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From: James Meiss <[e-mail]>
Newsgroups: sci.nonlinear,sci.answers,news.answers
Subject: Nonlinear Science FAQ
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                       Chaos, and Dynamical Systems
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This is version  2.0 (Sept. 2003) of the Frequently Asked Questions document
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This FAQ is maintained by Jim Meiss [e-mail].


Copyright (c) 1995-2003 by James D. Meiss, all rights reserved. This FAQ  may
be posted to any USENET newsgroup, on-line service, or BBS as long as it  is
posted in its entirety and includes this copyright statement. This FAQ may
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commercial collections or compilations without express permission from the
author.

[1.1] What's New?

      Fixed lots of broken and outdated links. A few sites seem to be gone,
and some new sites appeared.

   To some extent this FAQ is now been superseded by the Dynamical Systems site
run by SIAM. See http://www.dynamicalsystems.org There you will find a
glossary that contains most of the answers in this FAQ plus new ones. There is
also a growing software list. You are encouraged to contribute to this list,
and can do so interactively.



[1]   About Sci.nonlinear FAQ
[1.1] What's New?
[2]   Basic Theory
[2.1] What is nonlinear?
[2.2] What is nonlinear science?
[2.3] What is a dynamical system?
[2.4] What is phase space?
[2.5] What is a degree of freedom?
[2.6] What is a map?
[2.7] How are maps related to flows (differential equations)?
[2.8] What is an attractor?
[2.9] What is chaos?
[2.10] What is sensitive dependence on initial conditions?
[2.11] What are Lyapunov exponents?
[2.12] What is a Strange Attractor?
[2.13] Can computers simulate chaos?
[2.14] What is generic?
[2.15] What is the minimum phase space dimension for chaos?
[3]   Applications and Advanced Theory
[3.1] What are complex systems?
[3.2] What are fractals?
[3.3] What do fractals have to do with chaos?
[3.4] What are topological and fractal dimension?
[3.5] What is a Cantor set?
[3.6] What is quantum chaos?
[3.7] How do I know if my data are deterministic?
[3.8] What is the control of chaos?
[3.9] How can I build a chaotic circuit?
[3.10] What are simple experiments to demonstrate chaos?
[3.11] What is targeting?
[3.12] What is time series analysis?
[3.13] Is there chaos in the stock market?
[3.14] What are solitons?
[3.15] What is spatio-temporal chaos?
[3.16] What are cellular automata?
[3.17] What is a Bifurcation?
[3.18] What is a Hamiltonian Chaos?
[4]   To Learn More
[4.1] What should I read to learn more?
[4.2] What technical journals have nonlinear science articles?
[4.3] What are net sites for nonlinear science materials?
[5]   Computational Resources
[5.1] What are general computational resources?
[5.2] Where can I find specialized programs for  nonlinear science?
[6] Acknowledgments


[2]   Basic Theory
[2.1] What is nonlinear?

In geometry, linearity refers to Euclidean objects: lines, planes, (flat)
three-dimensional space, etc.--these objects appear the same no matter how we
examine them. A nonlinear object, a sphere for example, looks different on
different scales--when looked at closely enough it looks like a plane, and
from a far enough distance it looks like a point.

In algebra, we define linearity in terms of functions that have the property
f(x+y) = f(x)+f(y) and f(ax) = af(x). Nonlinear is defined as the negation of
linear. This means that the result f may be out of proportion to the input x
or y. The result may be more than linear, as when a diode begins to pass
current; or less than linear, as when finite resources limit Malthusian
population growth. Thus the fundamental simplifying tools of linear analysis
are no longer available: for example, for a linear system, if we have two
zeros, f(x) = 0 and f(y) = 0, then we automatically have a third zero f(x+y) =
0 (in fact there are infinitely many zeros as well, since linearity implies
that f(ax+by) = 0 for any a and b). This is called the principle of
superposition--it gives many solutions from a few. For nonlinear systems, each
solution must be fought for (generally) with unvarying ardor!


[2.2] What is nonlinear science?

Stanislaw Ulam reportedly said (something like) "Calling a science 'nonlinear'
is like calling zoology 'the study of non-human animals'. So why do we have a
name that appears to be merely a negative?

Firstly, linearity is rather special, and no model of a real system is truly
linear. Some things are profitably studied as linear approximations to the
real models--for example the fact that Hooke's law, the linear law of
elasticity (strain is proportional to stress) is approximately valid for a
pendulum of small amplitude implies that its period is approximately
independent of amplitude. However, as the amplitude gets large the period gets
longer, a fundamental effect of nonlinearity in the pendulum equations (see
http://monet.physik.unibas.ch/~elmer/pendulum/upend.htm and [3.10]).

(You might protest that quantum mechanics is the fundamental theory and that
it is linear! However this is at the expense of infinite dimensionality which
is just as bad or worse--and 'any' finite dimensional nonlinear model can be
turned into an infinite dimensional linear one--e.g. a map x' = f(x) is
equivalent to the linear integral equation often called the Perron-Frobenius
equation
p'(x) = integral [ p(y) \delta(x-f(y)) dy ])
Here p(x) is a density, which could be interpreted as the probability of
finding oneself at the point x, and the Dirac-delta function effectively moves
the points according to the map f to give the new density. So even a nonlinear
map is equivalent to a linear operator.)

Secondly, nonlinear systems have been shown to exhibit surprising and complex
effects that would never be anticipated by a scientist trained only in linear
techniques. Prominent examples of these include bifurcation, chaos, and
solitons. Nonlinearity has its most profound effects on dynamical systems (see
[2.3]).

Further, while we can enumerate the linear objects, nonlinear ones are
nondenumerable, and as of yet mostly unclassified. We currently have no
general techniques (and very few special ones) for telling whether a
particular nonlinear system will exhibit the complexity of chaos, or the
simplicity of order. Thus since we cannot yet subdivide nonlinear science into
proper subfields, it exists as a whole.

Nonlinear science has applications to a wide variety of fields, from
mathematics, physics, biology, and chemistry, to engineering, economics, and
medicine. This is one of its most exciting aspects--that it brings researchers
from many disciplines together with a common language.


[2.3] What is a dynamical system?

A dynamical system consists of an abstract phase space or state space, whose
coordinates describe the dynamical state at any instant; and a dynamical rule
which specifies the immediate future trend of all state variables, given only
the present values of those same state variables. Mathematically, a dynamical
system is described by an initial value problem.

Dynamical systems are "deterministic" if there is a unique consequent to every
state, and "stochastic" or "random" if there is more than one consequent
chosen from some probability distribution (the "perfect" coin toss has two
consequents with equal probability for each initial state). Most of nonlinear
science--and everything in this FAQ--deals with deterministic systems.

A dynamical system can have discrete or continuous time. The discrete case is
defined by a map, z_1 = f(z_0), that gives the state z_1 resulting from the
initial state z_0 at the next time value. The continuous case is defined by a
"flow", z(t) = \phi_t(z_0), which gives the state at time t, given that the
state was z_0 at time 0. A smooth flow can be differentiated w.r.t. time to
give a differential equation, dz/dt = F(z). In this case we call F(z) a
"vector field," it gives a vector pointing in the direction of the velocity at
every point in phase space.


[2.4] What is phase space?

Phase space is the collection of possible states of a dynamical system. A
phase space can be finite (e.g. for the ideal coin toss, we have two states
heads and tails), countably infinite (e.g. state variables are integers), or
uncountably infinite (e.g. state variables are real numbers). Implicit in the
notion is that a particular state in phase space specifies the system
completely; it is all we need to know about the system to have complete
knowledge of the immediate future. Thus the phase space of the planar pendulum
is two-dimensional, consisting of the position (angle) and velocity. According
to Newton, specification of these two variables uniquely determines the
subsequent motion of the pendulum.

Note that if we have a non-autonomous system, where the map or vector field
depends explicitly on time (e.g. a model for plant growth depending on solar
flux), then according to our definition of phase space, we must include time
as a phase space coordinate--since one must specify a specific time (e.g. 3PM
on Tuesday) to know the subsequent motion. Thus dz/dt = F(z,t) is a dynamical
system on the phase space consisting of (z,t), with the addition of the new
dynamics dt/dt = 1.

The path in phase space traced out by a solution of an initial value problem
is called an orbit or trajectory of the dynamical system. If the state
variables take real values in a continuum, the orbit of a continuous-time
system is a curve, while the orbit of a discrete-time system is a sequence of
points.


[2.5] What is a degree of freedom?

The notion of "degrees of freedom" as it is used for
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Hamiltonian systems means
one canonical conjugate pair, a configuration, q, and its conjugate momentum
p. Hamiltonian systems (sometimes mistakenly identified with the notion of
conservative systems) always have such pairs of variables, and so the phase
space is even dimensional.

In the study of dissipative systems the term "degree of freedom" is often used
differently, to mean a single coordinate dimension of the phase space. This
can lead to confusion, and it is advisable to check which meaning of the term
is intended in a particular context.

Those with a physics background generally prefer to stick with the Hamiltonian
definition of the term "degree of freedom." For a more general system the
proper term is "order" which is equal to the dimension of the phase space.

Note that a  dynamical system with N d.o.f. Hamiltonian nominally moves in a
2N dimensional phase space. However, if H(q,p) is time independent, then
energy is conserved, and therefore the motion is really on a 2N-1 dimensional
energy surface, H(q,p) = E. Thus e.g. the planar, circular restricted 3 body
problem is 2 d.o.f., and motion is on the 3D energy surface of constant
"Jacobi constant." It can be reduced to a 2D area preserving map by PoincarΘ
section (see [2.6]).

If the Hamiltonian is time dependent, then we generally say it has an
additional 1/2 degree of freedom, since this adds one dimension to the phase
space. (i.e. 1 1/2 d.o.f. means three variables, q, p and t, and energy is no
longer conserved).


[2.6] What is a map?

A map is simply a function, f, on the phase space that gives the next state,
f(z) (the image), of the system given its current state, z. (Often you will
find the notation z' = f(z), where the prime means the next point, not the
derivative.)

Now a function must have a single value for each state, but there could be
several different states that give rise to the same image. Maps that allow
every state in the phase space to be accessed (onto) and which have precisely
one pre-image for each state (one-to-one) are invertible. If in addition the
map and its inverse are continuous (with respect to the phase space coordinate
z), then it is called a homeomorphism. A homeomorphism that has at least one
continuous derivative (w.r.t. z) and a continuously differentiable inverse is
a diffeomorphism.

Iteration of a map means repeatedly applying the map to the consequents of the
previous application. Thus we get a sequence
                                                         n
                        z  = f(z   )  = f(f(z   )...) = f (z )
                         n      n-1          n-2            0

This sequence is the orbit or trajectory of the dynamical system with initial
condition z_0.


[2.7] How are maps related to flows (differential equations)?

Every differential equation gives rise to a map, the time one map, defined by
advancing the flow one unit of time. This map may or may not be useful. If the
differential equation contains a term or terms periodic in time, then the time
T map (where T is the period) is very useful--it is an example of a PoincarΘ
section. The time T map in a system with periodic terms is also called a
stroboscopic map, since we are effectively looking at the location in phase
space with a stroboscope tuned to the period T. This map is useful because it
permits us to dispense with time as a phase space coordinate: the remaining
coordinates describe the state completely so long as we agree to consider the
same instant within every period.

In autonomous systems (no time-dependent terms in the equations), it may also
be possible to define a PoincarΘ section and again reduce the phase space
dimension by one. Here the PoincarΘ section is defined not by a fixed time
interval, but by successive times when an orbit crosses a fixed surface in
phase space. (Surface here means a manifold of dimension one less than the
phase space dimension).

