Open Questions: Chaos Theory and Dynamical Systems

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See also: Self-organization and complex systems -- Mathematical analysis and differential equations

Mandelbrot Set Detail
See The Beauty of Chaos for more.



Introduction

Differing approaches

Chaos theory

Quantum chaos

Critical systems

Fractals

Non-linear dynamics

Turbulence

Solitons

Instantons

Percolation


Recommended references: Web sites

Recommended references: Magazine/journal articles

Recommended references: Books

Introduction

A better title for this page would probably have been "Non-linear mathematics". But then, far fewer people would recognize it as something perhaps interesting that they have already learned a little about.

Specifically, the central thing here is nonlinear ordinary and partial differential equations. A differential equation, in essence, simply expresses a relation between a function and some of the function's derivatives (rates of change). If the function has only one independent variable, the differential equation will be "ordinary". If it has two or more independent variables, the differential equation will be "partial" (involving derivatives with respect to more than one of the variables).

A differential equation is "linear" when neither the function nor any of its derivatives appears in the equation as a square, cube, or higher power, as the argument of another function, or in products involving derivatives of different orders. Stated differently, a linear differential equation involves a sum of the function and some of its derivatives standing alone or with coefficients that are constant or involve other functions.

A nonlinear differential equations is simply one that isn't in this category.

Generally, linear differential equations are relatively easy to solve, using one or more of a variety of well-studied techniques. Unfortunately, most real-world problems involve nonlinear differential equations, if indeed it's possible to find a suitable equation at all.

Historically, mathematics has lavished attention on linear equations, because there is so much theory that can be developed for them. Conversely, nonlinear equations have been largely neglected, because of the sheer difficulty of treating them with general techniques.

But clearly, one of the most significant, and challenging, "open questions" in mathematics is how to develop techniques for handling nonlinear equations, in spite of the difficulty. The challenge needs to be undertaken, because so much of the real world where one wants to apply mathematics is nonlinear. For instance, even Einstein's equation of general relativity is nonlinear. That's one thing that makes relativity theory so hard.

Why are linear equations so much easier to handle? It's mostly because if you have any two (or more) solutions, you can add them together to get additional solutions. You can even include constant or variable coefficients in the sum. This makes it possible to build up arbitrarily complicated solutions from a handful of initial ones. It may even be possible to represent all solutions in such a way -- that's where power series, Fourier series, and "eigenfunction expansions" come into the picture. Solving the equation in a new situation then just involves computing the right coefficients of the appropriate series expansion.

Here's another way to think about it. Instead of differential equations, consider algebraic equations. Linear algebraic equations are almost trivial to solve, alone or in systems when several variables are involved. It's a straightforward matter with determinants or matrix manipulation -- especially using a computer to handle the drudgery.

A nonlinear algebraic equation is a polynomial in one or more variables with at least one term of order degree greater than one, such as a quadratic equation. Quadratic equations with one variable are very easy to solve, since there's a simple formula. Third and fourth degree equations (cubics and quartics) in one variable also have formulas for the solution involving radicals, but if you've seen them, you realize they aren't so simple. And there are no explicit formulas involving radicals for the general equation of degree five or more.

In fact, the general treatment of higher-degree (i. e., nonlinear) algebraic equations in several variables is the subject matter of algebraic geometry, which is one of the hairiest subjects in mathematics -- a research area unto itself.

Just imagine how much more difficult it is, then, to deal with the general nonlinear differential equation, even with only one variable.


Differing approaches

How, then, have mathematicians and other scientists dealt with nonlinear equations when they had to?

The most common approach has been simply to try to avoid the difficulty by simplifying the problem in some way so that the equations become linear. In other words, make simplifying assumptions in order to be able to find solutions, and hope that real world situations satisfy the simplified assumptions closely enough to get useful results. This is, of course, a cop-out, and there isn't much to say about it mathematically.

Another common approach is to use "perturbations" and "perturbaton theory". That means, roughly, to take an approximate solution and add a small perturbation as a parameter. Sometimes it's possible to find a series expansion involving powers of the parameter that approximately solves the equation. Hopefully the approximation is better the more terms one takes in the series. (Usually one isn't so lucky, though.) This is what physicists usually do when working with their field equations. It tends to be a haphzard method, always skirting the abyss of non-convergence, but when it works... fine.

