |δxt| remains less than for all times. The reference
trajectory γs will then be qualified as unstable.
(ii) Such a vicinity can be found, in which case γs will be qualified as
stable.
(iii) γs is stable and, in addition, the system damps eventually the per-
turbations thereby returning to the reference state. γs will then be qualified
as asymptotically stable.
Typically, these different forms of stability are not manifested uniformly
in phase space: there are certain directions around the initial state x0 on
the reference trajectory along which there will be expansion, others along
which there will be contraction, still other ones along which distances neither
explode nor damp but simply remain in a vicinity of their initial values. This
classification becomes more transparent in the limit where |δx0| is taken to be
small. There is a powerful theorem asserting that instability or asymptotic

Fig. 2.4. Decomposition of an initial perturbation along the stable and un-
stable manifolds us and uunst of the reference trajectory γs.
stability in this limit of linear stability analysis guarantee that the same
properties hold true in the general case as well.
Figure 2.4 depicts a schematic representation of the situation. A generic
small perturbation δx0 possesses non-vanishing projections on directions us
and uunst along which there are, respectively, stabilizing and non-stabilizing
trends. One of the us’s lies necessarily along the local tangent of γs on x0, the
other us and uunst’s being transversal to γs. The hypersurface they define is
referred as the tangent space of γs, and is the union of the stable and unstable
manifolds associated to γs.
Analytically, upon expanding Ft
in (2.1) around x0 and neglecting terms
beyond the linear ones in |δx0| one has
δxt =
∂Ft
(x0, λ)
∂x0
· δx0
= M(t, x0) · δx0 (2.3)
Here M has the structure of an n×n matrix and is referred as the fundamental
matrix. An analysis of this equation shows that in the limit of long times
|δxt| increases exponentially along the uunst’s, and decreases exponentially or
follows a power law in t along the us’s. To express the privileged status of this
exponential dependence it is natural to consider the logarithm of |δxt|/|δx0|
divided by the time t,
σ(x0) =
1
t
ln
|δxt|
|δx0|
(2.4)
in the double limit where |δx0| tends to zero and t tends to infinity. A more
detailed description consists in considering perturbations along the uj’s and
evaluating the quantities σj(x0) corresponding to them. We refer to these

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[329]
Ononis Natrix. Yellow-Flowered Rest-Harrow.
Class and Order.
Diadelphia Decandria.
Generic Character.
Cal. 5-partitus: laciniis linearibus. Vexillum striatum. Legumen turgidum
sessile. Filamenta connata absque fissura.
Specific Character and Synonyms.
ONONIS Natrix pedunculis unifloris aristatis, foliis ternatis viscosis stipulis
integerrimis caule fruticoso. Linn. Syst. Vegetab. ed. 14. Murr. p. 653.
Ait. Kew. v. 3. p. 24.
ANONIS viscosa spinis carens lutea major. Bauh. Pin. 389.

No. 329
The Ononis Natrix, a plant usually to be met with in all general
collections of greenhouse plants, is a native of Spain, and the South
of France, where it is said to grow wild in the corn-fields.

The general practice sanctioned by that of Mr. Aiton, is to consider
this species as tender; Mr. Miller says it is very hardy, and
recommends it to be planted in the open border, a treatment likely
to suit it in mild winters; there is, however, one part of his account
evidently erroneous, he describes the root as perennial, and the
stem as herbaceous, this is not only contrary to Linnæus's specific
description, but to fact, the stalk being undoubtedly shrubby.
As this plant in the course of a year or two is apt to grow out of
form, it is advisable either to renew it frequently by seed, which it
produces in abundance, or to keep it closely cut in.
It flowers from the middle of summer till towards the close, and is
propagated readily either by seeds or cuttings.
Is no novelty in this country, having been cultivated by Mr. James
Sutherland in 1683[1].

[330]
Sida Cristata. Crested Sida.
Class and Order.
Monadelphia Polyandria.
Generic Character.
Cal. simplex, angulatus. Stylus multipartitus. Caps. plures 1-spermæ.
SIDA cristata foliis angulatis, inferioribus cordatis, superioribus
panduriformibus, capsulis multilocularibus. Sp. Pl. ed. 3. p. 964. Syst.
Veg. ed. 14. Murr. p. 623. Ait. Kew. v. 2. p. 444. Cavanill. Diss. 1. t.
11. f. 2.
ABUTILON Lavateræ flore, fructu cristato. Dill. Elth. t. 2.
ANODA hastata. Linn. Syst. Nat. ed. Gmel. p. 1040.

No. 330
Dillenius has figured and described this plant in his Hortus
Elthamensis as an Abutilon: Linnæus in his Sp. Pl. has ranked it with
the Sida's, in which he has been followed by Prof. Murray, Messrs.

Aiton and Cavanille; but Prof. Gmelin, in the last edition of Linnæus'
Syst. Nat. has made another new genus of it, by the name of Anoda;
as his reasons for so doing are by no means cogent, we join the
majority in continuing it a Sida.
It flowered in the garden of Mr. Sherard, at Eltham, in 1725, and
was introduced from Mexico, where it is a native: Mr. Aiton considers
it a stove plant, as he does the Tropæolum majus, and other natives
of South-America; strictly speaking they may be such, but if raised
early, and treated like other tender annuals, this plant will flower and
ripen its seeds in the open ground, as we have experienced at
Brompton.
It grows to the height of three feet, or more, producing during
the months of July and August a number of blossoms in succession,
which are large and shewy; the stigmata in this flower are curious
objects, resembling the heads of Fungi in miniature.