However, not every flow has a global PoincarΘ section (e.g. any flow with an
equilibrium point), which would need to be transverse to every possible orbit.

Maps arising from stroboscopic sampling or PoincarΘ section of a flow are
necessarily invertible, because the flow has a unique solution through any
point in phase space--the solution is unique both forward and backward in
time. However, noninvertible maps can be relevant to differential equations:
PoincarΘ maps are sometimes very well approximated by noninvertible maps. For
example, the Henon map (x,y) -> (-y-a+x^2,bx) with small |b| is close to the
logistic map, x -> -a+x^2.

It is often (though not always) possible to go backwards, from an invertible
map to a differential equation having the map as its PoincarΘ map. This is
called a suspension of the map. One can also do this procedure approximately
for maps that are close to the identity, giving a flow that approximates the
map to some order. This is extremely useful in bifurcation theory.

Note that any numerical solution procedure for a differential initial value
problem which uses discrete time steps in the approximation is effectively a
map. This is not a trivial observation; it helps explain for example why a
continuous-time system which should not exhibit chaos may have numerical
solutions which do--see [2.15].


[2.8] What is an attractor?

Informally an attractor is simply a state into which a system settles (thus
dissipation is needed). Thus in the long term, a dissipative dynamical system
may settle into an attractor.
   Interestingly enough, there is still some controversy in the mathematics
community as to an appropriate definition of this term. Most people adopt the
definition
Attractor: A set in the phase space that has a neighborhood in which every
point stays nearby and approaches the attractor as time goes to infinity.
Thus imagine a ball rolling inside of a bowl. If we start the ball at a point
in the bowl with a velocity too small to reach the edge of the bowl, then
eventually the ball will settle down to the bottom of the bowl with zero
velocity: thus this equilibrium point is an attractor. The neighborhood of
points that eventually approach the attractor is the basin of attraction for
the attractor. In our example the basin is the set of all configurations
corresponding to the ball in the bowl, and for each such point all small
enough velocities (it is a set in the four dimensional phase space [2.4]).
   Attractors can be simple, as the previous example. Another example of an
attractor is a limit cycle, which is a periodic orbit that is attracting
(limit cycles can also be repelling). More surprisingly, attractors can be
chaotic (see [2.9]) and/or strange (see [2.12]).
   The boundary of a basin of attraction is often a very interesting object
since it distinguishes between different types of motion. Typically a basin
boundary is a saddle orbit, or such an orbit and its stable manifold. A crisis
is the change in an attractor when its basin boundary is destroyed.
   An alternative definition of attractor is sometimes used because there
are systems that have sets that attract most, but not all, initial conditions
in their neighborhood (such phenomena is sometimes called riddling of the
basin). Thus, Milnor defines an attractor as a set for which a positive
measure (probability, if you like) of initial conditions in a neighborhood are
asymptotic to the set.


[2.9] What is chaos?

It has been said that "Chaos is a name for any order that produces confusion
in our minds." (George Santayana, thanks to Fred Klingener for finding this).
However, the mathematical definition is, roughly speaking,
Chaos: effectively unpredictable long time behavior arising in a deterministic
dynamical system because of sensitivity to initial conditions.
It must be emphasized that a deterministic dynamical system is perfectly
predictable given perfect knowledge of the initial condition, and is in
practice always predictable in the short term. The key to long-term
unpredictability is a property known as sensitivity to (or sensitive
dependence on) initial conditions.

For a dynamical system to be chaotic it must have a 'large' set of initial
conditions which are highly unstable. No matter how precisely you measure the
initial condition in these systems, your prediction of its subsequent motion
goes radically wrong after a short time. Typically (see [2.14] for one
definition of 'typical'), the predictability horizon grows only
logarithmically with the precision of measurement (for positive Lyapunov
exponents, see [2.11]). Thus for each increase in precision by a factor of 10,
say, you may only be able to predict two more time units (measured in units of
the Lyapunov time, i.e. the inverse of the Lyapunov exponent).

More precisely: A map f is chaotic on a compact invariant set S if
   (i) f is transitive on S (there is a point x whose orbit is dense in S), and
   (ii) f exhibits sensitive dependence on S (see [2.10]).
To these two requirements #DevaneyDevaney adds the requirement that periodic
points are dense in S, but this doesn't seem to be really in the spirit of the
notion, and is probably better treated as a theorem (very difficult and very
important), and not part of the definition.

Usually we would like the set S to be a large set. It is too much to hope for
except in special examples that S be the entire phase space. If the dynamical
system is dissipative then we hope that S is an attractor (see [2.8]) with a
large basin. However, this need not be the case--we can have a chaotic saddle,
an orbit that has some unstable directions as well as stable directions.

As a consequence of long-term unpredictability, time series from chaotic
systems may appear irregular and disorderly. However, chaos is definitely not
(as the name might suggest) complete disorder; it is disorder in a
deterministic dynamical system, which is always predictable for short times.

The notion of chaos seems to conflict with that attributed to Laplace: given
precise knowledge of the initial conditions, it should be possible to predict
the future of the universe. However, Laplace's dictum is certainly true for
any deterministic system, recall [2.3]. The main consequence of chaotic motion
is that given imperfect knowledge, the predictability horizon in a
deterministic system is much shorter than one might expect, due to the
exponential growth of errors. The belief that small errors should have small
consequences was perhaps engendered by the success of Newton's mechanics
applied to planetary motions. Though these happen to be regular on human
historic time scales, they are chaotic on the 5 million year time scale (see
e.g. "Newton's Clock", by Ivars Peterson (1993 W.H. Freeman).


[2.10] What is sensitive dependence on initial conditions?

Consider a boulder precariously perched on the top of an ideal hill. The
slightest push will cause the boulder to roll down one side of the hill or the
other: the subsequent behavior depends sensitively on the direction of the
push--and the push can be arbitrarily small.  Of course, it is of great
importance to you which direction the boulder will go if you are standing at
the bottom of the hill on one side or the other!

Sensitive dependence is the equivalent behavior for every initial condition--
every point in the phase space is effectively perched on the top of a hill.

More precisely a set S exhibits sensitive dependence if there is an r such
that for any epsilon > 0 and for each x in S, there is a y such that |x - y| <
epsilon, and |x_n - y_n| > r for some n > 0. Then there is a fixed distance r
(say 1), such that no matter how precisely one specifies an initial state
there are nearby states that eventually get a distance r away.

Note: sensitive dependence does not require exponential growth of
perturbations (positive Lyapunov exponent), but this is typical (see [2.14])
for chaotic systems. Note also that we most definitely do not require ALL
nearby initial points diverge--generically [2.14] this does not happen--some
nearby points may converge. (We may modify our hilltop analogy slightly and
say that every point in phase space acts like a high mountain pass.) Finally,
the words "initial conditions" are a bit misleading: a typical small
disturbance introduced at any time will grow similarly. Think of "initial" as
meaning "a time when a disturbance or error is introduced," not necessarily
time zero.


[2.11] What are Lyapunov exponents?
(Thanks to Ronnie Mainieri & Fred Klingener for contributing to this answer)

The hardest thing to get right about Lyapunov exponents is the spelling of
Lyapunov, which you will variously find as Liapunov, Lyapunof and even
Liapunoff. Of course Lyapunov is really spelled in the Cyrillic alphabet:
(Lambda)(backwards r)(pi)(Y)(H)(0)(B). Now that there is an ANSI standard of
transliteration for Cyrillic, we expect all references to converge on the
version Lyapunov.

Lyapunov was born in Russia in 6 June 1857. He was greatly influenced by
Chebyshev and was a student with Markov. He was also a passionate man:
Lyapunov shot himself the day his wife died. He died 3 Nov. 1918, three days
later. According to the request on a note he left, Lyapunov was buried with
his wife. [biographical data from a biography by A. T. Grigorian].

Lyapunov left us with more than just a simple note. He left a collection of
papers on the equilibrium shape of rotating liquids, on probability, and on
the stability of low-dimensional dynamical systems. It was from his
dissertation that the notion of Lyapunov exponent emerged. Lyapunov was
interested in showing how to discover if a solution to a dynamical system is
stable or not for all times. The usual method of studying stability, i.e.
linear stability, was not good enough, because if you waited long enough the
small errors due to linearization would pile up and make the approximation
invalid. Lyapunov developed concepts (now called Lyapunov Stability) to
overcome these difficulties.

Lyapunov exponents measure the rate at which nearby orbits converge or
diverge. There are as many Lyapunov exponents as there are dimensions in the
state space of the system, but the largest is usually the most important.
Roughly speaking the (maximal) Lyapunov exponent is the time constant, lambda,
in the expression for the distance between two nearby orbits, exp(lambda *
t).á If lambda is negative, then the orbits converge in time, and the
dynamical system is insensitive to initial conditions.á However, if lambda is
positive, then the distance between nearby orbits grows exponentially in time,
and the system exhibits sensitive dependence on initial conditions.

There are basically two ways to compute Lyapunov exponents. In one way one
chooses two nearby points, evolves them in time, measuring the growth rate of
the distance between them. This is useful when one has a time series, but has
the disadvantage that the growth rate is really not a local effect as the
points separate. A better way is to measure the growth rate of tangent vectors
to a given orbit.

More precisely, consider a map f in an m dimensional phase space, and its
derivative matrix Df(x). Let v be a tangent vector at the point x. Then we
define a function
                              1          n
        L(x,v)  =    lim     --- ln |( Df (x)v )|
                   n -> oo    n
Now the Multiplicative Ergodic Theorem of Oseledec states that this limit
exists for almost all points x and all tangent vectors v. There are at most m
distinct values of L as we let v range over the tangent space. These are the
Lyapunov exponents at x.

For more information on computing the exponents see

   Wolf, A., J. B. Swift, et al. (1985). "Determining Lyapunov Exponents from a
Time Series." Physica D 16: 285-317.
   Eckmann, J.-P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from
time series." Phys. Rev. A 34: 4971-4979.


[2.12] What is a Strange Attractor?
   Before Chaos (BC?), the only known attractors (see [2.8]) were fixed
points, periodic orbits (limit cycles), and invariant tori (quasiperiodic
orbits). In fact the famous PoincarΘ-Bendixson theorem states that for a pair
of first order differential equations,  only fixed points and limit cycles can
occur (there is no chaos in 2D flows).
   In a famous paper in 1963, Ed Lorenz discovered that simple systems of
three differential equations can have complicated attractors. The Lorenz
attractor (with its butterfly wings reminding us of sensitive dependence (see
[2.10])) is the "icon" of chaos
http://kong.apmaths.uwo.ca/~bfraser/version1/lorenzintro.html. Lorenz showed
that his attractor was chaotic, since it exhibited sensitive dependence.
Moreover, his attractor is also "strange," which means that it is a fractal
(see [3.2]).
   The term strange attractor was introduced by Ruelle and Takens in 1970
in their discussion of a scenario for the onset of turbulence in fluid flow.
They noted that when periodic motion goes unstable (with three or more modes),
the typical (see [2.14]) result will be a geometrically strange object.
   Unfortunately, the term strange attractor is often used for any chaotic
attractor. However, the term should be reserved for attractors that are
"geometrically" strange, e.g. fractal. One can have chaotic attractors that
are not strange (a trivial example would be to take a system like the cat map,
which has the whole plane as a chaotic set, and add a third dimension which is
simply contracting onto the plane). There are also strange, nonchaotic
attractors (see Grebogi, C., et al. (1984). "Strange Attractors that are not
Chaotic." Physica D 13: 261-268).