Yet another approach is to focus on specific types of nonlinear equations for which reasonable solutions are known just by good fortune. There are a number of instances of this, but only experts know much about them, since little generalization is possible.


Chaos theory

The main open question as far as "chaos" is concerned must be: is there really anything to it?

Here are some of the principal events that fueled the enthusiasm for chaos theory in the period around 1985-95:


Critical systems


Fractals


Non-linear dynamics


Turbulence


Solitons


Instantons


Percolation


Recommended references: Web sites

Site indexes

Open Directory Project: Chaos and Fractals
Categorized and annotated chaos and fractal links. A version of this list is at Google, with entries sorted in "page rank" order.
Galaxy: Chaos
Categorized site directory. Entries usually include descriptive annotations. Has a subcategory for fractals.
Systems Theory, Chaos Theory, Nonlinearity, etc.
Collection of links by Bill Huitt. Also links for artificial life and fuzzy logic.


Sites with general resources

Chaos at Maryland
Very good site produced by the Chaos Group at the University of Maryland at College Park. Resources include many book reviews, bibliographies, and links to other nonlinear sites.
Society for Chaos Theory in Psychology and Life Sciences
Resources include useful bibliographies and links to other sites.
Applied Nonlinear Mathematics
Site of the University of Bristol. Contains a sober assesment of the status of "chaos theory".
Sprott's Gateway
Home page of Julien Sprott, contains various resources related to chaos, complex systems, and fractals. Includes a good popular lecture on Strange Attractors From Art to Science.
The Chaos Center at the Visual Math Institute
Interactive chaos demonstrations. Includes list of publications by Ralph Abraham and other references on chaos theory.
Georgia Tech Applied Chaos Lab
Contents include list of reserach projects, paper archives, and external links.
The Chaos Experience
Various resources, including history, examples, glossary.


Surveys, overviews, tutorials

Chaos theory
Article from Wikipedia.
The Chaos Hypertextbook
Topics include the Mandelbrot set, strange attractors, fractal dimention, measuring chaos.
Chaos: Classical and Quantum
Complete online textbook, with supplementary material and links. PDF and PostScript formats.
Chaos Theory and Fractals
Nice one-page overview of the main ideas of fractals and chaos theory, by Jonathan Mendelson and Elana Blumenthal.
An Introduction to Mathematical Chaos Theory and Fractal Geometry
One-page overview, by Manus Donahue.
Chaos and Complexity Theory
Mainly consists of Chaos Without the Math -- a collection of essays on the history of chaos theory, instability, strange attractors, phase transitions, deep chaos, and self-organization. Good set of links. By Judy Petree.
What is Chaos? A five-part online course for everyone
Good course in five lessons, by Matthew Trump.
Feigenbaum Constants
Brief overview at the MathSoft site.
Applied Chaos Tutorial
From the Georgia Tech Applied Chaos Lab.
Emergence of Chaos
Interactive samples and tutorials.


Dynamical and nonlinear systems

Math Forum Internet Mathematics Library: Dynamical Systems
Alphabetized list of links with extensive annotations.
Galaxy: Dynamical Systems
Categorized site directory. Entries usually include descriptive annotations.
Mathematics Archives - Dynamical Systems
Extensive annotated list of links.
Mathematics Archives - Nonlinear Dynamics
Extensive annotated list of links.
The Dynamical Systems and Technology Project at Boston University
A collection of educational resources for teaching about chaos, fractals, and dynamics. Contents include software and interactive papers on dynamical systems and differential equations.
Dynamical Systems Homepage
At the Institute for Mathematical Sciences, SUNY at Stony Brook. Collections of various resources, such as a bibliography, a list of open problems in dynamical systems, links to online survey articles and a few other external links.
The Nonlinear Lab
Interactive material on chaos, nonlinearity, and related topics, by Blair Fraser.
Sci.nonlinear FAQ
Answers to frequently-asked questions.
Dynamical system
Article from Wikipedia.


