Chaos Introduction

Chaos theory is among the youngest of the sciences, and has rocketed from its obscure roots in the seventies to become one of the most fascinating fields in existence. At the forefront of much research on physical systems, and already being implemented in fields covering as diverse matter as arrhythmic pacemakers, image compression, and fluid dynamics, chaos science promises to continue to yield absorbing scientific information which may shape the face of science in the future.

This is an introduction to chaos theory designed for those who have an interest in chaos theory but are, very much like myself, confined by an as yet underdeveloped mathematical background to theory over mathematics. Seperate archives of seven varied topics in the area of chaos theory are included, with graphics where necessary; a bibliography is also included for independent research.

The rest of this homepage is a quasi-FAQ which covers much of the basic rationale and theory behind chaos theory.

Specific Topics

Bifurcation and Periodicity

Bifurcation Diagram

Physically Inspired Strange Attractors

Lorenz Attractor

Rossler Attractor

Henon Attractor

Astronomically Inspired Strange Attractors

Komogorov-Arnold-Moser Orbits

Fractal Geometry

Sierpinski Triangle

Von Koch Curve

Spleenwort Fern

Chaos Overview

What is chaos theory?

Formally, chaos theory is defined as the study of complex nonlinear dynamic systems. Complex implies just that, nonlinear implies recursion and higher mathematical algorithms, and dynamic implies nonconstant and nonperiodic. Thus chaos theory is, very generally, the study of forever changing complex systems based on mathematical concepts of recursion, whether in the form of a recursive process or a set of differential equations modeling a physical system.

For a more rigorous definition of chaos theory, it is advisable to visit the much more scientific, much more broad-reaching chaos network definition, in their excellent HTML document, What Is Chaos Theory?, also available in a text only version.

Misconceptions about chaos theory

Chaos theory has received some attention, beginning with its popularity in movies such as Jurassic Park; public awareness of a science of chaos has been steadily increasing. However, as with any media covered item, many misconceptions have arisen concerning chaos theory.

The most commonly held misconception about chaos theory is that chaos theory is about disorder. Nothing could be further from the truth! Chaos theory is not about disorder! It does not disprove determinism or dictate that ordered systems are impossible; it does not invalidate experimental evidence or claim that modelling complex systems is useless. The "chaos" in chaos theory is order--not simply order, but the very ESSENCE of order.

It is true that chaos theory dictates that minor changes can cause huge fluctuations. But one of the central concepts of chaos theory is that while it is impossible to exactly predict the state of a system, it is generally quite possible, even easy, to model the overall behavior of a system. Thus, chaos theory lays emphasis not on the disorder of the system--the inherent unpredictability of a system--but on the order inherent in the system--the universal behavior of similar systems.

Thus, it is incorrect to say that chaos theory is about disorder. To take an example, consider Lorenz's Attractor. The Lorenz Attractor is based on three differential equations, three constants, and three initial conditions. The attractor represents the behavior of gas at any given time, and its condition at any given time depends upon its condition at a previous time. If the initial conditions are changed by even a tiny amount, say as tiny as the inverse of Avogadro's number (a heinously small number with an order of 1E-24), checking the attractor at a later time will yield numbers totally different. This is because small differences will propagate themselves recursively until numbers are entirely dissimilar to the original system with the original initial conditions.

However, the plot of the attractor will look very much the same.
Both systems will have totally different values at any given time, and yet the plot of the attractor--the overall behavior of the system--will be the same.

Chaos theory predicts that complex nonlinear systems are inherently unpredictable--but, at the same time, chaos theory also insures that often, the way to express such an unpredictable system lies not in exact equations, but in representations of the behavior of a system--in plots of strange attractors or in fractals. Thus, chaos theory, which many think is about unpredictability, is at the same time about predictability in even the most unstable systems.

How is chaos theory applicable to the real world?

Everyone always wants to know one thing about new discoveries--what good are they? So what good is chaos theory?

First and foremost, chaos theory is a theory. As such, much of it is of use more as scientific background than as direct applicable knowledge. Chaos theory is great as a way of looking at events which happen in the world differently from the more traditional strictly deterministic view which has dominated science from Newtonian times. Moviegoers who watched Jurassic Park are surely aware that chaos theory can profoundly affect the way someone thinks about the world; and indeed, chaos theory is useful as a tool with which to interpret scientific data in new ways. Instead of a traditional X-Y plot, scientists can now interpret phase-space diagrams which--rather than describing the exact position of some variable with respect to time--represents the overall behavior of a system. Instead of looking for strict equations conforming to statistical data, we can now look for dynamic systems with behavior similar in nature to the statistical data--systems, that is, with similar attractors. Chaos theory provides a sound framework with which to develop scientific knowledge.

