Utility of chaos theory in product development

UTILITY OF CHAOS THEORY IN PRODUCT DEVELOPMENT
Erika Wetzel1
, Tapani Taskinen2
, and Jonathan Cagan1
1
Carnegie Mellon University, USA
2
VTT Industrial Systems, Finland
tapani.taskinen@vtt.fi
Abstract
External as well as internal influences are constantly applying pressure to
product development teams, making the best decisions along the way difficult to
grasp. Although several processes have been proposed for dealing with
uncertainty in new product development, it is not yet known how chaos theory
relates to and can be used to help companies develop better products, on time
and within budget. Traditionally, one of these processes is followed by teams in
hopes of creating successful new products. Due to influences outside of the
team’s control, this process may not be the best one based on the net present
value of the project, budget and time guidelines. It is difficult to know if the
team is making the correct decisions because it is impossible to know the
outcome without an infinitely precise knowledge of the present. To further
understand this uncertainty, we look at common characteristics between chaos
theory and product development. The overall goal is to determine how
understanding chaos theory can improve the effectiveness of product
development.
1. INTRODUCTION
The objective of this research is two fold: to explore how chaos theory relates to new
product development and how this understanding could improve the effectiveness of
product development.
This paper starts by examining literature on chaos theory to explain the properties
observed in chaotic systems that we think may apply to product development. Next,
product development theory is discussed with a focus on the internal and external
influences affecting teams. Finally, discussion of the relationship of chaos theory to
product development and how this understanding can be used to improve the efficiency
and effectiveness of the product development process is presented.
2. CHAOS THEORY
Chaos theory was originally used to attempt to define a structure in aperiodic, nonlinear,
unpredictable systems in physics and mathematics. Although there is no universally
agreed upon definition of chaos, it is defined in the mathematics world as “randomness”
generated by simple deterministic systems. This randomness is a result of the
sensitivity of chaotic systems to the initial conditions (Tsonis, 1992). Similarly, Singh
(2002) defines chaos as an aperiodic, unpredictable behavior arising in a system
extremely sensitive to initial conditions. In other words, chaos is used to describe a
system where the future state of the system cannot be known without precise
measurement at that future time (Peitgen et al., 1992).
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Systems which are traditionally described as erratic and random are now being
explained using chaos theory. For example, market fluctuations, biological populations
and weather forecasting have all looked to chaos theory to provide an explanation for
erratic behavior. Looking specifically at weather patterns, it is obvious that
meteorologists struggle to predict the weather a few days in advance, despite the
extensive analysis and deterministic equations known for atmospheric conditions. For
this reason, weather forecasting has become a central example for modern chaos theory.
2.1 PROPERTIES OF CHAOTIC SYSTEMS
According to Bradley (1990) and Peitgen (1992), there are three properties that are
absolutely necessary in a chaotic system: (1) sensitive dependence to initial conditions,
(2) mixing, (3) dense periodic points. Table 1 displays these properties as well as other
behaviors observed in chaotic systems.
Characteristic Definition
Sensitive Dependence
Starting from very close initial conditions a chaotic system very
rapidly moves to different states.
Mixing You can get everywhere in the system from anywhere
Dense Periodic Points Points throughout the system that yield periodic behavior
Bounded instability System bounded on a global scale, instable locally
Short term predictability/long
term unpredictability
Chaotic systems are predictable only in the very short term
Predictability horizon of a
system
How far in the future one can predict the future state with some
degree of certainty
Causal relationships
Chaotic systems do follow a set of rules, and the system is
indeed deterministic
Feedback Relationship between current and future states of a system
TABLE 1 – CHARACTERISTICS OF CHAOS THEORY
Sensitive dependence to initial conditions means that small disturbances (or errors) in
the initial conditions will grow to produce a difference as large as the signal itself. This
characteristic is central to the definition of chaos by most authors (Tsonis 1992, Peitgen
et. al. 1992, Singh 2002) and is the most well known property of chaotic systems. The
implication of sensitive dependence is that it is not possible to know the future state of a
chaotic system without an infinitely precise picture of the present state. For example,
looking at weather forecasting, it is not possible to know the current state to the
necessary accuracy, due to many things, such as location and accuracy of measuring
devices. It is this characteristic that causes the commonly known phenomenon of
chaotic systems, the butterfly effect, according to which a butterfly flapping its’ wings
in Africa can cause a hurricane in New York two weeks later.