[2.13] Can computers simulate chaos?

Strictly speaking, chaos cannot occur on computers because they deal with
finite sets of numbers. Thus the initial condition is always precisely known,
and computer experiments are perfectly predictable, in principle. In
particular because of the finite size, every trajectory computed will
eventually have to repeat (an thus be eventually periodic). On the other hand,
computers can effectively simulate chaotic behavior for quite long times (just
so long as the discreteness is not noticeable). In particular if one uses
floating point numbers in double precision to iterate a map on the unit
square, then there are about 10^28 different points in the phase space, and
one would expect the "typical" chaotic orbit to have a period of about 10^14
(this square root of the number of points estimate is given by Rannou for
random diffeomorphisms and does not really apply to floating point operations,
but nonetheless the period should be a big number). See, e.g.,

   Earn, D. J. D. and S. Tremaine, "Exact Numerical Studies of Hamiltonian
Maps: Iterating without Roundoff Error," Physica D 56, 1-22 (1992).
   Binder, P. M. and R. V. Jensen, "Simulating Chaotic Behavior with Finite
State Machines," Phys. Rev. 34A, 4460-3 (1986).
   Rannou, F., "Numerical Study of Discrete Plane Area-Preserving Mappings,"
Astron. and Astrophys. 31, 289-301 (1974).


[2.14] What is generic?
(Thanks to Hawley Rising for contributing to this answer)

Generic in dynamical systems is intended to convey "usual" or, more properly,
"observable". Roughly speaking, a property is generic over a class if any
system in the class can be modified ever so slightly (perturbed), into one
with that property.

The formal definition is done in the language of topology: Consider the class
to be a space of systems, and suppose it has a topology (some notion of a
neighborhood, or an open set). A subset of this space is dense if its closure
(the subset plus the limits of all sequences in the subset) is the whole
space. It is open and dense if it is also an open set (union of
neighborhoods). A set is countable if it can be put into 1-1 correspondence
with the counting numbers. A countable intersection of open dense sets is the
intersection of a countable number of open dense sets. If all such
intersections in a space are also dense, then the space is called a Baire
space, which basically means it is big enough. If we have such a Baire space
of dynamical systems, and there is a property which is true on a countable
intersection of open dense sets, then that property is generic.

If all this sounds too complicated, think of it as a precise way of defining a
set which is near every system in the collection (dense), which isn't too big
(need not have any "regions" where the property is true for every system).
Generic is much weaker than "almost everywhere" (occurs with probability 1),
in fact, it is possible to have generic properties which occur with
probability zero. But it is as strong a property as one can define
topologically, without having to have a property hold true in a region, or
talking about measure (probability), which isn't a topological property (a
property preserved by a continuous function).


[2.15] What is the minimum phase space dimension for chaos?

This is a slightly confusing topic, since the answer depends on the type of
system considered. First consider a flow (or system of differential
equations). In this case the PoincarΘ-Bendixson theorem tells us that there is
no chaos in one or two-dimensional phase spaces. Chaos is possible in three-
dimensional flows--standard examples such as the Lorenz equations are indeed
three-dimensional, and there are mathematical 3D flows that are provably
chaotic (e.g. the 'solenoid').

Note: if the flow is non-autonomous then time is a phase space coordinate, so
a system with two physical variables + time becomes three-dimensional, and
chaos is possible (i.e. Forced second-order oscillators do exhibit chaos.)

For maps, it is possible to have chaos in one dimension, but only if the map
is not invertible. A prominent example is the Logistic map
                    x' = f(x) = rx(1-x).
This is provably chaotic for r = 4, and many other values of r as well (see
e.g. #DevaneyDevaney). Note that every point x < f(1/2) has two preimages, so
this map is not invertible.

For homeomorphisms, we must have at least two-dimensional phase space for
chaos. This is equivalent to the flow result, since a three-dimensional flow
gives rise to a two-dimensional homeomorphism by PoincarΘ section (see [2.7]).

Note that a numerical algorithm for a differential equation is a map, because
time on the computer is necessarily discrete. Thus numerical solutions of two
and even one dimensional systems of ordinary differential equations may
exhibit chaos. Usually this results from choosing the size of the time step
too large. For example Euler discretization of the Logistic differential
equation, dx/dt = rx(1-x), is equivalent to the logistic map. See e.g. S.
Ushiki, "Central difference scheme and chaos," Physica 4D (1982) 407-424.



[3]   Applications and Advanced Theory
[3.1] What are complex systems?
(Thanks to Troy Shinbrot for contributing to this answer)

Complex systems are spatially and/or temporally extended nonlinear systems
characterized by collective properties associated with the system as a whole--
and that are different from the characteristic behaviors of the constituent
parts.

While, chaos is the study of how simple systems can generate complicated
behavior, complexity is the study of how complicated systems can generate
simple behavior. An example of complexity is the synchronization of biological
systems ranging from fireflies to neurons (e.g. Matthews, PC, Mirollo, RE &
Strogatz, SH "Dynamics of a large system of coupled nonlinear oscillators,"
Physica 52D (1991) 293-331). In these problems, many individual systems
conspire to produce a single collective rhythm.

The notion of complex systems has received lots of popular press, but it is
not really clear as of yet if there is a "theory" about a "concept". We are
withholding judgment. See

   http://www.calresco.org/index.htm The Complexity & Artificial Life Web Site
   http://www.calresco.org/sos/sosfaq.htm The self-organized systems FAQ


[3.2] What are fractals?

One way to define "fractal" is as a negation: a fractal is a set that does not
look like a Euclidean object (point, line, plane, etc.) no matter how closely
you look at it. Imagine focusing in on a smooth curve (imagine a piece of
string in space)--if you look at any piece of it closely enough it eventually
looks like a straight line (ignoring the fact that for a real piece of string
it will soon look like a cylinder and eventually you will see the fibers, then
the atoms, etc.). A fractal, like the Koch Snowflake, which is topologically
one dimensional, never looks like a straight line, no matter how closely you
look. There are indentations, like bays in a coastline; look closer and the
bays have inlets, closer still the inlets have subinlets, and so on. Simple
examples of fractals include Cantor sets (see [3.5], Sierpinski curves, the
Mandelbrot set  and (almost surely) the Lorenz attractor (see [2.12]).
Fractals also approximately describe many real-world objects, such as clouds
(see http://makeashorterlink.com/?Z50D42C16)  mountains, turbulence,
coastlines, roots and branches of trees and veins and lungs of animals.

"Fractal" is a term which has undergone refinement of definition by a lot of
people, but was first coined by B. Mandelbrot,
http://physics.hallym.ac.kr/reference/physicist/Mandelbrot.html,  and defined
as a set with fractional (non-integer) dimension (Hausdorff dimension, see
[3.4]). Mandelbrot defines a fractal in the following way:

    A geometric figure or natural object is said to be fractal if it
    combines the following characteristics: (a) its parts have the same
    form or structure as the whole, except that they are at a different
    scale and may be slightly deformed; (b) its form is extremely irregular,
    or extremely interrupted or fragmented, and remains so, whatever the scale
    of examination; (c) it contains "distinct elements" whose scales are very
    varied and cover a large range." (Les Objets Fractales 1989, p.154)

See the extensive FAQ from sci.fractals at
   <ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq


[3.3] What do fractals have to do with chaos?

Often chaotic dynamical systems exhibit fractal structures in phase space.
However, there is no direct relation. There are chaotic systems that have
nonfractal limit sets (e.g. Arnold's cat map) and fractal structures that can
arise in nonchaotic dynamics (see e.g. Grebogi, C., et al. (1984). "Strange
Attractors that are not Chaotic." Physica 13D: 261-268.)


[3.4] What are topological and fractal dimension?

See the fractal FAQ:
   ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq
or the site
   http://pro.wanadoo.fr/quatuor/mathematics.htm


[3.5] What is a Cantor set?
(Thanks to Pavel Pokorny for contributing to this answer)

A Cantor set is a surprising set of points that is both infinite (uncountably
so, see [2.14]) and yet diffuse. It is a simple example of a fractal, and
occurs, for example as the strange repellor in the logistic map (see [2.15])
when r>4. The standard example of a Cantor set is the "middle thirds" set
constructed on the interval between 0 and 1. First, remove the middle third.
Two intervals remain, each one of length one third. From each remaining
interval remove the middle third. Repeat the last step infinitely many times.
What remains is a Cantor set.

More generally (and abstrusely) a Cantor set is defined topologically as a
nonempty, compact set which is perfect (every point is a limit point) and
totally disconnected (every pair of points in the set are contained in
disjoint covering neighborhoods).

See also
   http://www.shu.edu/html/teaching/math/reals/topo/defs/cantor.html
   http://personal.bgsu.edu/~carother/cantor/Cantor1.html
   http://mizar.uwb.edu.pl/JFM/Vol7/cantor_1.html

Georg Ferdinand Ludwig Philipp Cantor was born 3 March 1845 in St Petersburg,
Russia, and died 6 Jan 1918 in Halle, Germany. To learn more about him see:
   http://turnbull.dcs.st-and.ac.uk/history/Mathematicians/Cantor.html
   http://www.shu.edu/html/teaching/math/reals/history/cantor.html

To read more about the Cantor function (a function that is continuous,
differentiable, increasing, non-constant, with a derivative that is zero
everywhere except on a set with length zero) see
http://www.shu.edu/projects/reals/cont/fp_cantr.html


[3.6] What is quantum chaos?
(Thanks to Leon Poon for contributing to this answer)

 According to the correspondence principle, there is a limit where classical
behavior as described by Hamilton's equations becomes similar, in some
suitable sense, to quantum behavior as described by the appropriate wave
equation. Formally, one can take this limit to be h -> 0, where h is Planck's
constant; alternatively, one can look at successively higher energy levels.
Such limits are referred to as "semiclassical". It has been found that the
semiclassical limit can be highly nontrivial when the classical problem is
chaotic. The study of how quantum systems, whose classical counterparts are
chaotic, behave in the semiclassical limit has been called quantum chaos. More
generally, these considerations also apply to elliptic partial differential
equations that are physically unrelated to quantum considerations. For
example, the same questions arise in relating classical waves to their
corresponding ray equations. Among recent results in quantum chaos is a
prediction relating the chaos in the classical problem to the statistics of
energy-level spacings in the semiclassical quantum regime.

Classical chaos can be used to analyze such ostensibly quantum systems as the
hydrogen atom, where classical predictions of microwave ionization thresholds
agree with experiments. See Koch, P. M. and K. A. H. van Leeuwen (1995).
"Importance of Resonances in Microwave Ionization of Excited Hydrogen Atoms."
Physics Reports 255: 289-403.

See also:
   http://sagar.physics.neu.edu/qchaos/qc.html Quantum Chaos
   http://www.mpipks-dresden.mpg.de/~noeckel/microlasers.html  Microlaser
Cavities



[3.7] How do I know if my data are deterministic?
(Thanks to Justin Lipton for contributing to this answer)

How can I tell if my data is deterministic? This is a very tricky problem. It
is difficult because in practice no time series consists of pure 'signal.'
There will always be some form of corrupting noise, even if it is present as
round-off or truncation error or as a result of finite arithmetic or
quantization. Thus any real time series, even if mostly deterministic, will be
a stochastic processes

All methods for distinguishing deterministic and stochastic processes rely on
the fact that a deterministic system will always evolve in the same way from a
given starting point. Thus given a time series that we are testing for
determinism we
   (1) pick a test state
   (2) search the time series for a similar or 'nearby' state and
   (3) compare their respective time evolution.

