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 See reader questions & answers on this topic!  Help others by sharing your knowledge Html format from: http://amath.colorado.edu/faculty/jdm/faq.html Colorado, http://wwwchaos.engr.utk.edu/faq.html Tennessee, http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html England, http://www.sci.usq.edu.au/mirror/sci.nonlinear.faq/faq.html Australia, http://www.faqs.org/faqs/sci/nonlinearfaq/ Hypertext FAQ Archive Or in other formats: http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.pdf PDF Format, http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.rtf RTF Format, http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.tex old version in TeX, http://www.faqs.org/ftp/faqs/sci/nonlinearfaq the FAQ's site ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/ text format. This FAQ is maintained by Jim Meiss jdm@boulder.colorado.edu. Copyright (c) 19952003 by James D. Meiss, all rights reserved. This FAQ may be posted to any USENET newsgroup, online service, or BBS as long as it is posted in its entirety and includes this copyright statement. This FAQ may not be distributed for financial gain. This FAQ may not be included in commercial collections or compilations without express permission from the author. [1.1] What's New? Fixed lots of broken and outdated links. A few sites seem to be gone, and some new sites appeared. To some extent this FAQ is now been superseded by the Dynamical Systems site run by SIAM. See http://www.dynamicalsystems.org There you will find a glossary that contains most of the answers in this FAQ plus new ones. There is also a growing software list. You are encouraged to contribute to this list, and can do so interactively. [1] About Sci.nonlinear FAQ [1.1] What's New? [2] Basic Theory [2.1] What is nonlinear? [2.2] What is nonlinear science? [2.3] What is a dynamical system? [2.4] What is phase space? [2.5] What is a degree of freedom? [2.6] What is a map? [2.7] How are maps related to flows (differential equations)? [2.8] What is an attractor? [2.9] What is chaos? [2.10] What is sensitive dependence on initial conditions? [2.11] What are Lyapunov exponents? [2.12] What is a Strange Attractor? [2.13] Can computers simulate chaos? [2.14] What is generic? [2.15] What is the minimum phase space dimension for chaos? [3] Applications and Advanced Theory [3.1] What are complex systems? [3.2] What are fractals? [3.3] What do fractals have to do with chaos? [3.4] What are topological and fractal dimension? [3.5] What is a Cantor set? [3.6] What is quantum chaos? [3.7] How do I know if my data are deterministic? [3.8] What is the control of chaos? [3.9] How can I build a chaotic circuit? [3.10] What are simple experiments to demonstrate chaos? [3.11] What is targeting? [3.12] What is time series analysis? [3.13] Is there chaos in the stock market? [3.14] What are solitons? [3.15] What is spatiotemporal 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) threedimensional space, etc.these objects appear the same no matter how we examine them. A nonlinear object, a sphere for example, looks different on different scaleswhen 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 superpositionit 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 nonhuman 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 modelsfor 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 worseand 'any' finite dimensional nonlinear model can be turned into an infinite dimensional linear onee.g. a map x' = f(x) is equivalent to the linear integral equation often called the PerronFrobenius equation p'(x) = integral [ p(y) \delta(xf(y)) dy ]) Here p(x) is a density, which could be interpreted as the probability of finding oneself at the point x, and the Diracdelta 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 aspectsthat 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 scienceand everything in this FAQdeals 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 twodimensional, 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 nonautonomous 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 coordinatesince 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 continuoustime system is a curve, while the orbit of a discretetime 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 2N1 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 preimage for each state (onetoone) 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 n1 n2 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 usefulit 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 timedependent 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 spacethe 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) > (ya+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 continuoustime system which should not exhibit chaos may have numerical solutions which dosee [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 longterm 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 casewe can have a chaotic saddle, an orbit that has some unstable directions as well as stable directions. As a consequence of longterm 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 pushand 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 divergegenerically [2.14] this does not happensome 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 lowdimensional 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: 285317. Eckmann, J.P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from time series." Phys. Rev. A 34: 49714979. [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 PoincaréBendixson 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: 261268). [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, 122 (1992). Binder, P. M. and R. V. Jensen, "Simulating Chaotic Behavior with Finite State Machines," Phys. Rev. 34A, 44603 (1986). Rannou, F., "Numerical Study of Discrete Plane AreaPreserving Mappings," Astron. and Astrophys. 