Mixing, according to Peitgen (1992), defines the characteristic that a small interval of
initial values will eventually become spread over the whole interval of the system as
they are iterated. Basically, the idea is that you can get everywhere in the system from
any initial point. Also called transitivity by Bradley (1990), this means that the regions
in a chaotic system are connected.
Periodic points, as defined in mathematics, are conditions which will yield periodic
behavior. In chaotic systems, these points are said to be dense because they are found
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throughout the system’s state space. Because they present a sense of “order”, these
dense periodic points appear out of place in a chaotic system (Bradley, 1990). However,
although periodic points are known to exist in chaotic systems, it is difficult (impossible
on a computer) to detect a periodic point. Because of sensitivity, after one iteration,
rounding errors throw off what would be a periodic orbit, and chaos is observed.
Although the above characteristics are used to define a chaotic system by some authors,
there are other behaviors observed in chaotic systems that appear to be relevant to
product development (see Table 1). As applied in the world of physics, chaos
represents nonlinear systems that do indeed follow rules, and sometimes these rules can
even be described by mathematical equations. However, due to sensitive dependence,
and the inability to get an infinitely precise picture of the present, the system’s behavior
appears irregular and is unpredictable.
Although a chaotic system is locally unstable, there is a global stability due to the
hidden order of the system. It is within this “bounded instability”, the area between
stable equilibrium and complete instability, that chaos is observed (Caulkin, 1995).
Due to the amplification of small errors, the system is locally unstable. However,
looking at the system from a global perspective, there is a sort of stability, as the future
state of the system is known within some boundaries.
On the other hand, chaotic systems are predictable in the very short term, but
unpredictable in the long term (Peitgen et. al., 1992). In the very short term, the error
due to sensitive dependence is relatively low. However, as the error grows, the system
becomes unpredictable in the long term.
In a chaotic system, the predictability horizon of a system defines how far in the
future one can predict the state of the system with some degree of certainty (Peitgen et.
al., 1992). For example, weather forecasts can only be certain to a week or two in
advance. Although more precise measurements improve the forecast, even very
accurate measurements do not extend the length of this horizon. In order to determine
the forecast past the predictability horizon, the system must be measured again as a new
starting point.
Feedback, used generally in mathematics and science as a tool to compare processes
and their resulting behavior, is central to understanding relationships between current
and past states of a system, and is important in chaotic systems. Because chaotic
systems are nonlinear by definition, effects are fed back to the system, sometimes
having drastic effects on system’s state. Caulkin (1995) discusses the nonlinear
relationships in chaos theory as causes become effects and vice versa. Related again to
sensitive dependence, seemingly insignificant variations in chaotic systems are
magnified by positive feedback into huge consequences.
It is important to remember that chaotic systems do have causal relationships and are
not random. Usually, complex systems do not follow a pattern and are more random in
nature. The causal relationships mean that chaotic systems do follow a set of rules, and
that the system is indeed deterministic (Singh, 2002). The unique thing is that this
determinism is about a state of inequillibrium, rather than equilibrium as would
traditionally be expected from a deterministic system; therefore, the rules of behavior
sometimes change as they evolve (Peitgen et. al., 1992). For example, Paul
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Glendinning (1994) compared a pinball machine to a chaotic system. It is nearly
impossible to produce the exact same scores consistently on a pinball machine, even
though it functions on a simple display of Newton’s deterministic laws. Small
differences in the balls motion are magnified as the ball continues its path to the point
that consistent scores become impossible.
2.2 APPLICATIONS OF CHAOS THEORY
Chaos theory has been used to explain complex, non-linear systems across many
disciplines, ranging from social work to environmental crisis in the rainforest (see Table
2). Several authors have recognized and used chaos in their respective field to introduce
order or understanding in areas that are not traditionally understood.
Application Area How Applied
Social Work
Increase the understanding of social workers (Bolland and
Atherton, 1999)
Environmental Disasters Simplify the concerns to key issues (Kakonge, 2001)
Project Management Simplify the project to a few key principles (Caulkin, 1995)
Management Theory
Managers need to be prepared for chaos (Singh and Singh,
2002)
Educational Administration
Increase the understanding and practice of leadership in
organizations (Galbraith, 2004)
TABLE 2 – APPLICATIONS OF CHAOS THEORY
Bolland and Atherton (1999) use chaos theory to explain the complicated relationships
involved in the field of social work. Although they conclude that chaos theory does not
imply a distinctive practice model for social work, understanding of chaos theory does
transform the social worker’s understanding of human relationships and thus does have
a profound effect (Bolland and Atherton, 1999).