Define the error as the difference between the time evolution of the 'test'
state and the time evolution of the nearby state. A deterministic system will
have an error that either remains small (stable, regular solution) or increase
exponentially with time (chaotic solution). A stochastic system will have a
randomly distributed error.

Essentially all measures of determinism taken from time series rely upon
finding the closest states to a given 'test' state (i.e., correlation
dimension, Lyapunov exponents, etc.). To define the state of a system one
typically relies on phase space embedding methods, see [3.14].

Typically one chooses an embedding dimension, and investigates the propagation
of the error between two nearby states. If the error looks random, one
increases the dimension. If you can increase the dimension to obtain a
deterministic looking error, then you are done. Though it may sound simple it
is not really! One complication is that as the dimension increases the search
for a nearby state requires a lot more computation time and a lot of data (the
amount of data required increases exponentially with embedding dimension) to
find a suitably close candidate. If the embedding dimension (number of
measures per state) is chosen too small (less than the 'true' value)
deterministic data can appear to be random but in theory there is no problem
choosing the dimension too large--the method will work. Practically, anything
approaching about 10 dimensions is considered so large that a stochastic
description is probably more suitable and convenient anyway.

See e.g.,
   Sugihara, G. and R. M. May (1990). "Nonlinear Forecasting as a Way of
      Distinguishing Chaos from Measurement Error in Time Series." Nature
344: 734-740.


[3.8] What is the control of chaos?

Control of chaos has come to mean the two things:
   stabilization of unstable periodic orbits,
   use of recurrence to allow stabilization to be applied locally.
Thus term "control of chaos" is somewhat of a misnomer--but the name has
stuck. The ideas for controlling chaos originated in the work of Hubler
followed by the Maryland Group.

   Hubler, A. W. (1989). "Adaptive Control of Chaotic Systems." Helv. Phys.
Acta 62: 343-346.
   Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review
Letters 64(11): 1196-1199. http://www-
chaos.umd.edu/publications/abstracts.html#prl64.1196

The idea that chaotic systems can in fact be controlled may be
counterintuitive--after all they are unpredictable in the long term.
Nevertheless, numerous theorists have independently developed methods which
can be applied to chaotic systems, and many experimentalists have demonstrated
that physical chaotic systems respond well to both simple and sophisticated
control strategies. Applications have been proposed in such diverse areas of
research as communications, electronics, physiology, epidemiology, fluid
mechanics and chemistry.

The great bulk of this work has been restricted to low-dimensional systems;
more recently, a few researchers have proposed control techniques for
application to high- or infinite-dimensional systems. The literature on the
subject of the control of chaos is quite voluminous; nevertheless several
reviews of the literature are available, including:

   Shinbrot, T. Ott, E., Grebogi, C. & Yorke, J.A., "Using Small Perturbations
to Control Chaos," Nature, 363 (1993) 411-7.
   Shinbrot, T., "Chaos: Unpredictable yet Controllable?" Nonlin. Sciences
Today, 3:2 (1993) 1-8.
   Shinbrot, T., "Progress in the Control of Chaos," Advance in Physics (in
press).
   Ditto, WL & Pecora, LM "Mastering Chaos," Scientific American (Aug. 1993),
78-84.
   Chen, G. & Dong, X, "From Chaos to Order -- Perspectives and Methodologies
in Controlling Chaotic Nonlinear Dynamical Systems," Int. J. Bif. & Chaos 3
(1993) 1363-1409.

It is generically quite difficult to control high dimensional systems; an
alternative approach is to use control to reduce the dimension before applying
one of the above techniques. This approach is in its infancy; see:

   Auerbach, D., Ott, E., Grebogi, C., and Yorke, J.A. "Controlling Chaos in
   High Dimensional Systems," Phys. Rev. Lett. 69  (1992) 3479-82
   http://www-chaos.umd.edu/publications/abstracts.html#prl69.3479


[3.9] How can I build a chaotic circuit?
(Thanks to Justin Lipton and Jose Korneluk for contributing to this answer)

There are many different physical systems which display chaos, dripping
faucets, water wheels, oscillating magnetic ribbons etc. but the most simple
systems which can be easily implemented are chaotic circuits. In fact an
electronic circuit was one of the first demonstrations of chaos which showed
that chaos is not just a mathematical abstraction. Leon Chua designed the
circuit 1983.

The circuit he designed, now known as Chua's circuit, consists of a piecewise
linear resistor as its nonlinearity (making analysis very easy) plus two
capacitors, one resistor and one inductor--the circuit is unforced
(autonomous). In fact the chaotic aspects (bifurcation values, Lyapunov
exponents, various dimensions etc.) of this circuit have been extensively
studied in the literature both experimentally and theoretically. It is
extremely easy to build and presents beautiful attractors (see [2.8]) (the
most famous known as the double scroll attractor) that can be displayed on a
CRO.

For more information on building such a circuit try: see

   http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html  Chua's Circuit  Applet

References
   Matsumoto T. and Chua L.O. and Komuro M. "Birth and Death of the Double
      Scroll" Physica D24 97-124, 1987.
   Kennedy M. P., "Robust OP Amp Realization of Chua's Circuit", Frequenz
      46, no. 3-4, 1992
   Madan, R. A., Chua's Circuit: A paradigm for chaos, ed. R. A. Madan,
      Singapore: World Scientific, 1993.
   Pecora, L. and Carroll, T. Nonlinear Dynamics in Circuits, Singapore:
      World Scientific, 1995.
   Nonlinear Dynamics of Electronic Systems, Proceedings of the Workshop
      NDES 1993, A.C.Davies and W.Schwartz, eds., World Scientific, 1994,
      ISBN 981-02-1769-2.
   Parker, T.S., and L.O.Chua, Practical Numerical Algorithms for Chaotic
      Systems, Springer-Verlag, 1989, ISBN's: 0-387-96689-7
      and 3-540-96689-7.


[3.10] What are simple experiments to demonstrate chaos?

 There are many "chaos toys" on the market. Most consist of some sort of
pendulum that is forced by an electromagnet. One can of course build a simple
double pendulum to observe beautiful chaotic behavior see
   http://quasar.mathstat.uottawa.ca/~selinger/lagrange/doublependulum.html
Experimental Pendulum Designs
   http://www.maths.tcd.ie/~plynch/SwingingSpring/doublependulum.html  Java
Applet
   http://monet.physik.unibas.ch/~elmer/pendulum/ Java Applets Pendulum Lab

My favorite double pendulum consists of two identical planar pendula, so that
you can demonstrate sensitive dependence [2.10], for a Java applet simulation
see http://www.cs.mu.oz.au/~mkwan/pendulum/pendulum.html. Another cute toy is
the "Space Circle" that you can find in many airport gift shops. This is
discussed in the article:

   A. Wolf & T. Bessoir, Diagnosing Chaos in the Space Circle, Physica 50D,
1991.

One of the simplest chemical systems that shows chaos is the Belousov-
Zhabotinsky reaction. The book by Strogatz [4.1] has a good introduction to
this subject,. For the recipe see
http://www.ux.his.no/~ruoff/BZ_Phenomenology.html. Chemical chaos is modeled
(in a generic sense) by the "Brusselator" system of differential equations.
See

   Nicolis, Gregoire & Prigogine, (1989) Exploring Complexity: An
      Introduction W. H. Freeman

The Chaotic waterwheel, while not so simple to build, is an exact realization
of Lorenz famous equations. This is nicely discussed in Strogatz book [4.1] as
well.

Billiard tables can exhibit chaotic motion, see
http://www.maa.org/mathland/mathland_3_3.html, though it might be hard to see
this next time you are in a bar, since a rectangular table is not chaotic!


[3.11] What is targeting?
(Thanks to Serdar Iplikτi for contributing to this answer)

Targeting is the task of steering a chaotic system from any initial point to
the target, which can be either an unstable equilibrium point or an unstable
periodic orbit, in the shortest possible time, by applying relatively small
perturbations. In order to effectively control chaos, [3.8] a targeting
strategy is important. See:

   Kostelich, E., C. Grebogi, E. Ott, and J. A. Yorke, "Higher
      Dimensional Targeting," Phys Rev. E,. 47, , 305-310 (1993).
   Barreto, E., E. Kostelich, C. Grebogi, E. Ott, and J. A. Yorke, "Efficient
      Switching Between Controlled Unstable Periodic Orbits in Higher
      Dimensional Chaotic Systems," Phys Rev E, 51, 4169-4172 (1995).

One application of targeting is to control a spacecraft's trajectory so that
one can find low energy orbits from one planet to another. Recently targeting
techniques have been used in the design of trajectories to asteroids and even
of a grand tour of the planets. For example,

   E. Bollt and J. D. Meiss, "Targeting Chaotic Orbits to the Moon
      Through Recurrence," Phys. Lett. A  204, 373-378 (1995).
   http://www.cds.caltech.edu/~marsden/software/spacecraft_orbits.html


 [3.12] What is time series analysis?
(Thanks to Jim Crutchfield for contributing to this answer)

This is the application of dynamical systems techniques to a data series,
usually obtained by "measuring" the value of a single observable as a function
of time. The major tool in a dynamicist's toolkit is "delay coordinate
embedding" which creates a phase space portrait from a single data series. It
seems remarkable at first, but one can reconstruct a picture equivalent
(topologically) to the full Lorenz attractor (see [2.12])in three-dimensional
space by measuring only one of its coordinates, say x(t), and plotting the
delay coordinates (x(t), x(t+h), x(t+2h)) for a fixed h.

It is important to emphasize that the idea of using derivatives or delay
coordinates in time series modeling is nothing new. It goes back at least to
the work of Yule, who in 1927 used an autoregressive (AR) model to make a
predictive model for the sunspot cycle. AR models are nothing more than delay
coordinates used with a linear model. Delays, derivatives, principal
components, and a variety of other methods of reconstruction have been widely
used in time series analysis since the early 50's, and are described in
several hundred books. The new aspects raised by dynamical systems theory are
(i) the implied geometric view of temporal behavior and (ii) the existence of
"geometric invariants", such as dimension and Lyapunov exponents. The central
question was not whether delay coordinates are useful for time series
analysis, but rather whether reconstruction methods preserve the geometry and
the geometric invariants of dynamical systems. (Packard, Crutchfield, Farmer &
Shaw)

   G.U. Yule, Phil. Trans. R. Soc. London A 226 (1927) p. 267.
   N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, "Geometry
      from a time series", Phys. Rev. Lett. 45, no. 9 (1980) 712.
   F. Takens, "Detecting strange attractors in fluid turbulence", in: Dynamical
      Systems and Turbulence, eds. D. Rand and L.-S. Young
      (Springer, Berlin, 1981)
   Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.Sh.T.
      "The analysis of observed chaotic data in physical systems",
      Rev. Modern Physics 65 (1993) 1331-1392.
   D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics,
      Springer-Verlag http://www.cnd.mcgill.ca/books_understanding.html
   E. Peters (1994) Fractal Market Analysis : Applying Chaos Theory to
      Investment and Economics, Wiley
      http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471585246.html


[3.13] Is there chaos in the stock market?
(Thanks to Bruce Stewart for Contributions to this answer)

In order to address this question, we must first agree what we mean by chaos,
see [2.9].

