Chaos and Complexity
Resources for Students and Teachers (Fonte)
our latest project. Please consider it a work in progress. The goal here is
to provide you with concise descriptions of the concepts underlying Nonlinear
Dynamical Systems Theory and the many avenues down which these studies have
taken us so far. We begin with a guided tour that starts with the basics and
allows you to branch off and explore many of the ideas in further depth in
additional tutorials and readings.
- How do I analyze data
from nonlinear phenomena?
- Are there data sets
I can use?
- Any advice for a graduate
students who is looking for a dissertation?
(c) 2006 Society for Chaos Theory in Psychology & Life Sciences.
Other authors are listed individually where their work is linked.
Chaos and Complexity
Resources for Students and Teachers
here is to provide you with concise descriptions of the concepts underlying
Nonlinear Dynamical Systems Theory and the many avenues down which these studies
have taken us so far. We begin with a guided tour that starts with the basics
and allows you to branch off and explore many of the ideas in further depth
in additional tutorials and readings.
CHAOS THEORY? Chaos theory is one of a set of approaches to study
nonlinear phenomena. Specifically, chaos is a particular nonlinear dynamic
wherein seemingly random events are actually predictable from simple deterministic
equations. Thus, a phenomenon that appears locally unpredictable may indeed
be globally stable, exhibit clear boundaries and display sensitivity to initial
conditions. Small differences in initial states eventually compound to produce
markedly different end states later on in time. The latter property is also
known as The Butterfly Effect. Chaos has a close relationship to other dynamics,
however, such as attractors, bifurcations, fractals, and self-organization.
SYSTEMS THEORY? This is an umbrella term for the study of phenomena such
as attractors, bifurcations, chaos, fractals, catastrophes, and self-organization,
all of which describe systems as they change over time, and there is a large
variety of patterns by which change can occur. The basic concepts are grounded
in mathematical theory and analysis, notably from the realm of differential
topology. We study the products of that math, however, and are proportionately
less concerned with equation-solving or proofs. A good proportion of the ideas
that we work with are encapsulated in some precise yet provocative pictures
of many examples of chaotic attractors. To find out more about how chaos
and chaotic attractors work, please visit some of the tutorials in Menu
1. Dr. Steven Strogatz' video on chaos can also be reached from Menu
1. There you can see some chaos in action with a double pendulum, water
flows, aircraft wing design, electronic signal processing, and chaotic
of a fractal, which is geometric form in a non-integer number of dimensions.
This example came from Dr. J. C. Sprott's Fractal of the Day archives.
Sprott's daily fractal image appears on the home page of this web site.
To learn more about fractals, their relationship to chaos, and their
prominence in nature, please visit some of the tutorials in Menu 1,
particularly the contributions by Dr. Larry Liebovitch.
logistic map is a simple quadratic function with the special property
of describing the change in a system from a fixed point attractor, to
a periodic attractor, to chaos as a bifurcation variable increases in
value. It originated in studies of population dynamics. To learn more
about the logistic map and other properties of attractors and bifurcations,
please visit some of the tutorial listed in Menu 1.
IS CATASTROPHE THEORY? Catastrophes are sudden changes in
events; they are not necessarily bad or unwanted events as the word
"catastrophe" might suggest. Catastrophes involve combinations of attractors
and bifurcations, and are operating in some self-organizing events.
To learn more about the cusp and other catastrophes, please visit tutorials
on Menu 1 by Dr. Lucien Dujardin and Dr. Stephen Guastello. For more
about self-organizing event, please continue.
cusp catastrophe is a simple, but one of the more useful models of discontinuous
COMPLEXITY THEORY? Complexity theory is concerned with self-organizing
phenomena and the effect of one subsystem behavior on another. Self-organization
is sometimes known as "order for free" because systems acquire patterns of
behavior without any input from outside sources. Self-organization is also
the basis underlying agent-based models. Agent-based models are computer simulations
of the behavior of many virtual decision makers that act according to their
own decision rules, but produce a collective pattern if they interact long
enough and often enough.
output from an agent-based model showing the distribution of four types
of entities after a period of agents interacting according to specific
rules. This particular example appear as part of a research paper by
S. Bankes and R. Lempert (2004, "Robust Reasoning with Agent-Based Models,
Nonlinear Dynamics, Psychology, and Life Sciences, 8, 259-278). -
are closely related to several other computational systems that illustrate
self-organization dynamics such cellular automata, genetic algorithms,
and the spin-glass models that formed the basis of NK or Rugged Landscape
models of self-organizing behavior. To learn more about agent-based
modeling and to see some examples in action, please visit some of the
items that appear in Menu 2. Menu 2 also contains a narrative account
of Rugged Landscape dynamics, by Dr. Kevein Dooley and the Sugarscape
model for artificial societies, which was developed by the researchers
ar Brookings Institute.
