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This is version 2.0 (Sept. 2003) of the Frequently Asked Questions document

for the newsgroup sci.nonlinear. This document can also be found in

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Copyright (c) 1995-2003 by James D. Meiss, all rights reserved. This FAQ may

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posted in its entirety and includes this copyright statement. This FAQ may

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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Fixed lots of broken and outdated links.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To some extent this FAQ is now been superseded by the Dynamical Systems site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 Poincaré

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 Poincaré

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 Poincaré section and again reduce the phase space

dimension by one. Here the Poincaré 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 Poincaré 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 Poincaré 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:

Poincaré 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 Poincaré 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). If lambda is negative, then the orbits converge in time, and the

dynamical system is insensitive to initial conditions. 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 Poincaré-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 Poincaré-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 Poincaré 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 Iplikçi 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 Poincaré 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. Fraser¹s 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 Netscape¹s 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,

Poincaré 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 DSP card, 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, Poincaré 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 Löffelmann H., Gröller 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, Poincaré 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 a.r.ch...@bristol.ac.uk

Jim Crutchfield ch...@gojira.Berkeley.EDU

S. H. Doole Stuart...@Bristol.ac.uk

David Elliot dell...@isr.umd.edu

Fred Klingener klin...@BrockEng.com

Matt Kennel ken...@msr.epm.ornl.gov

Jose Korneluk jose.k...@sfwmd.gov

Wayne Hayes wa...@cs.toronto.edu

Justin Lipton J...@basil.eng.monash.edu.au

Ronnie Mainieri ron...@cnls.lanl.gov

Zhen Mei mei...@mathematik.uni-marburg.de

Gerard Middleton midd...@mcmail.CIS.McMaster.CA

Andy de Paoli andrea....@mail.esrin.esa.it

Lou Pecora pec...@zoltar.nrl.navy.mil

Pavel Pokorny pok...@tiger.vscht.cz,

Leon Poon lp...@Glue.umd.edu

Hawley Rising ris...@crl.com,

Michael Rosenstein MT...@aol.com

Harold Ruhl h...@connix.com

Troy Shinbrot shin...@bart.chem-eng.nwu.edu

Viorel Stancu vst...@sb.tuiasi.ro

Jaroslav Stark j.s...@ucl.ac.uk

Bruce Stewart bste...@bnlux1.bnl.gov

Richard Tasgal tas...@math.tau.ac.il

Anyone else who would like to contribute, please do! Send me your comments:

http://amath.colorado.edu/appm/faculty/jdm/ Jim Meiss at

j...@boulder.colorado.edu

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