In dynamical systems theory, chaos means irregular fluctuations in a
deterministic system (see [2.3] and [3.7]). This means the system behaves
irregularly because of its own internal logic, not because of random forces
acting from outside. Of course, if you define your dynamical system to be the
socio-economic behavior of the entire planet, nothing acts randomly from
outside (except perhaps the occasional meteor), so you have a dynamical
system. But its dimension (number of state variables--see [2.4]) is vast, and
there is no hope of exploiting the determinism. This is high-dimensional
chaos, which might just as well be truly random behavior. In this sense, the
stock market is chaotic, but who cares?

To be useful, economic chaos would have to involve some kind of collective
behavior which can be fully described by a small number of variables. In the
lingo, the system would have to be self-organizing, resulting in low-
dimensional chaos. If this turns out to be true, then you can exploit the low-
dimensional chaos to make short-term predictions. The problem is to identify
the state variables which characterize the collective modes. Furthermore,
having limited the number of state variables, many events now become external
to the system, that is, the system is operating in a changing environment,
which makes the problem of system identification very difficult.

If there were such collective modes of fluctuation, market players would
probably know about them; economic theory says that if many people recognized
these patterns, the actions they would take to exploit them would quickly
nullify the patterns. Market participants would probably not need to know
chaos theory for this to happen. Therefore if these patterns exist, they must
be hard to recognize because they do not emerge clearly from the sea of noise
caused by individual actions; or the patterns last only a very short time
following some upset to the markets; or both.

A number of people and groups have tried to find these patterns. So far the
published results are negative. There are also commercial ventures involving
prominent researchers in the field of chaos; we have no idea how well they are
succeeding, or indeed whether they are looking for low-dimensional chaos. In
fact it seems unlikely that markets remain stationary long enough to identify
a chaotic attractor (see [2.12]). If you know chaos theory and would like to
devote yourself to the rhythms of market trading, you might find a trading
firm which will give you a chance to try your ideas. But don't expect them to
give you a share of any profits you may make for them :-) !

In short, anyone who tells you about the secrets of chaos in the stock market
doesn't know anything useful, and anyone who knows will not tell. It's an
interesting question, but you're unlikely to find the answer.

On the other hand, one might ask a more general question: is market behavior
adequately described by linear models, or are there signs of nonlinearity in
financial market data? Here the prospect is more favorable. Time series
analysis (see [3.14]) has been applied these tests to financial data; the
results often indicate that nonlinear structure is present. See e.g. the book
by Brock, Hsieh, LeBaron, "Nonlinear Dynamics, Chaos, and Instability", MIT
Press, 1991; and an update by B. LeBaron, "Chaos and nonlinear forecastability
in economics and finance," Philosophical Transactions of the Royal Society,
Series A, vol 348, Sept 1994, pp 397-404. This approach does not provide a
formula for making money, but it is stimulating some rethinking of economic
modeling. A book by Richard M. Goodwin, "Chaotic Economic Dynamics," Oxford
UP, 1990, begins to explore the implications for business cycles.


[3.14] What are solitons?

The process of obtaining a solution of a linear (constant coefficient)
differential equations is simplified by the Fourier transform (it converts
such an equation to an algebraic equation, and we all know that algebra is
easier than calculus!); is there a counterpart which similarly simplifies
nonlinear equations? The answer is No. Nonlinear equations are qualitatively
more complex than linear equations, and a procedure which gives the dynamics
as simply as for linear equations must contain a mistake. There are, however,
exceptions to any rule.

Certain nonlinear differential equations can be fully solved by, e.g., the
"inverse scattering method." Examples are the Korteweg-de Vries, nonlinear
Schrodinger, and sine-Gordon equations. In these cases the real space maps, in
a rather abstract way, to an inverse space, which is comprised of continuous
and discrete parts and evolves linearly in time. The continuous part typically
corresponds to radiation and the discrete parts to stable solitary waves, i.e.
pulses, which are called solitons. The linear evolution of the inverse space
means that solitons will emerge virtually unaffected from interactions with
anything, giving them great stability.

More broadly, there is a wide variety of systems which support stable solitary
waves through a balance of dispersion and nonlinearity. Though these systems
may not be integrable as above, in many cases they are close to systems which
are, and the solitary waves may share many of the stability properties of true
solitons, especially that of surviving interactions with other solitary waves
(mostly) unscathed. It is widely accepted to call these solitary waves
solitons, albeit with qualifications.

Why solitons? Solitons are simply a fundamental nonlinear wave phenomenon.
Many very basic linear systems with the addition of the simplest possible or
first order nonlinearity support solitons; this universality means that
solitons will arise in many important physical situations. Optical fibers can
support solitons, which because of their great stability are an ideal medium
for transmitting information. In a few years long distance telephone
communications will likely be carried via solitons.

The soliton literature is by now vast. Two books which contain clear
discussions of solitons as well as references to original papers are
   A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia,
      Penn. (1985)
   M.J. Ablowitz and P.A.Clarkson, Solitons, nonlinear evolution equations and
inverse
      scattering, Cambridge (1991).
http://www.cup.org/titles/catalogue.asp?isbn=0521387302
   See  http://www.ma.hw.ac.uk/solitons/


[3.15] What is spatio-temporal chaos?

   Spatio-temporal chaos occurs when system of coupled dynamical systems
gives rise to dynamical behavior that exhibits both spatial disorder (as in
rapid decay of spatial correlations) and temporal disorder (as in nonzero
Lyapunov exponents). This is an extremely active, and rather unsettled area of
research. For an introduction see:
   Cross, M. C. and P. C. Hohenberg (1993). "Pattern Formation outside of
       Equilibrium."  Rev. Mod. Phys. 65: 851-1112.
   http://www.cmp.caltech.edu/~mcc/st_chaos.html Spatio-Temporal Chaos

An interesting application which exhibits pattern formation and spatio-
temporal chaos is to excitable media in biological or chemical systems. See

   Chaos, Solitions and Fractals 5 #3&4 (1995) Nonlinear Phenomena in Excitable
      Physiological System,  http://www.elsevier.nl/locate/chaos
   http://ojps.aip.org/journal_cgi/dbt?KEY=CHAOEH&Volume=8&Issue=1
      Chaos focus issue on Fibrillation


[3.16] What are cellular automata?
(Thanks to Pavel Pokorny for Contributions to this answer)

   A Cellular automaton (CA) is a dynamical system with discrete time (like
a map, see [2.6]), discrete state space and discrete geometrical space (like
an ODE), see [2.7]). Thus they can be represented by a state s(i,j) for
spatial state i, at time j, where s is taken from some finite set. The update
rule is that the new state is some function of the old states, s(i,j+1) =
f(s). The following table shows the distinctions between PDE's, ODE's, coupled
map lattices (CML) and CA in taking time, state space or geometrical space
either continuous (C) of discrete (D):
        time   state space    geometrical space
 PDE      C          C              C
 ODE      C          C              D
 CML      D          C              D
 CA       D          D              D

   Perhaps the most famous CA is Conway's game "life." This CA evolves
according to a deterministic rule which gives the state of a site in the next
generation as a function of the states of neighboring sites in the present
generation. This rule is applied to all sites.

For further reading see

   S. Wolfram (1986) Theory and Application of Cellular Automata, World
Scientific Singapore.
   Physica 10D (1984)--the entire volume

Some programs that do CA, as well as more generally "artificial life" are
available at
   http://www.alife.org/links.html
   http://www.kasprzyk.demon.co.uk/www/ALHome.html


[3.17] What is a Bifurcation?
(Thanks to Zhen Mei for Contributions to this answer)

A bifurcation is a qualitative change in dynamics upon a small variation in
the parameters of a system.

Many dynamical systems depend on parameters, e.g. Reynolds number, catalyst
density, temperature, etc. Normally a gradually variation of a parameter in
the system corresponds to the gradual variation of the solutions of the
problem. However, there exists a large number of problems for which the number
of solutions changes abruptly and the structure of solution manifolds varies
dramatically when a parameter passes through some critical values. For
example, the abrupt buckling of a stab when the stress is increased beyond a
critical value, the onset of convection and turbulence when the flow
parameters are changed, the formation of patterns in certain PDE's, etc. This
kind of phenomena is called bifurcation, i.e. a qualitative change in the
behavior of solutions of a dynamics system, a partial differential equation or
a delay differential equation.

Bifurcation theory is a method for studying how solutions of a nonlinear
problem and their stability change as the parameters varies. The onset of
chaos is often studied by bifurcation theory. For example, in certain
parameterized families of one dimensional maps, chaos occurs by infinitely
many period doubling bifurcations
   (See http://www.stud.ntnu.no/~berland/math/feigenbaum/)

There are a number of well constructed computer tools for studying
bifurcations. In particular see [5.2] for AUTO and DStool.


[3.18] What is a Hamiltonian Chaos?

The transition to chaos for a Hamiltonian (conservative) system is somewhat
different than that for a dissipative system (recall [2.5]). In an integrable
(nonchaotic) Hamiltonian system, the motion is "quasiperiodic", that is motion
that is oscillatory, but involves more than one independent frequency (see
also [2.12]). Geometrically the orbits move on tori, i.e. the mathematical
generalization of a donut. Examples of integrable Hamiltonian systems include
harmonic oscillators (simple mass on a spring, or systems of coupled linear
springs), the pendulum, certain special tops (for example the Euler and
Lagrange tops), and the Kepler motion of one planet around the sun.

It was expected that a typical perturbation of an integrable Hamiltonian
system would lead to "ergodic" motion, a weak version of chaos in which all of
phase space is covered, but the Lyapunov exponents [2.11] are not necessarily
positive. That this was not true was rather surprisingly discovered by one of
the first computer experiments in dynamics, that of Fermi, Pasta and Ulam.
They showed that trajectories in nonintegrable system may also be surprisingly
stable. Mathematically this was shown to be the case by the celebrated theorem
of Kolmogorov Arnold and Moser (KAM), first proposed by Kolmogorov in 1954.
The KAM theorem is rather technical, but in essence says that many of the
quasiperiodic motions are preserved under perturbations. These orbits fill out
what are called KAM tori.

An amazing extension of this result was started with the work of John Greene
in 1968. He showed that if one continues to perturb a KAM torus, it reaches a
stage where the nearby phase space [2.4] becomes self-similar (has fractal
structure [3.2]). At this point the torus is "critical," and any increase in
the perturbation destroys it. In a remarkable sequence of papers, Aubry and
Mather showed that there are still quasiperiodic orbits that exist beyond this
point, but instead of tori they cover cantor sets [3.5]. Percival actually
discovered these for an example in 1979 and named them "cantori."
Mathematicians tend to call them "Aubry-Mather" sets. These play an important
role in limiting the rate of transport through chaotic regions.

Thus, the transition to chaos in Hamiltonian systems can be thought of as the
destruction of invariant tori, and the creation of cantori. Chirikov was the
first to realize that this transition to "global chaos" was an important
physical phenomena. Local chaos also occurs in Hamiltonian systems (in the
regions between the KAM tori), and is caused by the intersection of stable and
unstable manifolds in what PoincarΘ called the "homoclinic trellis."

To learn more: See the introductory article by Berry, the text by Percival and
Richards and the collection of articles on Hamiltonian systems by MacKay and
Meiss [4.1]. There are a number of excellent advanced texts on Hamiltonian
dynamics, some of which are listed in [4.1], but we also mention

   Meyer, K. R. and G. R. Hall (1992), Introduction to Hamiltonian dynamical
systems and the N-body problem  (New York, Springer-Verlag).