In other fields of work, researchers use the application of chaos theory as a means to
bring order by simplifying the system (Kakonge, 2001, Caulkin, 1995). In the field of
environmental disasters, Kakonge (2001) maps out the key problems and their
relationships. Although this appears complex and random, he shows that he can
simplify this to ten key points that are all interrelated (Kakonge, 2001). Similarly,
Caulkin (1995) discusses simplifying project management to a few key principles that
adapt to changing conditions swiftly and organically. This relates to the “boids”
phenomenon, where computer generated birds would start at a random distribution and
assemble themselves into flocks, based on the simple rules (Caulkin, 1995).
Singh and Singh (2002) relate chaos theory to modern management theory. They
conclude that chaos theory provides a theory to explain events in project management,
and project managers must be prepared for subtle, non-linear influences on a regular
basis (Singh and Singh 2002). Galbraith (2004) discusses the use of chaos theory in
organizational leadership, specifically around educational administration.
3. PRODUCT DEVELOPMENT
Product development is a driving force across corporate success today. On average, 33
percent of company sales come from new products, defined as those not sold 5 years
ago (Cooper, 2001). It is clear that new product innovation is a driver for corporate
success. Rigorous international competition, the explosion of market segments and
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niches, and accelerating technological change has created a challenging environment for
product development teams and corporations to navigate (Wheelwright and Clark,
1992).
Speed to market, coupled with fluid markets has required teams to speed up
development work and be more flexible in their process. Regardless what process is
followed, the fact remains that projects are often behind schedule, produced out of
budget and initial project goals are at times not reached. The question remains, how can
product development teams control these important problems to ensure products are
released on time and within budget at the end of the development period?
Several processes for product development have been proposed to answer this question.
Specifically, authors in product development have discussed uncertainties and risk
involved with product development. Several authors have recognized the uncertainty
involved in product development but the utility of using chaos theory has not been
explored.
The Stage-Gate process developed by Cooper (2001) focuses on providing a structured
development in seven stages, with specific activities and approval criteria that must be
reached at each gate before moving to the next phase. Cooper describes the risk in
product development as a necessity with innovation. There are two components to risk:
amount at stake (money invested) and uncertainties. Cooper explains that at the start of
a product development project, the amount at stake is low but uncertainties are high
(technical feasibility and market success are huge question marks). As more is put at
stake in the project, the uncertainties must decrease at the same rate in order to manage
risk. Cooper proposes the Stage-Gate process to help teams navigate this risk to ensure
a successful product.
Wheelwright and Clark (1992) provide a management framework for managing the
product development process with focus on project organization, management, review
and corrections. They talk about certainty in product development as an obstacle to
achieving high-quality development. This is due to the fact that product development
must predict the future, which is often several years away. Additional complexity is
added to product development when the actual product complexity is considered. Even
relatively simple products, such as a Bic pen, are manufactured in complex
environments. Wheelwright and Clark also touch on the added difficulty when a
product development project must interact with the corporation managing it. Often,
several projects are developed at the same time and must share resources.
Cagan and Vogel (2002) propose a user centered integrated product development
process that tackles many of the uncertainties in traditional development activities.
They argue that allocating more resources to the front end of development will help
eliminate many of the problems later in the process. Because upstream product
specifications often lacks understanding of the needs, wants and desires of the target
market, downstream detailed design runs into expensive redesign and complex product
patching. Cagan and Vogel instead show that more up front research reduces
downstream uncertainty.
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3.1 PRODUCT DEVELOPMENT PROCESS
The product development process presented by Cagan and Vogel (2002) is split into
seven phases, as shown in Table 3: Problem Identification (Identifying and
Understanding Opportunities), Concept Generation, Transition, Product Refinement,
Production Prototypes and Launch Preparation.
Phase Action Items Uncertainty
Identify Opportunities
Determine Product Opportunity
Identify Target Markets
Identify Potential Stakeholders and Advisors
Understanding Opportunities
In-Depth understanding of the user
Product Characteristics and Constraints
Concept Generation
Product Concept
Visual and Functional Prototypes
Clear Market Definition
Transition: Concept Refinement
Form and Functional Models
Manufacturing Plan
Clear Marketing Plan, including financials
Product Refinement
Refined Concept
Prototype Testing
Supplier Recognition
Production Prototypes
Prototypes to prepare for volume manufacture
Packaging, Final design and manufacture decisions
Launch Preparation
Production Ramp-Up
Marketing Implementation
Post-Launch Review
TABLE 3 – PRODUCT DEVELOPMENT PROCESS
The product development process can be described as a funnel (see Table 3) where in
the beginning there is greater uncertainty which, if the process is followed properly, will
decrease as the project progresses (Cagan and Vogel, 2002, Cooper, 2001, Wheelwright
and Clark, 1992). At each phase, although the overall uncertainty in product definition
decreases, unexpected events continue to occur that can push a project off schedule or
out of budget.