Fractals

Fractal
Article from Wikipedia.
Fractal Frequently Asked Questions and Answers
From the sci.fractals group of Usenet.


Recommended references: Magazine/journal articles

The Quadratic Family as a Qualitatively Solvable Model of Chaos
Mikhail Lyubich
Notices of the AMS, October 2000, pp. 1042-1052
This technical article uses the quadratic family of maps f(x) = x2 + c, where c is a parameter, to illustrate ideas of chaotic dynamics, such as universality and renormalization.
[Article in PDF format]
What's New on Lorenz Strange Attractors?
Marcelo Viana
Mathematical Intelligencer, Summer 2000, pp. 6-19
Excellent survey article which covers many topics in dynamical systems through a consideration of the concept of strange attractors.
Instabilities in Fluid Motion
Susan Friedlander; Victor Yudovich
Notices of the AMS, December 1999, pp. 1358-1367
The motion of an ideal fluid is governed by the Euler equations, a system of two nonlinear partial differential equations. The discussion here illustrates the phenomenon of instability.
[Article in PDF format]
Developments in Chaotic Dynamics
Lai-Sang Young
Notices of the AMS, November 1998, pp. 1318-1328
Survey article that gives some of the flavor of the theory of chaotic dynamics through a look at developments since the 1960s. Among other things, it introduces the concepts of Lyapunov exponents, entropy, and dimension.
[Article in PDF format]
Finding a Horseshoe on the Beaches of Rio
Steve Smale
Mathematical Intelligencer, Winter 1998, pp. 39-44
The title alludes to an incident in the social history of mathematics involving the author, but the article itself helps illuminate the ideas of chaos in dynamical systems through Smale's concept of the "horseshoe" map.
Symmetric Chaos: How and Why
Mike Field; Martin Golubitsky
Notices of the AMS, February 1995, pp. 240-244
Numerical experiments hint at the existence of symmetries in the orbits of points under iterations of a mapping. These experiments suggest interesting questions about symmetry groups of dynamical systems.
[Article in PDF format]
Quantum Chaos
Martin C. Gutzwiller
Scientific American, January 1992, pp. 78-84
Chaotic behavior in classical dynamical systems has been recognized for a hundred years. There are now indications it exists in quantum systems as well, such as the energy levels of some atomic systems and the scattering of electrons from small molecules.
Disorder, Dynamical Chaos, and Structures
Andrei V. Gaponov-Grekhov; Mikhail I. Rabinovich
Physics Today, July 1990, pp. 30-38
Even very simple systems can give rise to complex, nearly random behavior. Nevertheless, orderly regular structures can emerge from disordered initial states in systems far from equilibrium. Such systems may exhibit particle-like excitations, as in planetary systems, lattice dislocations, and turbulence.
Chaos: How Regular Can It Be?
Alexander A. Chernikov; Roald Z. Sagdeev; George M. Zaslavsky
Physics Today, November 1988, pp. 27-35
Dynamical chaos -- the appearance of seemingly random motion in a deterministic dynamical system -- can occur even in systems with few degrees of freedom. But underlying symmetries may guarantee order within the randomness.
Chaos
James P. Crutchfield; J. Doyne Farmer; Norman H. Packard; Robert S. Shaw
Scientific American, December 1986, pp. 46-57
The apparent randomness in chaos has an underlying geometric form. Although chaos places fundamental limits on predictability, it can also reveal cause-effect relationships which explain certain phenomena with simple laws.