[4]   To Learn More
[4.1] What should I read to learn more?
   Popularizations
1  Gleick, J. (1987). Chaos, the Making of a New Science.
      London, Heinemann. http://www.around.com/chaos.html
2  Stewart, I. (1989). Does God Play Dice? Cambridge, Blackwell.
      http://www.amazon.com/exec/obidos/ASIN/1557861064
3  Devaney, R. L. (1990). Chaos, Fractals, and Dynamics: Computer
      Experiments in Mathematics. Menlo Park, Addison-Wesley
      http://www.amazon.com/exec/obidos/ASIN/1878310097
4  Lorenz, E., (1994) The Essence of Chaos, Univ. of Washington Press.
      http://www.amazon.com/exec/obidos/ASIN/0295975148
5  Schroeder, M. (1991) Fractals, Chaos, Power: Minutes from an infinite paradise
      W. H. Freeman New York:
   Introductory Texts
1  Abraham, R. H. and C. D. Shaw (1992) Dynamics: The Geometry of
      Behavior, 2nd ed. Redwood City, Addison-Wesley.
2  Baker, G. L. and J. P. Gollub (1990). Chaotic Dynamics.
      Cambridge, Cambridge Univ. Press.
      http://www.cup.org/titles/catalogue.asp?isbn=0521471060
3  DevaneyDevaney, R. L. (1986). An Introduction to Chaotic Dynamical
      Systems. Menlo Park, Benjamin/Cummings.
      http://math.bu.edu/people/bob/books.html
4  Kaplan, D. and L. Glass (1995). Understanding Nonlinear Dynamics,
      Springer-Verlag New York.  http://www.cnd.mcgill.ca/books_understanding.html
5  Glendinning, P. (1994). Stability, Instability and Chaos.
      Cambridge, Cambridge Univ Press.
       http://www.cup.org/Titles/415/0521415535.html
6  Jurgens, H., H.-O. Peitgen, et al. (1993). Chaos and Fractals: New
      Frontiers of Science. New York, Springer Verlag.
      http://www.springer-ny.com/detail.tpl?isbn=0387979034
7  Moon, F. C. (1992). Chaotic and Fractal Dynamics. New York, John Wiley.
      http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471545716.html
8  Percival, I. C. and D. Richard (1982). Introduction to Dynamics. Cambridge,
      Cambridge Univ. Press.
      http://www.cup.org/titles/catalogue.asp?isbn=0521281490
9  Scott, A. (1999). NONLINEAR SCIENCE: Emergence and Dynamics of
      Coherent Structures, Oxford http://www4.oup.co.uk/isbn/0-19-850107-2
      http://www.imm.dtu.dk/documents/users/acs/BOOK1.html
10 Smith, P (1998) Explaining Chaos, Cambridge
       http://us.cambridge.org/titles/catalogue.asp?isbn=0521477476
11 Strogatz, S. (1994). Nonlinear Dynamics and Chaos. Reading,
      Addison-Wesley
      http://www.perseusbooksgroup.com/perseus-cgi-bin/display/0-7382-0453-6
12 Thompson, J. M. T. and H. B. Stewart (1986) Nonlinear Dynamics and
      Chaos. Chichester, John Wiley and Sons.
      http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471876844.html
13 Tufillaro, N., T. Abbott, et al. (1992). An Experimental Approach
      to Nonlinear Dynamics and Chaos. Redwood City, Addison-Wesley.
      http://www.amazon.com/exec/obidos/ASIN/0201554410/
14 Turcotte, Donald L. (1992). Fractals and Chaos in Geology and
      Geophysics, Cambridge Univ. Press.
      http://www.cup.org/titles/catalogue.asp?isbn=0521567335

   Introductory Articles
1  May, R. M. (1986). "When Two and Two Do Not Make Four."
      Proc. Royal Soc. B228: 241.
2  Berry, M. V. (1981). "Regularity and Chaos in Classical Mechanics,
      Illustrated by Three Deformations of a Circular Billiard."
      Eur. J. Phys. 2: 91-102.
3  Crawford, J. D. (1991). "Introduction to Bifurcation Theory."
      Reviews of Modern Physics 63(4): 991-1038.
3  Shinbrot, T., C. Grebogi, et al. (1992). "Chaos in a Double Pendulum."
      Am. J. Phys 60: 491-499.
5  David Ruelle. (1980). "Strange Attractors,"
      The Mathematical Intelligencer 2:  126-37.

   Advanced Texts
1  Arnold, V. I. (1978). Mathematical Methods of Classical Mechanics.
     New York, Springer.
        http://www.springer-ny.com/detail.tpl?isbn=038796890
2  Arrowsmith, D. K. and C. M. Place (1990),  An Introduction to Dynamical Systems.
      Cambridge, Cambridge University Press.
      http://us.cambridge.org/titles/catalogue.asp?isbn=0521316502
3  Guckenheimer, J. and P. Holmes (1983), Nonlinear Oscillations, Dynamical
      Systems, and Bifurcation of Vector Fields, Springer-Verlag New York.
4  Kantz, H., and T. Schreiber (1997). Nonlinear time series analysis.
      Cambridge, Cambridge University Press
      http://www.mpipks-dresden.mpg.de/~schreibe/myrefs/book.html
5  Katok, A. and B. Hasselblatt (1995), Introduction to the Modern
      Theory of Dynamical Systems, Cambridge, Cambridge Univ. Press.
       http://titles.cambridge.org/catalogue.asp?isbn=0521575575
6  Hilborn, R. (1994), Chaos and Nonlinear Dyanamics: an Introduction for
      Scientists and Engineers, Oxford Univesity Press.
       http://www4.oup.co.uk/isbn/0-19-850723-2
7  Lichtenberg, A.J. and M. A. Lieberman (1983), Regular and Chaotic Motion,
      Springer-Verlag, New York .
8  Lind, D. and Marcus, B.  (1995) An Introduction to Symbolic Dynamics and
       Coding, Cambridge University Press, Cambridge
       http://www.math.washington.edu/SymbolicDynamics/
9  MacKay, R.S and  J.D. Meiss  (eds) (1987), Hamiltonian Dynamical Systems
      A reprint selection, , Adam Hilger, Bristol
10 Nayfeh, A.H.  and B. Balachandran (1995), Applied Nonlinear Dynamics:
      Analytical, Computational and Experimental Methods
      John Wiley& Sons Inc., New York
      http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471593486.html
11 Ott, E. (1993). Chaos in Dynamical Systems. Cambridge University Press,
      Cambridge. http://us.cambridge.org/titles/catalogue.asp?isbn=0521010845
12 L.E. Reichl, (1992), The Transition to Chaos, in Conservative and
      Classical Systems: Quantum Manifestations  Springer-Verlag, New York
13 Robinson, C. (1999), Dynamical Systems: Stability, Symbolic
      Dynamics, and Chaos, 2nd Edition, Boca Raton, CRC Press.
      http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=8495
14 Ruelle, D. (1989), Elements of Differentiable Dynamics and Bifurcation
    Theory, Academic Press Inc.
15 Tabor, M. (1989), Chaos and Integrability in Nonlinear Dynamics:
      an Introduction, Wiley, New York.
      http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471827282.html
16 Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems
and Chaos, Springer-Verlag New York.
17 Wiggins, S. (1988), Global Bifurcations and Chaos, Springer-Verlag New
York.



[4.2] What technical journals have nonlinear science articles?

Physica D                    The premier journal in Applied Nonlinear Dynamics
Nonlinearity                 Good mix, with a mathematical bias
Chaos                        AIP Journal, with a good physical bent
SIAM J. of Dynamical Systems Online Journal with multimedia
                              http://www.siam.org/journals/siads/siads.htm
Chaos Solitons and Fractals  Low quality, some good applications
Communications in Math Phys  an occasional paper on dynamics
Comm. in Nonlinear Sci.      New Elsevier journal
     and Num. Sim.              http://www.elsevier.com/locate/cnsns
Ergodic Theory and           Rigorous mathematics, and careful work
      Dynamical Systems
International J of           lots of color pictures, variable quality.
    Bifurcation and Chaos
J Differential Equations     A premier journal, but very mathematical
J Dynamics and Diff. Eq.     Good, more focused version of the above
J Dynamics and Stability     Focused on Eng. applications. New editorial
      of Systems              board--stay tuned.
J Fluid Mechanics            Some expt. papers, e.g. transition to turbulence
J Nonlinear Science          a newer journal--haven't read enough yet.
J Statistical Physics        Used to contain seminal dynamical systems papers
Nonlinear Dynamics           Haven't read enough to form an opinion
Nonlinear Science Today      Weekly News: http://www.springer-ny.com/nst/
Nonlinear Processes in       New, variable quality...may be improving
    Geophysics
Physics Letters A            Has a good nonlinear science section
Physical Review E            Lots of Physics articles with nonlinear emphasis
Regular and Chaotic Dynamics Russian Journal http://web.uni.udm.ru/~rcd/



[4.3] What are net sites for nonlinear science materials?

   Bibliography
   http://www.uni-mainz.de/FB/Physik/Chaos/chaosbib.html Mainz http site
   ftp://ftp.uni-mainz.de/pub/chaos/chaosbib/ Mainz ftp site
   http://www-chaos.umd.edu/publications/searchbib.html Seach the Mainz Site
   http://www-chaos.umd.edu/publications/references.html Maryland
   http://www.cpm.mmu.ac.uk/~bruce/combib/  Complexity Bibliography
   http://www.mth.uea.ac.uk/~h720/research/ Ergodic Theory and Dynamical Systems
   http://www.drchaos.net/drchaos/intro.html Nonlinear Dynamics Resources (pdf file)
   http://www.nonlin.tu-muenchen.de/chaos/Projects/miguelbib Sanjuan's Bibliography

   Preprint Archives
   http://www.math.sunysb.edu/dynamics/preprints/  StonyBrook
   http://cnls.lanl.gov/People/nbt/intro.html Los Alamos Preprint Server
   http://xxx.lanl.gov/  Nonlinear Science Eprint Server
   http://www.ma.utexas.edu/mp_arc/mp_arc-home.html   Math-Physics Archive
   http://www.ams.org/global-preprints/ AMS Preprint Servers List

   Conference Announcements
   http://at.yorku.ca/amca/conferen.htm Mathematics Conference List
   http://www.math.sunysb.edu/dynamics/conferences/conferences.html

   StonyBrook List
   http://www.nonlin.tu-muenchen.de/chaos/termine.html Munich List
   http://xxx.lanl.gov/Announce/Conference/ Los Alamos List
   http://www.tam.uiuc.edu/Events/conferences.html Theoretical & Applied Mechanics
   http://www.siam.org/meetings/ds99/index.htm SIAM Dynamical Systems 1999

   Newsletters
   gopher://gopher.siam.org:70/11/siag/ds   SIAM Dynamical Systems Group
   http://www.amsta.leeds.ac.uk/Applied/news.dir/  UK Nonlinear News

   Education Sites
   http://math.bu.edu/DYSYS/  Devaney's Dynamical Systems Project

   Electronic Journals
   http://www.springer-ny.com/nst/   Nonlinear Science Today
   http://www3.interscience.wiley.com/cgi-bin/jtoc?ID=38804 Complexity
   http://journal-ci.csse.monash.edu.au/  Complexity International Journal

   Electronic Texts
   http://cnls.lanl.gov/People/nbt//Book/node1.html An experimental approach
            to nonlinear dynamics and chaos
   http://www.nbi.dk/~predrag/QCcourse/  Lecture Notes on Periodic Orbits
   http://hypertextbook.com/chaos/ The Chaos HyperTextBook

   Institutes and Academic Programs
   http://physicsweb.org/resources/dsearch.phtml Physics Institutes
   http://ip-service.com/WiW/institutes.html   Nonlinear Groups
   http://www-chaos.engr.utk.edu/related.html Research Groups in Chaos