4. INTERNAL AND EXTERNAL INFLUENCES IN PRODUCT DEVELOPMENT
Throughout the product development process, teams are put under pressure from
external and internal influences that are both controllable and uncontrollable (see Table
4). Internal influences are usually controllable by the company as a whole, if not
controllable by the team. On the other side, external influences are usually difficult to
control for the company and the team. In the table, disciplines that are most influenced
or responsible are noted; P = Project Management, E = Engineering, D = Design, M =
Marketing. However, it is important to note that the whole team is influenced by these
influences and management of these influences is ultimately a team responsibility.
Internal controllable pressures facing the product development team include those of
timing and budget constraints, resource allocation, team dynamic and manufacturing
capability. Setting a realistic schedule and budget is important to ensure all the
necessary work is completed correctly. Resource allocation is a challenge across the
company and may be out of the hands of the product development team. For example,
several concurrent projects by one company may be forced to share testing facilities,
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prototype labs and even man-hours. The team dynamic can have a positive or negative
influence on the project, depending on the type of relationship team members have. A
team that is not well integrated will suffer due to the lack of understanding of needs
across disciplines. Finally, manufacturing capabilities of the company can play a huge
role in product definition.
TABLE 4 – INFLUENCES TO PRODUCT DEVELOPMENT AND PRIMARY OWNERSHIP
The internal influences which are uncontrollable by the product development team
directly include the corporate culture, brand image and specific supplier constraints.
Corporate culture can be responsible for driving certain projects and designs through
development, sometimes without the proper research. The current brand image, which
can be controlled on a corporate level, is usually out of the control of the development
team though it is important to remember that product development can have a major
influence on future brand image. Coordinating with suppliers and supplier schedules
adds a new dynamic and uncertainty for development teams.
Teams must be prepared to deal with the external controllable influences such as the
introduction of applicable new technologies as well as government regulations. New
technology introduced in the current market or that may be applied to the current market
impact the new product. This impact can be positive if the technology can improve the
current functionality. Additionally, government regulations can sometimes determine
the success of a new product before it is introduced to the market.
External uncontrollable influences include some constant uncertainties such as
competition and changing market demands. Often times, teams must make decisions
based on best guess estimates as to what condition the market will be in at the end of the
development period, including competitive products and new technologies, when the
product is released. Additionally, as customer needs change, it is important that the
team be prepared to make the necessary changes. The user centered development
process recognizes this concern and provides tools to deal with it, e.g., Social,
Economic and Technical factors for trend analysis and the Value Opportunity Analysis
for user analysis (Cagan and Vogel, 2002). Finally, sociopolitical events which fall
completely out of the control of the team, can almost instantly cause major changes to a
market.
Internal Influences P E D M External Influences P E D M
Timing • Regulations • •
Budget • New Technology •
Resource Allocation •
Team Dynamic • • • •
Controllable
Manufacturing Capability • •
Corporate Culture • Socio Political Events •
Brand Image • • • Competition •
Supplier Constraints • • Disruptive Tech •
Dynamic/Global Market •
Uncontrollable
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5. CHAOS THEORY RELATED TO PRODUCT DEVELOPMENT
The characteristics of chaotic systems that relate to the uncertainty and risk involved in
product development are examined in this section. A summary of these relationships is
shown in Table 5.
Characteristics of Chaos Theory Product Development
Sensitive Dependence Dependence on decision making
Local instability/global stability Local uncertainties in the process
Predictability/unpredictability Impossible to predict the future success with accuracy
Causal relationships Casual relationships exist in PD, unpredictable occurrences
Predictability horizon Longer for stable markets, shorter for dynamic markets
Feedback Control, Change and Decision Making
TABLE 5 – CHARACTERISTICS OF CHAOS THEORY RELATED TO PRODUCT
DEVELOPMENT
Just as sensitive dependence to initial conditions is an important part of the definition
of chaos theory, it is central to product development as well. Decisions made
throughout development, no matter how insignificant they appear, could have drastic
effects on the outcome of the product and product success. This is because as each
decision is made, it plays a role in all subsequent decisions.