31, 289301 (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 11 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 PoincaréBendixson theorem tells us that there is no chaos in one or twodimensional phase spaces. Chaos is possible in three dimensional flowsstandard examples such as the Lorenz equations are indeed threedimensional, and there are mathematical 3D flows that are provably chaotic (e.g. the 'solenoid'). Note: if the flow is nonautonomous then time is a phase space coordinate, so a system with two physical variables + time becomes threedimensional, and chaos is possible (i.e. Forced secondorder 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(1x). 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 twodimensional phase space for chaos. This is equivalent to the flow result, since a threedimensional flow gives rise to a twodimensional 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(1x), is equivalent to the logistic map. See e.g. S. Ushiki, "Central difference scheme and chaos," Physica 4D (1982) 407424. [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) 293331). 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 selforganized 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 realworld 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 (noninteger) 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/ [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: 261268.) [3.4] What are topological and fractal dimension? See the fractal FAQ: ftp://rtfm.mit.edu/pub/usenet/news.answers/ 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.stand.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, nonconstant, 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 energylevel 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: 289403. See also: http://sagar.physics.neu.edu/qchaos/qc.html Quantum Chaos http://www.mpipksdresden.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 roundoff 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 largethe 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: 734740. [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 misnomerbut 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: 343346. Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review Letters 64(11): 11961199. http://www chaos.umd.edu/publications/abstracts.html#prl64.1196 The idea that chaotic systems can in fact be controlled may be counterintuitiveafter 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 lowdimensional systems; more recently, a few researchers have proposed control techniques for application to high or infinitedimensional 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) 4117. Shinbrot, T., "Chaos: Unpredictable yet Controllable?" Nonlin. Sciences Today, 3:2 (1993) 18. Shinbrot, T., "Progress in the Control of Chaos," Advance in Physics (in press). Ditto, WL & Pecora, LM "Mastering Chaos," Scientific American (Aug. 1993), 7884. Chen, G. & Dong, X, "From Chaos to Order  Perspectives and Methodologies in Controlling Chaotic Nonlinear Dynamical Systems," Int. J. Bif. & Chaos 3 (1993) 13631409. 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) 347982 http://wwwchaos.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 inductorthe 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 97124, 1987. Kennedy M. P., "Robust OP Amp Realization of Chua's Circuit", Frequenz 46, no. 34, 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 9810217692. Parker, T.S., and L.O.Chua, Practical Numerical Algorithms for Chaotic Systems, SpringerVerlag, 1989, ISBN's: 0387966897 and 3540966897. [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, , 305310 (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, 41694172 (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, 373378 (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 threedimensional 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) 13311392. D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics, SpringerVerlag 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/productCd0471585246.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 socioeconomic 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 variablessee [2.4]) is vast, and there is no hope of exploiting the determinism. This is highdimensional 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 selforganizing, resulting in low dimensional chaos. If this turns out to be true, then you can exploit the low dimensional chaos to make shortterm 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 lowdimensional 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 397404. 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 Kortewegde Vries, nonlinear Schrodinger, and sineGordon 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 spatiotemporal chaos? Spatiotemporal 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: 8511112. http://www.cmp.caltech.edu/~mcc/st_chaos.html SpatioTemporal 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 selfsimilar (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 "AubryMather" 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 Nbody problem (New York, SpringerVerlag). [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, AddisonWesley 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, AddisonWesley. 