   Java Applets Sites
   http://physics.hallym.ac.kr/education/TIPTOP/VLAB/about.html Virtual Laboratory
   http://monet.physik.unibas.ch/~elmer/pendulum/ Java Pendulum
   http://kogs-www.informatik.uni-hamburg.de/~wiemker/applets/fastfrac/fastfrac.html
     Java Fractal Explorer
   http://www.apmaths.uwo.ca/~bfraser/index.html B. Fraser╣s Nonlinear Lab
   http://www.cmp.caltech.edu/~mcc/Chaos_Course/ Mike Cross' Demos

   Who is Who in Nonlinear Dynamics
   http://www.chaos-gruppe.de/wiw/wiw.html Munich List
   http://www.math.sunysb.edu/dynamics/people/list.html  Stonybrook List

   Lists of Nonlinear sites
   http://makeashorterlink.com/?C58C23C16 Netscape╣s List
   http://cnls.lanl.gov/People/nbt/sites.html  Tufillaro's List
   http://cires.colorado.edu/people/peckham.scott/chaos.html Peckham's List
   http://members.tripod.com/~IgorIvanov/physics/nonlinear.html Physics Encyclopedia
   http://www.maths.ex.ac.uk/~hinke/dss/index.html  Osinga's Software List

   Dynamical Systems
   http://www.math.sunysb.edu/dynamics/  Dynamical Systems Home Page
   http://www.math.psu.edu/gunesch/entropy.html  Entropy and Dynamics

   Chaos sites
   http://www.industrialstreet.net/chaosmetalink/  Chaos Metalink
   http://bofh.priv.at/ifs/  Iterated Function Systems Playground
   http://www.xahlee.org/PageTwo_dir/more.html Xah Lee's dynamics and Fractals pages
   http://acl2.physics.gatech.edu/tutorial/outline.htm Tutorial on Control of Chaos
   http://www.mathsoft.com/mathresources/constants/wellknown/article/0,,2090,00.html
             All about  Feigenbaum Constants
   http://www.stud.ntnu.no/~berland/math/feigenbaum/ The Feigenbaum Fractal
   http://members.aol.com/MTRw3/index.html  Mike Rosenstein's Chaos Page.
   http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/cspls.html Chaos in Psychology
   http://www.eie.polyu.edu.hk/~cktse/NSR/  Movies and Demonstrations

   Time Series
   http://www.drchaos.net/drchaos/refs.html Dynamics and Time Series
   http://astro.uni-tuebingen.de/groups/time/ Time series Analysis
   http://www-personal.buseco.monash.edu.au/~hyndman/TSDL/index.htm
                 Time Series Data Library

   Complex Systems Sites
   http://www.math.upatras.gr/~mboudour/nonlin.html  Complexity Home Page
   http://www.calresco.org/ The Complexity & Artificial Life Web Site
   http://www.physionet.org/  Complexity and Physiology Site

   Fractals Sites
   http://forum.swarthmore.edu/advanced/robertd/index.html#frac A Fractal Gallery
   http://spanky.triumf.ca/www/welcome1.html  The Spanky Fractal DataBase
   http://sprott.physics.wisc.edu/fractals.htm  Sprott's Fractal Gallery
   http://fractales.inria.fr/ Projet Fractales
   http://force.stwing.upenn.edu/~lau/fractal.html  Lau's Fractal Stuff
   http://skal.planet-d.net/quat/f_gal.html 3D Fractals
   http://www.cnam.fr/fractals.html  Fractal Gallery
   http://www.fractaldomains.com/ Fractal Domains Gallery
   http://home1.swipnet.se/~w-17723/fracpro.html Fractal Programs
   http://xahlee.org/PageTwo_dir/MathPrograms_dir/mathPrograms.html#Fractals
         Fractal Programs



[5]   Computational Resources

[5.1] What are general computational resources?
   CAIN Europe Archives
      http://www.can.nl/education/material/software.html  Software Area
   FAQ guide to packages from sci.math.num-analysis
      ftp://rtfm.mit.edu/pub/usenet/news.answers/num-analysis/faq/part1
   NIST Guide to Available Mathematical Software
      http://gams.cam.nist.gov/
   Mathematics Archives Software
      http://archives.math.utk.edu/software.html
   Matpack, C++ numerical methods and data analysis library
       http://www.matpack.de/
   Numerical Recipes Home Page
      http://www.nr.com/

[5.2] Where can I find specialized programs for  nonlinear
science?

   The Academic Software Library:
   Chaos Simulations
Bessoir, T., and A. Wolf, 1990. Demonstrates logistic map, Lyapunov exponents,
billiards in a stadium, sensitive dependence, n-body gravitational motion.
   Chaos Data Analyser
A PC program for analyzing time series. By Sprott, J.C. and G. Rowlands.
For more info:http://sprott.physics.wisc.edu/cda.htm
   Chaos Demonstrations
A PC program for demonstrating chaos, fractals, cellular automata, and related
nonlinear phenomena.  By J. C. Sprott and G. Rowlands.
System: IBM PC or compatible with at least 512K of memory.
Available: The Academic Software Library, (800) 955-TASL. $70.
   Chaotic Dynamics Workbench
Performs interactive numerical experiments on systems modeled by ordinary
differential equations, including: four versions of driven Duffing
oscillators, pendulum, Lorenz, driven Van der Pol osc., driven Brusselator,
and the Henon-Heils system.  By R. Rollins.
System: IBM PC or compatible, 512 KB memory.
Available: The Academic Software Library, (800) 955-TASL, $70

   Applied Chaos Tools
Software package for time series analysis based on the UCSD group's, work.
This package is a companion for Abarbanel's book Analysis of Observed Chaotic
Data, Springer-Verlag.
System: Unix-Motif, Windows 95/NT
For more info see: http://www.zweb.com/apnonlin/csp.html

   AUTO
Bifurcation/Continuation Software (THE standard). The latest version is
AUTO97. The GUI requires X and Motif to be present. There is also a command
line version AUTO86. The software is transported as a compressed file called
auto.tar.Z.
System: versions to run under X windows--SUN or sgi or LINUX
Available: anonymous ftp from ftp://ftp.cs.concordia.ca/pub/doedel/auto

   BZphase
Models Belousov- Zhabotinsky reaction based on the scheme of Ruoff and Noyes.
The dynamics ranges from simple quasisinusoidal oscillations to quasiperiodic,
bursting, complex periodic and chaotic.
System: DOS 6 and higher + PMODE/W DOS Extender. Also openGL version
Available: http://members.tripod.com/~RedAndr/BZPhase.htm

   Chaos
Visual simulation in two- and three-dimensional phase space; based on visual
algorithms rather than canned numerical algorithms; well-suited for
educational use; comes with tutorial exercises. By Bruce Stewart
System: Silicon Graphics workstations, IBM RISC workstations with GL
Available: http://msg.das.bnl.gov/~bstewart/software.html

   Chaos
A Program Collection for the PC by Korsch, H.J. and H-J. Jodl, 1994, A
book/disk combo that gives a hands-on, computer experiment approach to
learning nonlinear dynamics. Some of the modules cover billiard systems,
double pendulum, Duffing oscillator, 1D iterative maps, an "electronic chaos-
generator", the Mandelbrot set, and ODEs.
System: IBM PC or compatible.
Available: $$http://www.springer-ny.com/catalog/np/updates/0-387-57457-3.html

   CHAOS II
Chaos Programs to go with Baker, G. L. and J. P. Gollub (1990) Chaotic
Dynamics. Cambridge, Cambridge Univ.
http://www.cup.org/titles/catalogue.asp?isbn=0521471060
System: IBM, 512K memory, CGA or EGA graphics, True Basic
For more info: contact Gregory Baker, P.O. Box 278 ,Bryn Athyn, PA, 19009

   Chaos Analyser
Programs to Time delay embedding, Attractor (3d) viewing and animation,
PoincarΘ sections, Mutual information, Singular Value Decomposition embedding,
Full Lyapunov spectra (with noise cancellation), Local SVD analysis (for
determining the systems dimension). By Mike Banbrook.
System: Unix, X windows
For more info: http://www.ee.ed.ac.uk/~mb/analysis_progs.html

   Chaos Cookbook
These programs go with J. Pritchard's book, The Chaos Cookbook System:
Programs written in Visual Basic & Turbo Pascal
Available: $$http://www.amazon.com/exec/obidos/ASIN/0750617772

   Chaos Plot
ChaosPlot is a simple program which plots the chaotic behavior of a damped,
driven anharmonic oscillator.
System: Macintosh
For more info:
http://archives.math.utk.edu/software/mac/diffEquations/.directory.html

   Cubic Oscillator Explorer
The CUBIC OSCILLATOR EXPLORER is a Macintosh application which allows
interactive exploration of the chaotic processes of the Cubic Oscillator,
i.e..Duffing's equation.
System: Macintosh + Digidesign DSPácard, Digisystem init 2.6 and (optional)
MIDI Manager
Available: (Missing??) Fractal Music

   DataPlore
Signal and time series analysis package. Contains standard facilities for
signal processing as well as advanced features like wavelet techniques and
methods of nonlinear dynamics.
Systems: MS Windows, Linux, SUN Solaris 2.6
Available: $$http://www.datan.de/dataplore/

   dstool
Free software from Guckenheimer's group at Cornell; DSTool has lots of
examples of chaotic systems, PoincarΘ sections, bifurcation diagrams.
System: Unix, X windows.
Available: ftp://cam.cornell.edu/pub/dstool/

   Dynamical Software Pro
Analyze non-linear dynamics and chaos. Includes ODEs, delay differential
equations, discrete maps, numerical integration, time series embedding, etc.
System: DOS. Microsoft Fortran compiler for user defined equations.
Available: SciTech http://www.scitechint.com/

   Dynamics: Numerical Explorations.
A book + disk by H. Nusse,  and J.Yorke. A hands on approach to learning the
concepts and the many aspects in computing relevant quantities in chaos
System: PC-compatible computer or X-windows system on Unix computers
Available: $$ http://www.springer-ny.com/detail.tpl?isbn=0387982647

   Dynamics Solver
Dynamics Solver solve numerically both initial-value problems and boundary-
value problems for continuous and discrete dynamical systems.
System: Windows 3.1 or Windows 95/98/NT
Available: http://tp.lc.ehu.es/jma/ds/ds.html

   DynaSys
Phase plane portraits of 2D ODEs by Etienne Dupuis
System: Windows 95/98
Available: (Missing??)