Due to this “waterfall effect” of decision making, early decisions can have a larger
effect on the final outcome of the product. For example, choosing a target market early
on in the process will play a drastic role throughout the rest of development. For this
reason, it is important for teams to understand the full effects of their decisions and
avoid making rash decisions.
If the measurable state of product development is taken to be product success in the
market, it is always difficult to predict the future success of a project with accuracy at
any given time in development. However, on a more broad perspective, teams have a
general idea of the product function and key target markets through development,
despite the local instability, i.e., uncertainties throughout development. This is similar
to the global stability/local instability analogy in chaos theory. It is important that a
clear product goal is defined and that everybody in the development team understands
this goal, throughout development, to control some of the instability. Product
development falls within a “bounded instability”, that is to say that as decisions are
made in the process, uncertainties should decrease, assuming the project is well
researched, and you know what the final product will be, within a certain boundary.
On the other hand, is the short term predictability, long term unpredictability of
chaos theory (Peitgen et. al., 1992). In product development, a team can understand the
current state of the market and create a successful product, in the short term, but in the
long term, the uncertainties become overwhelming and it’s not possible to guarantee
success. Events can happen that are outside of the team’s control that could completely
change a projects outcome. Although there are relationships between causes and
effects, it is difficult to predict some of these causes. This is what makes predictability
of success difficult.
The predictability horizon of the development of a product depends on the dynamic
market and industry that the product is intended for. A highly volatile market will have
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a short predictability horizon where a more stable market may have a longer
predictability horizon. There is also a predictability horizon related to new technology,
as the development and acceptance can only be predicted to a limited time in the future.
As feedback is an important consideration in chaos theory, it is important to consider
the feedback systems in product development, that is, the systems that take information
and apply them to affect the outcome. In product development, most of the decisions
made play this role. Although they cannot be defined in a mathematical equation, teams
make decisions based on research and analysis which affect product definition and
future decisions. For this reason, decisions are amplified in importance as the project
progress.
6. UTILITY OF CHAOS THEORY IN PRODUCT DEVELOPMENT
Although there is an obvious connection between properties of chaotic systems and
product development, the utility of chaos theory in product development is not obvious.
However, simply understanding that there is a chaos in product development can
improve the effectiveness of teams developing new products. Teams that are prepared
for and embrace changes in their environment are more apt to create successful products,
on time and within budget. The overarching goal of embracing this chaos is controlling
it. That is to say, keep the chaotic boundaries within the control of the team, and
decrease these boundaries continually as the project progresses, but not too early.
Our primary attention in dealing with chaos is focused on the property of sensitive
dependence. To review, sensitive dependence means that small changes in initial
conditions can cause drastic changes on the outcome. This is the most central
characteristic of chaos and all other properties of chaos are a result of sensitive
dependence.
There are several key principles of product development that play an important role in
controlling the sensitive dependence in product development: focus on up-front research,
interdisciplinary representation, common and well defined product goals, complete
understanding of the goal by all team members, good communication between team
members, flexibility in the process, user centered process, speed and efficiency to
market, strategy and process measurement techniques (Cooper, 2001, Wheelwright and
Clark, 1995, Cagan and Vogel, 2002).
Understanding the importance of good product development principles is only part of
the battle in controlling sensitive dependence. We propose a “project forecast” as a tool
to use through development to determine where chaos has and can cause difficulties.
Similar to weather forecasts that can only be predicted a few days in advance with
accuracy, product development forecasts are only accurate for a limited time in the
future. The product development forecast includes the influences to product
development,, e.g., timing, budget, regulations, brand image, competition. It is
important to remember that similar to weather forecasting, it is impossible to predict the
future state with accuracy unless you have precise knowledge of the current state.
Particularly in the initial stages of development, it is important for teams to qualitatively
evaluate the sensitive dependence in each area of the forecast to determine where the
highest risk of chaos is. Additionally, with evaluation of the sensitive dependence in
the project forecast at each milestone, the team can assess the project goals and
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determine if changes need to be made before it is too late. This evaluation will help
teams assess the risk of certain decisions and determine which decisions are more
important. This will also require a flexible process that will allow for changes.
The level of sensitive dependence can and will vary for different aspects of the project
forecast, depending on the project. For example, the team may determine that the
financial situation is stable for the next phase of development, and thus chaos is not
such a concern and decisions that deal with the budget will not be as critical. However,
the same team may determine that the team itself is subject to sensitive dependence.
