Open Questions: Chaos Theory and Dynamical Systems
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Self-organization and complex systems --
Mathematical analysis and differential equations
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.
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.
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:
- Discovery of strange attractors by Lorenz
- Discovery of the Feigenbaum constant
- Discovery of "1/f noise" in many time series
- Development of fractal geometry by Mandelbrot and others
- Modeling of many natural processes and phenomena using ideas of chaos
and fractals
- Better understanding of turbulence
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.
- 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.
- 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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