Yes indeed, just as there are many applications of nonlinear dynamics in physics,
engineering and biology, psychological theory has been transforming rapidly
with the nonlinear influence. It would be correct to call chaos theory in
psychology a new paradigm in psychological thought: Nonlinear theory introduces
new concepts to psychology for understanding change, new questions that can
be asked, and offers new explanations for phenomena. Application areas include
physiological psychology, perception and cognition, motivation and emotion,
group dynamics and leadership, abnormal psychology and psychotherapy, and
organizational behavior, to name a few broad areas. Reading recommendations
appear below in the section, "What to Read."
IN LIFE SCIENCES?
Some of the first applications explored the comparisons of the fractal dimensions
of healthy and unhealthy hearts, lungs, and other organs. Larger dimensions,
which indicate greater complexity, were observed for the healthy specimens.
This finding gave rise to the concept of a complex adaptive system in other
living and social systems.
In the area of cognitive
neuroscience, memory is now viewed as a distributed process that involves
many relatively small groupings of neurons, and that the temporal patterns
of neuron firing contain a substantial amount of information about memory
storage processing. The temporal dynamics of memory experiments can elucidate
how the response to one experimental trial would impact on subsequent responses
and provide information on the cue encoding, retrieval, and decision processes.
One might examine behavioral response times and rates, the transfer of local
electroencephalogram (EEG) field potentials, similar local transfers in functional
magnetic resonance images. In light of the complex relationships that must
exist in processes that are driven by both bottom-up and top-down dynamics,
the meso-level neuronal circuitry has become a new focus of attention from
the perspective of nonlinear dynamics.
Ecology is another area
of nonlinear prominence. The discovery of the logistic map function actually
arose in the context of population dynamics, and birth rates and environmental
carrying capabity change. But see below under ecological economics.
IN ECONOMICS AND POLICY
SCIENCES? The relationship between economic concepts and those from physics,
nonlinear and otherwise, dates back to the early 20th century. The Walrasian
equilibrium and the Nash equilibrium decades later are two notable examples.
Game theory became a key concept in psychological, economic, and other agent-based
dynamics ever since.
Finance is one of four
broad areas of economics that is replete with nonlinear dynamics. Macroeconomics
is another challenging dynamical domain. Some schools of economic thought
attempt to reduce system-level events to the decisions of individual financial
agents. NDS studies in macroeconomics, however, are more consistent with other
schools of economic thought that assume that one cannot assume a particular
system-level outcomes, such as inflation or unemployment from knowledge of
forces acting on individual economic agents.
Ecological economics is
a third broad area that is replete with potential for NDS analysis. Topics
in this group include environmental resource protection and utilization, agricultural
management, and land use and the fractal growth of urban areas. Cellular automata
have been useful tools for studying urban expansion.
is the study of behavior change on the part of microeconomic agents, institutions,
or macroeconomic structures. Here one finds chaos and elementary dynamics,
game-theoretical experiments with evolutionarily stable states, and multi-agent
simulations based on genetic algorithms or related computational strategies.
An important part of policy making in any of these area is the ability of
the decision makers to recognize the "time signatures" of various dynamical
READ? In addition to the classic overviews presented earlier we
have prepared some reading lists for people who would like to explore particular
topics more fully. The Top 40 book list was based on the most frequently cited
books in Nonlinear Dynamics, Psychology, and Life Sciences during the years
1999-2000. Obviously we need to update it, but we'll get around to it soon.
Meanwhile, enjoy the abstracts and articles in NDPLS. We also have a list
of books that were written by SCTPLS authors. We keep this list current with
no limits on the earliest dates. A subset of books by SCTPLS authors that
have been published most recently appears on the NEWS ARCHIVES page of this
web site and in Menu 3 below.