That is to say, small changes in representation or understanding by team members can
cause major changes in the final product design.
Because the level of sensitive dependence can change throughout the project, we
propose the use of a spider diagram (see Figure 1) to ensure the whole team understands
the importance of sensitive dependence. The forecast consists of each of the influences
affecting product development teams. Each forecasted item appears on a scale from the
center of the diagram outward. At each phase, the team or project manager should
qualitatively assess what level of sensitive dependence that each item has and mark it on
each scale. The marks are connected and as the project progresses, the total area
enclosed, as well as the boundary, should decrease, meaning that sensitive dependence,
and thus chaos, is being controlled. The advantage of the spider diagram is that each
phase can be compared by overlapping the diagram for each one, as the example below
shows.
In practice teams should first evaluate how well they know the current situation in each
of the measurement dimensions. Second, they should consider sensitive dependence in
each of the dimensions, i.e., what are the changes that could occur and have negative
influence on the project, and what is the probability of those changes. Third, the team
should react based on the evaluation results.
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FIGURE 1 – SPIDER DIAGRAM FOR MANAGING SENSITIVE DEPENDENCE
As can be seen from the hypothetical example shown in Figure 1, the forecast was
evaluated at each segment through the first three phases of product development. It is
important to note that in the 3rd
phase of the process, a new competitive product was
introduced, causing the forecast for sensitive dependence to raise for the market and
competitive segments. However, other segments of the forecast remained ‘low’, which
continued to reduced the overall area for that phase, and showed an overall more
controlled phase. Additionally, in the second phase, there was a considerable reduction
in the sensitive dependence, much more than the third phase. This is because the team
focused on understanding their product and market in the first phase, answering many
of the unknowns for the second.
The important consideration in measuring the project forecast is that the team is aware
of the sensitive dependence in each area of the forecast. Although the team may not
know the current state to predict the future accurately, consciously considering each
forecast area will prepare the team for the inevitable sensitive dependence. This
forecast will also point out what areas present the highest concern for sensitive
dependence, and thus the team will need to be flexible to changes that occur within
these areas.
7. CONCLUSIONS
This paper presents chaos theory in relationship to product development. Based on
characteristics in chaos theory that relate to product development, it is proposed that
product development is indeed a chaotic process. As internal and external influences
have a major affect on product development teams, chaotic properties are visible
throughout development. Understanding that the relationship between chaos theory and
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product development exists is the first step to being able to control the process and help
ensure successful products. In addition to understanding that chaos is present, teams
that measure the sensitive dependence in each of the project forecast areas, and act
based on the measurements, will be more prepared and more capable of controlling the
inevitable chaos in the project. This is especially important in the beginning stages of
development, as the number of unknowns is considerably higher and a lot of important
decisions are made in these crucial phases. The better the team knows the current
situation the better they are able to forecast the future project performance, i.e., the
longer the predictability horizon for the project is.
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Bolland, K.A. and Atherton, C.R. 1999, “Chaos Theory: An Alternative Approach to Social Work
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Cagan, J and Vogel, C. 2002, Creating Breakthrough Products: Innovation from Product Planning to
Program Approval. Financial Times Prentice Hall, Upper Saddle River, NJ.
Caulkin, S. 1995, “Chaos Inc.,” Across the Board July/August, pp. 32-36.
Cooper, R.G. 2001, Winning at New Products. Perseus Publishing, Cambridge, MA.
Galbraith, P. 2004, “Organizational Leadership and Chaos Theory,” Journal of Educational
Administration vol. 42, pp. 9-28.
Glendinning, P. 1994, Stability, Instability and Chaos. Cambridge University Press, New York, NY.
Kakonge, J. 2002, “Application of Chaos Theory to Solving the Problems of Social and Environmental
Decline in Lesotho,” Journal of Environmental Management 65, pp. 63-78.
Peitgen, H., et. al. 1992, Chaos and Fractals: New Frontiers of Science Springer-Verlag, New York, NY.
Singh, H. and Singh, A. 2002, “Principles of Complexity and Chaos Theory in Project Execution: A New
Approach to Management,” Cost Engineering December, pp. 23.
Wheelwright, S.C. and Clark, K.B. 1992, Revolutionizing Product Development, Quantum Leaps in
Speed, Efficiency, and Quality. The Free Press, New York.
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Utility of chaos theory in product development

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