   FD3
A program to estimate fractal dimensions of a set. By DiFalco/Sarraille
System: C source code, suitable for compiling for use on a Unix or DOS
platform.
Available: ftp://ftp.cs.csustan.edu/pub/fd3/

   FracGen
FracGen is a freeware program  to create fractal images using Iterated
Function Systems. A tutorial is provided with the program. By Patrick Bangert
System: PC-compatible computer, Windows 3.1
Available: http://212.201.48.1/pbangert/site/fracgen.html

   Fractal Domains
Generates of Mandelbrot and Julia sets. By Dennis C. De Mars
System: Power Macintosh
Available: http://www.fractaldomains.com/

   Fractal Explorer
Generates Mandelbrot and Newton's method fractals. By Peter Stone
System: Power Macintosh
Available: http://usrwww.mpx.com.au/~peterstone/index.html

   GNU Plotutils
The GNU plotutils package contains C/C++ function library for exporting 2-D
vector graphics in many file formats, and for doing vector graphics
animations. The package also contains several command-line programs for
plotting scientific data, such as GNU graph, which is based on libplot, and
ODE integration software.
System: GNU/Linux, FreeBSD, and Unix systems.
Available: http://www.gnu.org/software/plotutils/plotutils.html

   Ilya
A program to visually study a reaction-diffusion model based on the
Brusselator from Future Skills Software, Herber Sauro.
System: Requires Windows 95, at least 256 colours
Available : http://www.fssc.demon.co.uk/rdiffusion/ilya.htm

   INSITE
(It's a Nonlinear Systems Investigative Toolkit for Everyone) is a collection
for the simulation and characterization of dynamical systems, with an emphasis
on chaotic systems. Companion software for T.S. Parker and L.O. Chua (1989)
Practical Numerical Algorithms for Chaotic Systems  Springer Verlag. See their
paper "INSITE A Software Toolkit for the Analysis of Nonlinear Dynamical
Systems," Proc. of the IEEE, 75, 1081-1089 (1987).
System: C codes in Unix Tar or DOS format (later requires QuickWindowC
             or MetaWINDOW/Plus 3.7C. and  MS C compiler 5.1)
Available: INSITE SOFTWARE, p.o. Box 9662, Berkeley, CA , U.S.A.

   Institut fur ComputerGraphik
A collection of programs for developing advanced visualization techniques in
the field of three-dimensional dynamical systems. By L÷ffelmann H., Gr÷ller E.
System: various, requires AVS
Available: http://www.cg.tuwien.ac.at/research/vis/dynsys/

   KAOS1D
A tool for studying one-dimensional (1D) discrete dynamical systems. Does
bifurcation diagrams, etc. for a number of maps
System: PC compatible computer, DOS, VGA graphics
Available: http://www.if.ufrgs.br/~arenzon/jsoftw.html

   LOCBIF
An interactive tool for bifurcation analysis of non-linear ordinary
differential equations ODE's and maps. By Khibnik, Nikolaev, Kuznetsov and V.
Levitin
System: Now part of XPP (See below)
Available: http://www.math.pitt.edu/~bard/classes/wppdoc/locbif.html

   Lyapunov Exponents
Keith Briggs Fortran codes for Lyapunov exponents
System: any with a Fortran compiler
Available: http://more.btexact.com/people/briggsk2/

   Lyapunov Exponents and Time Series
Based on Alan Wolf's algorithm, see [2.11], but a more efficient version.
System: Comes as C source, Fortran source, PC executable, etc
Available: http://www.cooper.edu/engineering/physics/wolf/  (Seems to be
missing?)

   Lyapunov Exponents and Time Series
Michael Banbrook's C codes for Lyapunov exponents & time series analysis
System: Sun with X windows.
Available: http://www.see.ed.ac.uk/~mb/analysis_progs.html

   Lyapunov Exponents Toolbox (LET)
A user-contributed MATLAB toolbox  that provides a graphical user interface
for users to determine the full sets of Lyapunov exponents and Lyapunov
dimensions of discrete and continuous chaotic systems.
System: MATLAB 5
Available: ftp://ftp.mathworks.com/pub/contrib/v5/misc/let

   Lyapunov.m
A Matlab program based on the QR Method , by von Bremen, Udwadia, and
Proskurowski, Physica D, vol. 101, 1-16, (1997)
System: Matlab
Available: http://www.usc.edu/dept/engineering/mecheng/DynCon/

   Macintosh Dynamics Programs
Lists available at: http://hypertextbook.com/chaos/92.shtml
and http://www.xahlee.org/PageTwo_dir/MathPrograms_dir/mathPrograms.html

   MacMath
Comes on a disk with the book MacMath, by Hubbard and West. A collection of
programs for dynamical systems (1 & 2 D maps, 1 to 3D flows). Version 9.2 is
the current version, but West is working on a much improved update.
System: Macintosh
For more info: http://www.math.hmc.edu/codee/solvers/mac-math.html
Available: $$ Springer-Verlag http://www.springer-
ny.com/detail.tpl?isbn=0387941355

   Madonna
Solves Differential and Difference Equations. Runs STELLA. Has a parser with a
control language. By Robert Macey and George Oster at Berkeley
System: Macintosh or Windows 95 or later
Available : $$  http://www.berkeleymadonna.com/

   MatLab Chaos
A collection of routines for generate diagrams which illustrate chaotic
behavior associated with the logistic equation.
System: Requires MatLab.
Available : ftp://ftp.mathworks.com/pub/contrib/misc/chaos/

   MTRChaos
MTRCHAOS and MTRLYAP compute correlation dimension and largest Lyapunov
exponents, delay portraits. By Mike Rosenstein.
System: PC-compatible computer running DOS 3.1 or higher, 640K RAM, and EGA
display. VGA & coprocessor recommended
Available:  ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/

   Nonlinear Dynamics Toolbox
Josh Reiss' NDT includes routines for the analysis of chaotic data, such as
power spectral analyses, determination of the Lyapunov spectrum, mutual
information function, prediction, noise reduction, and dimensional analysis.
System: Windows 95, 98, or NT
Available : Missing??

   NLD Toolbox
This toolbox has many of the standard dynamical systems, By Jeff Brush
System: PC, MS-DOS.
Available: http://www.physik.tu-darmstadt.de/nlp/nldtools/nldtools.html

   ODECalc
A program for integrating boundary value and initial value Problems for up to
9th order ODEs. By Optimal Designs.
System: PC 386+, DOS 3.3+, 16 bit arch.
Available : ftp://ftp.mecheng.asme.org/pub/EDU_TOOL/Ode200.exe

   PHASER
Kocak, H., 1989. Differential and Difference Equations through Computer
Experiments: with a supplementary diskette containing PHASER: An
Animator/Simulator for Dynamical Systems. Demonstrates a large number of 1D-4D
differential equations--many not chaotic--and 1D-3D difference equations.
System: PC-compatible
Available: Springer-Verlag http://www.springer-
ny.com/detail.tpl?isbn=0387142029

   PhysioToolkit
Software for physiologic signal processing and analysis, detection of
physiologically significant events using both classical techniques and novel
methods based on statistical physics and nonlinear dynamics
System: Unix
Available: http://www.physionet.org/physiotools/

   Recurrence Quantification Analysis
Recurrence plots give a visual indication of deterministic behavior in complex
time series. The program, by Webber and Zbilut creates the plots and
quantifies the determinism with five measures.
System: DOS executable
Available:http://homepages.luc.edu/~cwebber/

   SciLab
A simulation program similar in intent to MatLab. It's primarily designed for
systems/signals work, and is large. From INRIA in France.
System: Unix, X Windows, 20 Meg Disk space.
Available : ftp://ftp.inria.fr/INRIA/Projects/Meta2/Scilab

   StdMap
Iterates Area Preserving Maps, by J. D. Meiss. Iterates 8 different maps. It
will find periodic orbits, cantori, stable and unstable manifolds, and allows
you to iterate curves.
System: Macintosh
Available: http://amath.colorado.edu/faculty/jdm/stdmap.html

   STELLA
Simulates dynamics for Biological and Social systems modelling. Uses a
building block metaphor constructing models.
System: Macintosh and Windows PC
Available: $$ http://www.hps-inc.com/edu/stella/stella.htm

   Time Series Tools
An extensive list of Unix tools for Time Series analysis
System: Unix
For more info: http://chuchi.df.uba.ar/guille/TS/tools/tools.html (Link
down??)

   Time Series Analysis from Darmstadt
Four prgrams Time Series analysis and Dimension calculation from the Institute
of Applied Physics at Darmstadt.
System: OS2 or Solaris/Linux/Win9X/NT + Fortran source
For more info: http://www.physik.tu-darmstadt.de/nlp/distribution.html

   Time Series Analysis from Kennel
The program mkball  finds the minimum embedding dimension using the false
strands enhancement of the false neighbors algorithm of Kennel & Abarbanel.
System: any C compiler
Available: ftp://lyapunov.ucsd.edu/pub/nonlinear/mbkall.tar.gz

   TISEAN Time Series Analysis
Agorithms for data representation, prediction, noise reduction, dimension and
Lyapunov estimation, and nonlinearity testing. By Rainer Hegger, Holger Kantz
and Thomas Schreiber
System:  C, C++ and Fortran Codes for Unix,
Available: http://www.mpipks-dresden.mpg.de/~tisean/

   Tufillaro's Programs
From the book Nonlinear Dynamics and Chaos by Tufillaro, Abbot and Reilly
(1992) (for a sample section see
http://www.drchaos.net/drchaos/Book/node1.html). A collection of programs for
the Macintosh.
System: Macintosh
Available: http://www.drchaos.net/drchaos/bb.html

   Unified Life Models (ULM)
ULM, by Stephane Legendre, is a program to study population dynamics and more
generally, discrete dynamical systems. It models any species life cycle graph
(matrix models) inter- and intra-specific competition (non linear systems),
environmental stochasticity, demographic stochasticity (branching processes),
and  metapopulations, migrations (coupled systems).
System: PC/Windows 3.X
Available: from http://www.snv.jussieu.fr

   Virtual Laboratory
Simulations of 2D active media by the Complex Systems Group at the Max Planck
Inst. in Berlin.
System: Requires PV-Wave by Visual Numerics
$$http://www.vni.com/products/wave/
Available: $$ http://w3.rz-berlin.mpg.de/~mik/oertzen/vlm/m_contents.htm

   VRA (Visual Recurrence Analysis)
VRA is a software to display and Study the recurrence plots, first described
by Eckmann, Oliffson Kamphorst And Ruelle in 1987. With RP, one can
graphically detect hidden patterns and structural changes in data or see
similarities in patterns across the time series under study. By Eugene Kononov
Stystem: Windows 95
Available: http://pweb.netcom.com/~eugenek/download.html

   Xphased
Phase 3D plane program for X-windows systems (for systems like Lorenz,
Rossler). Plot, rotate in 3-d, PoincarΘ sections, etc. By Thomas P. Witelski
System: X-windows, Unix, SunOS 4 binary
Available: http://www.alumni.caltech.edu/~witelski/xphased.html

   XPP-Aut
Differential equations and maps for x-windows systems. Links to Auto for
bifurcation analysis. By Bard Ermentrout
System: X-windows, Binaries for many unix systems
Available : ftp://ftp.math.pitt.edu/pub/bardware/tut/start.html

   XSpiral
Simulate pattern formation in 2-D excitable media (in particular 2 models, one
of them the  FitzHugh-Nagumo). By Flavio Fenton.
System: X-windows
Available : (Missing??)




[6] Acknowledgments

    Alan Champneys [e-mail]
    Jim Crutchfield [e-mail]
    S. H. Doole [e-mail]
    David Elliot [e-mail]
    Fred Klingener [e-mail]
    Matt Kennel [e-mail]
    Jose Korneluk [e-mail]
    Wayne Hayes [e-mail]
    Justin Lipton [e-mail]
    Ronnie Mainieri [e-mail]
    Zhen Mei [e-mail]
    Gerard Middleton [e-mail]
    Andy de Paoli [e-mail]
    Lou Pecora [e-mail]
    Pavel Pokorny [e-mail],
    Leon Poon [e-mail]
    Hawley Rising [e-mail],
    Michael Rosenstein [e-mail]
    Harold Ruhl [e-mail]
    Troy Shinbrot [e-mail]
    Viorel Stancu [e-mail]
    Jaroslav Stark [e-mail]
    Bruce Stewart [e-mail]
    Richard Tasgal [e-mail]

Anyone else who would like to contribute, please do! Send me your comments:
http://amath.colorado.edu/appm/faculty/jdm/ Jim Meiss at
[e-mail]