Recommended references: Books

Hiroyuki Nagashima, Yoshikazu Baba -- Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena
Institute of Physics Publishing, 1999
A short and mostly mathematical presentation of some concepts in chaos theory. Not the best choice as a first book on chaos for the uninitiated, it goes into some of the more sophisticated topics such as Lyapunov number and chaos in realistic systems.
Peter Smith -- Explaining Chaos
Cambridge University Press, 1998
This is a different kind of book on chaos. Written by a philosopher, it does discuss some of the elementary mathematics of chaos and dynamical systems. But then it goes into a more detailed analysis of the philosophical issues dealing with the kinds of explanations of empirical phenomena that can be proivided by chaos theory.
David Acheson -- From Calculus to Chaos: An Introduction to Dynamics
Oxford University Press, 1997
If you know introductory calculus, this is a good book to learn more about the actual mathematics of chaos and dynamical systems. It covers topics like oscillations, planetary motion, waves and diffusion, fluid flow, catastrophe, and non-simple pendulums. Simple computational programs are available from:
[Book home page]
Kathleen T. Alligood, Tim D. Sauer, James A Yorke -- Chaos: An Introduction to Dynamical Systems
Springer-Verlag, 1996
Although it's a textbook, and necessarily uses calculus-level mathematics to explain chaos and dynamical systems, the book is very clear and has many helpful illustrations, diagrams, and a few color plates. Topics covered include the usual: one- and two-dimensional maps, the logistic map, fractals, attractors, differential equations, periodic orbits, and bifurcations.
Gregory L. Baker; Jerry P. Golub -- Chaotic Dynamics: An Introduction
Cambridge University Press, 1996
This is a genuinely mathematical introduction to the subject that doesn't hesitate to use equations and calculus, yet the mathematics required isn't all that demanding. You will find here real mathematical discussion of topics like phase space, time series, Lyapunov exponents, and chaotic attractors. Also included are some applications to such things as lasers, chemical reactions, fluid dynamics, turbulence, and earthquakes.
H. G. Solari, M. A. Natiello, G. B. Mindlin -- Nonlinear Dynamics: A Two Way Trip from Physics to Math
Institute of Physics Publishing, 1996
The authors offer a more advanced and detailed textbook of nonlinear dynamics, emphasizing bifurcation theory and topological analysis of nonlinear systems.
Nina Hall, ed. -- Exploring Chaos: A Guide to the New Science of Disorder
W. W. Norton & Company, 1993
This is a stimulating collection of articles on the applications of chaos theory for a general audience. Topics covered include weather, biology, stability of the solar system, chemistry, fractals, fluid dynamics, stock markets, electronic systems, and quantum physics.
Jan Frøland -- Introduction to Chaos and Coherence
Institute of Physics Publishing, 1992
Provides a concise, straightforward introduction to chaos and associated topics, including fractals, the logistic map, higher dimensional maps, differential equations, the Lorenz model, and time series analysis. The presentation is clear, but calculus-level mathematics is required.
Heinz-Otto Peitgen; Hartmut Jürgens; Dietmar Saupe -- Chaos and Fractals: New Frontiers of Science
Springer-Verlag, 1992
This massive volume (almost 1000 pages) is an encyclopedic introduction to almost anything and everything to do with fractals and chaos. It covers many related topics as well, such as cellular automata, L-systems, percolation, and strange attractors. Yet the mathematical requirements aren't extreme -- little more than basic calculus here and there.
David Ruelle -- Chaos and Chance
Princeton University Press, 1991
Ruelle is a noted theoretical physicist who has made important contributions to thermodynamics and statistical mechanics. In this relatively short book for a general audience, he covers many topics, including probability, turbulence, chaos, quantum theory, entropy, statistical mechanics, information, and algorithmic complexity.
Ian Stewart -- Does God Play Dice? The Mathematics of Chaos
Basil Blackwell, 1989
The title may be misleading -- there's nothing about religion in the book. And although it's written by a professional mathematician (and mathematical popularizer), there are hardly any equations worth mentioning either. Nevertheless, it's a good introduction to the mathematical ideas behind chaos theory, where the emphasis is on concepts rather than calculations.
Ivar Ekeland -- Mathematics and the Unexpected
University of Chicago Press, 1988
Ekeland, a mathematician, presents a very concise overview for a general audience (i. e., using very few equations) of the main ideas of mathematical chaos theory. Topics covered include dynamical systems, dissipative systems, and catastrophe theory.
James Gleick -- Chaos: Making a New Science
Viking Penguin, 1987
The classic, and most noteworthy, introduction to chaos for general readers. The book's eleven chapters deal with topics such as chaos, fractals, dynamical systems, and applications to meteorology, biology, and fluid flow by focusing on the individuals and groups that have made the main contributions.
[Book home page]

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