[331]
Kalmia Angustifolia. Narrow-Leav'd Kalmia.
Class and Order.
Decandria Monogynia.
Generic Character.
Cal. 5-partitus. Cor. hypocrateriformis, limbo subtus quinque corni. Caps. 5-
locularis.
KALMIA angustifolia foliis lanceolatis, corymbis lateralibus. Linn. Syst. Veget.
ed. 14. Murr. p. 404. Ait. Kew. v. 2. p. 64. Gronov. Fl. Virg. p. 65.
CHAMÆDAPHNE sempervirens, foliis oblongis angustis, foliorum fasciculis
oppositis e foliorum alis. Catesb. Carol. app. t. 17. f. 1.
LEDUM floribus bullatis fasciculatim ex alis foliorum oppositis nascentibus,
foliis lanceolatis integerrimis glabris. Trew. Ehr. t. 38.

No. 331
In this work we have already given three different species of
Kalmia, two commonly, and one more rarely cultivated with us, we
mean the hirsuta, and which indeed we are sorry to find is scarcely

to be kept alive in this country by the most skilfull management; to
these we now add another species, a native also of North-America,
introduced by Peter Collinson, Esq. in 1736, two years after he had
introduced the latifolia; Catesby mentions its having flowered at
Peckham in 1743; it is a low shrub, rarely rising above the height of
two feet, growing spontaneously in swampy ground, and flowering
with us from May to July; there are two principal varieties of it, one
with pale and another with deep red flowers; these two plants differ
also in their habits, the red one, the most humble of the two, not
only produces the most brilliant flowers, but those in greater
abundance than the other; Mr. Whitley, who has these plants in
great perfection, assures me that it usually blows in the autumn as
well as summer.
This shrub is extremely hardy, thriving best in bog earth, and is
propagated most commonly by layers.
Like the latifolia, it is regarded in America as poisonous to sheep.

[332]
Oenothera Fruticosa. Shrubby Oenothera.
Class and Order.
Octandria Monogynia.
Generic Character.
Calix 4-fidus. Petala 4. Capsula cylindrica infera. Semina nuda.
OENOTHERA fruticosa foliis lanceolatis subdentatis, capsulis pedicellatis
acutangulis, racemo pedunculato. Linn. Syst. Veget. ed. 14. Murr. p.
358. Ait. Kew. v. 2. p. 4. L'Herit. Stirp. nov. t. 2. t. 5.
OENOTHERA florum calyce monophyllo, hinc tantum, aperto. Gron. virg. 42.
LYSIMACHIA lutea caule rubente, foliis salicis alternis nigro maculatis, flore
specioso amplo, vasculo seminali eleganter striato insidente, Clayt. n.
36.

No. 332
Most of the Oenothera tribe are annual, have large yellow flowers,
which open once only, and that in the evening, displaying their
beauty, and exhaling their fragrance at a time which will not admit of

their being much enjoyed; the present species in some respects
deviates from many of the others, the root is perennial, the flowers
which are large and shewy, though they open in the evening, remain
expanded during most of the ensuing day; the flower-buds, the
germen, and the stalk are enlivened by a richness of colour which
contributes to render this species one of the most ornamental and
desirable of the tribe.
It is a hardy perennial, growing to the height of three or four feet,
with us altogether herbaceous, and therefore improperly called
fruticosa; a native of Virginia, flowering from June to August: was
cultivated in 1739 by Mr. Miller.
May be propagated by seeds, by parting of the roots, and also by
cuttings.

[333]
Cerinthe Major. Great Honey-wort.
Class and Order.
Pentandria Monogynia.
Generic Character.
Corollæ limbus tubulato ventricosus: fauce pervia. Semina 2, bilocularia.
CERINTHE major foliis amplexicaulibus, corollis obtusiusculis patulis. Linn.
Syst. Vegetab. ed. 14. Murr. p. 187. Ait. Kew. v. 1. p. 183.
CERINTHE glaber foliis oblongo-ovatis glabris amplexicaulibus, corollis
obtusiusculis patulis. Mill. Dict. ed. 6. 4to.
CERINTHE flore ex rubro purpurascente. Bauh. pin. p. 258.
CERINTHE major. Great Honiewoort. Ger. Herb.

No. 333
Ancient writers on plants, supposing that the flowers of this genus
produced abundantly the material of which bees form their wax,
gave it the name of Cerinthe, which rendered into English would be

wax-flower or waxwort, not honeywort, by which the genus has long
been, and is now, generally called.
Of this genus there are only two species known, the major and
the minor, both happily distinguished by the different form of their
flowers, a part from which it is not common to draw specific
differences, though in some instances they afford the best.
The major varies much, the leaves being sometimes spotted, very
rough, and the flowers of a more yellow hue; this is the sort figured
by Gerard in his Herbal, who mentions its growing in his garden
(1597). Miller considers this as a species but Linnæus, Haller, Aiton,
and others, regard it as a variety; our figure represents the Cerinthe
glaber of Miller.
This is an annual, remarkable for the singular colour of its foliage;
its flowers, though not very brilliant, possess a considerable share of
beauty; both combined render it worthy a place in our gardens,
more especially as it is a plant of easy culture, coming up
spontaneously from self-sown seeds, and being a native of
Switzerland, as well as the more southern parts of Europe, seedling
plants produced in the Autumn rarely suffer by our winters. It
flowers in July, August, and September.

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Foundations of complex systems Nonlinear dynamic statistical physics information and prediction Gregoire Nicolis

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