Complexity Theory

Nonlinear Dynamics and Complex Adaptive Systems

The Complexity Theory class was originally created in 1993 for the Science Research Fellows program, and was taught by C. Fornari of the Biology dept., and H. Brooks of the Physics department. By theoretical discussions and concrete projects, the students in the course examined the dynamical behavior of complex systems ranging from earthquakes, sand piles, volcanoes and chemical reactions, to genetic algorithims, evolution, genomic and neural networks, immune system responses, and whole ecosystems. Any such complex system can be theoretically analyzed in terms of the interactions of large numbers of discrete elements, or participating units, that constitute the system and communicate with each other by defined, logical functions to produce complex, often adaptive, behaviors.

Complexity/Chaos Theory = the non-linear dynamics of complex adaptive systems; it is an attempt to explain the dynamical behavior of complex systems by discovering the laws governing such behavior, which can range from the chaotic to the rigidly ordered, with a critical "edge-of-chaos" region in the middle where order and disorder co-exist. At the edge-of-chaos region, complex dynamical systems become adaptive, solve problems and evolve by mutation and selection. Such systems are called Complex Adaptive Systems, or CAS.

The underlying hypotheses of CAS are predicated upon an idealized mathematical model devised and popularized by (among others) Stuart Kauffman of the Santa Fe Institute for Complexity, and Stephen Wolfram, now the CEO of Wolfram Research at the Univ. of Illinois. Kauffman's model is called the Autonomous, Synchronous, discrete, Boolean Network and can be tested rigorously and exhaustively by computer simulation to uncover regularities and patterns which apply to all complex systems. These Boolean Networks are characterized by certain properties and features which make them particularly useful for analyzing the dynamical behavior of real-world complex systems.

Kauffman also asserts that order can arise spontaneously in complex systems, without any selection operating to move the system in one direction or the other. He calls this well-documented phenomenon "order for free" and it occurs without any violation of the 2nd law of thermodynamics. Complex systems represented by Random (autonomous, synchronous, discrete) Boolean networks with particular properties (low connectivity among the participating elements, and a bias towards a particular logical switching rule, the OR function), show extraordinary spontaneous order.

Kauffman maintains that this spontaneous order is highly susceptible to selective factors in the environments of complex systems, and so the emergence of even more complex systems readily occurs. He hypothesizes with loads of experimental evidence accumulated from both computer simulations with Boolean Networks, and from naturally occurring complex systems such as cellular genomes, that a vast readily available reservoir of spontaneous order not only preceded selection but was a necessary "substrate" for selection. He asserts that the evolution of life was highly probable, and not the extremely improbable event assumed by most.

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Bibliography for Complexity Theory Class

(Books and Scientific American articles)

Kauffman, S.

At Home in the Universe.
 The Origins of Order.

Waldrop, Mitchell M.

Complexity : The Emerging Science at the Edge of Order and Chaos

Holland, John H.

Hidden Order : How Adaptation Builds Complexity

Coveney,Peter and Roger Highfield

Frontiers of Complexity : The Search for Order in a Chaotic World

Casti, John L.

Complexification : Explaining a Paradoxical World Through the Science of Surprise

Cohen, J., and I. Stewart.

The Collapse of Chaos : Discovering  Simplicity in a Complex World

Kelly, K.

Out of Control : The New Biology of Machines, Social Systems and the Economic World

Mitchell, M.

An Introduction to Genetic Algorithms (Complex Adaptive Systems)

Michalewicz, Z.

Genetic Algorithms + Data Structures = Evolution Programs

Wolfram, Stephen

Cellular Automata and Complexity

Wolfram, Stephen

A New Kind of Science

William Roetzheim

Why Things Are: How Complexity Theory answers Life's Toughest Questions

Roger Lewin

Complexity: Life at the Edge of Chaos

Scientific American Articles:
Adapting to Complexity. Jan., 1993
Antichaos and Adaptation. Aug., 1991
Self-Organized Criticality. Jan., 1991
Genetic Algorithms. July, 1992
Artificial Intelligence. Jan., 1990 (two opposing articles)
From Complexity to Perplexity. June, 1995
Computing with DNA. Aug., 1998 

Science Apr 2 1999, Volume 284 No.5411,
"Complex Systems"

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Links to Complexity Theory sites:

 Attractors and Their Types
Complexity Theory - Wikipedia
Complexity - a non-technical introduction
Complexity Theory - Wolfram Institute
Complex Systems Links in the WWW
Complex Systems Modeling
Defining a CAS or Complex Adaptive System
Complex Systems
Serendip Complex Systems
Center for Complex Systems Research
Center for the Study of Complex Systems
Nonlinear Dynamics and Complex Systems Theory (glossary)
 Limit Cycles and Chaos
 Complexity International
 Society for Chaos Theory in Psychology & Life Sciences
 Complexity Pages
 Welcome to the Santa Fe Institute
 Yahoo! - Science:Complex Systems
Complexity Theory and Network Centric Warfare
 Practical Guide to Genetic Algorithms
 DNA computer
 Complex Systems Journal
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