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Welcome to 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.
Coming Soon:
Our 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.
WHAT IS 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.
NONLINEAR DYNAMICAL 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 and graphics.
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Left: Two 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 music. |
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Left: The 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. |
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WHAT 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. |
Above: A cusp catastrophe is a simple, but one of the more useful models of discontinuous change. |
WHAT IS 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.
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Left: The 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). - Agent-based models 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. |
IN PSYCHOLOGY? 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.
Evolutionary economics 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 processes.
WHAT TO 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.