However, this is not to say that chaos theory has no applications in real life.

Chaos theory techniques have been used to model biological systems, which are of course some of the most chaotic systems imaginable. Systems of dynamic equations have been used to model everything from population growth to epidemics to arrhythmic heart palpitations.

In fact, almost any chaotic system can be readily modeled--the stock market provides trends which can be analyzed with strange attractors more readily than with conventional explicit equations; a dripping faucet seems random to the untrained ear, but when plotted as a strange attractor, reveals an eerie order unexpected by conventional means.

Fractals have cropped up everywhere, most notably in graphic applications like the highly successful Fractal Design Painter series of products. Fractal image compression techniques are still under research, but promise such amazing results as 600:1 graphic compression ratios. The movie special effects industry would have much less realistic clouds, rocks, and shadows without fractal graphic technology.

And of course, chaos theory gives people a wonderfully interesting way to become more interested in mathematics, one of the more unpopular pursuits of the day.

Who were the pioneers of chaos theory?

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Under Construction
Please stay with us as we continue to construct this page.

Bibliography

This is simply a list of books which address topics in chaos theory. This list does not pretend to be either comprehensive or complete; it simply contains the paper-and-cardboard equivalent of hypertext links. It is not presented in formal bibliographic format; you probably will not require that and I certainly do not. Click on bibliography.

This page was selected as selected as an outstanding website by Webcrawler Select.

This page was cited in the "Tips and Links" section of the Discovery Channel School program "Wonders of Weather: Forecasting," scheduled to air on Assignment Discovery on 2/14/96 with repeat airings on 4/3/96 and 5/22/96.

Assignment Discovery airs on Discovery Channel Monday-Friday, 9-10am ET/PT.

This site is referenced in the sci.nonlinear FAQ under the "Chaos Sites" heading.
This site is referenced by the wonderful Chaos at Maryland site run by the University of Maryland at College Park under "Nonlinear Sites."

This site is linked to by Andrew Ho, its creator. He thinks it is cool.
This site is linked to by Cyberina Flux under "stuff" as a religion. I sort of like that.
This site is linked to by Ray and Jim from Unplugged Software. They click-n program.
This site is linked to by Jon Nelson, designer of integrated Customer Interactive Systems.
This site is linked to by Mingchao Shen of Susquehanna University.
This site is linked to by Eric Andersen of Brown University.
This site is linked to by Craig A. Hunter, professor at William & Mary and researcher at Langley.
This site is linked to by Gabor Fenyes, math nut, in his "Gabor's Favorite World Wide Web Links" section.
This site is linked to by Steve Lee, member of the educational institute Tripod.
This site is linked to by Zachary Harrison, yet another member of the Doom religion.
This site is linked to by Derrel Blain, who maintains a beautiful site about chaos and symmetry.
This site is linked to by Judy Ptre, who considers the religious implications of chaos theory.
This site is linked to by Diane Nyholt, who has an intimate knowledge of chaos as a mother, in her educational links.
This site is linked to by the Pomona, California chapter of the Tau Alpha Pi Engineering Technology Honor Society.
This site is linked to by the Harrisburg, Pennsylvania chapter of ACM under "Favorite Science Sites."
This site is linked to by Bobby Hardenbrook under his "Chaos in Science" section.
This site was rated a link of the week by Professor John Dutcher of the University of Guelph in Ontario, Canada.
This page is linked to by French grad student David Aubin of Princeton.
This page was cited in Chris Braun's final exam for some class... :)
This page appears as a science "book" at the Old Town University library.
Interesting comments on chaos theory as politics appear at this page on Political Elements of Strategic Planning for Technology.
This page is linked to by Bonnie Lenore, whose dissertation links chaos theory to composition and rhetoric.
This page is linked to by Austrian researcher Rudolf H. Winger on his page of science links.
This page is linked to by Shane Beattie, fellow Mathematica user.
This page is linked to by Ronnie Joe Record under Mathematics & Science.
This page is linked to by PhD student Warwick Tucker of Uppsala, Sweden, who flatters me by using my marbled background. :)
This page is linked to in the vast "Chaos, Complexity, and Everything Else" list of Lisa Landers of Mercyhurst College.
This page is linked to by the Visual Math Institute under "Related Pages."
This page is linked to by Jeff of Brock University in his discussion of reality.

John Childs of Ontario has a fascinating chaos theory "info-pak" available, which includes an 8-page introduction to chaos theory and fractals, an MS-DOS format disk of ten varied fractal-related programs, a Mandelbrot set postcard, and a Mandelbrot set pin. The set is available for a mere $5; see http://www.grenvillecc.ca/jchilds.

This page has been accessed

times since November 27, 1995.

Andrew Ho

Netscape 1.1 or up recommended.