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, SpringerVerlag 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.springerny.com/detail.tpl?isbn=0387979034 7 Moon, F. C. (1992). Chaotic and Fractal Dynamics. New York, John Wiley. http://www.wiley.com/WileyCDA/WileyTitle/productCd0471545716.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/0198501072 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, AddisonWesley http://www.perseusbooksgroup.com/perseuscgibin/display/0738204536 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/productCd0471876844.html 13 Tufillaro, N., T. Abbott, et al. (1992). An Experimental Approach to Nonlinear Dynamics and Chaos. Redwood City, AddisonWesley. 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: 91102. 3 Crawford, J. D. (1991). "Introduction to Bifurcation Theory." Reviews of Modern Physics 63(4): 9911038. 3 Shinbrot, T., C. Grebogi, et al. (1992). "Chaos in a Double Pendulum." Am. J. Phys 60: 491499. 5 David Ruelle. (1980). "Strange Attractors," The Mathematical Intelligencer 2: 12637. Advanced Texts 1 Arnold, V. I. (1978). Mathematical Methods of Classical Mechanics. New York, Springer. http://www.springerny.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, SpringerVerlag New York. 4 Kantz, H., and T. Schreiber (1997). Nonlinear time series analysis. Cambridge, Cambridge University Press http://www.mpipksdresden.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/0198507232 7 Lichtenberg, A.J. and M. A. Lieberman (1983), Regular and Chaotic Motion, SpringerVerlag, 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/productCd0471593486.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 SpringerVerlag, 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/productCd0471827282.html 16 Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, SpringerVerlag New York. 17 Wiggins, S. (1988), Global Bifurcations and Chaos, SpringerVerlag 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 boardstay tuned. J Fluid Mechanics Some expt. papers, e.g. transition to turbulence J Nonlinear Science a newer journalhaven'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.springerny.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.unimainz.de/FB/Physik/Chaos/chaosbib.html Mainz http site ftp://ftp.unimainz.de/pub/chaos/chaosbib/ Mainz ftp site http://wwwchaos.umd.edu/publications/searchbib.html Seach the Mainz Site http://wwwchaos.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.tumuenchen.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_archome.html MathPhysics Archive http://www.ams.org/globalpreprints/ 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.tumuenchen.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.springerny.com/nst/ Nonlinear Science Today http://www3.interscience.wiley.com/cgibin/jtoc?ID=38804 Complexity http://journalci.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://ipservice.com/WiW/institutes.html Nonlinear Groups http://wwwchaos.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://kogswww.informatik.unihamburg.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.chaosgruppe.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.unituebingen.de/groups/time/ Time series Analysis http://wwwpersonal.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.planetd.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/~w17723/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.numanalysis ftp://rtfm.mit.edu/pub/usenet/news.answers/numanalysis/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, nbody 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) 955TASL. $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 HenonHeils system. By R. Rollins. System: IBM PC or compatible, 512 KB memory. Available: The Academic Software Library, (800) 955TASL, $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, SpringerVerlag. System: UnixMotif, 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 windowsSUN or sgi or LINUX Available: anonymous ftp from ftp://ftp.cs.concordia.ca/pub/doedel/ 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 threedimensional phase space; based on visual algorithms rather than canned numerical algorithms; wellsuited 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 HJ. Jodl, 1994, A book/disk combo that gives a handson, 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: $$http://www.springerny.com/catalog/np/updates/0387574573.html CHAOS II Chaos Programs to go with Baker, G. L. and J. P. Gollub (1990) Chaotic Dynamics. Cambridge, Cambridge Univ. http://www.cup.org/titles/catalogue.asp?isbn=0521471060 System: IBM, 512K memory, CGA or EGA graphics, True Basic For more info: contact Gregory Baker, P.O. Box 278 ,Bryn Athyn, PA, 19009 Chaos Analyser Programs to Time delay embedding, Attractor (3d) viewing and animation, Poincaré sections, Mutual information, Singular Value Decomposition embedding, Full Lyapunov spectra (with noise cancellation), Local SVD analysis (for determining the systems dimension). By Mike Banbrook. System: Unix, X windows For more info: http://www.ee.ed.ac.uk/~mb/analysis_progs.html Chaos Cookbook These programs go with J. Pritchard's book, The Chaos Cookbook System: Programs written in Visual Basic & Turbo Pascal Available: $$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: $$http://www.datan.de/dataplore/ dstool Free software from Guckenheimer's group at Cornell; DSTool has lots of examples of chaotic systems, Poincaré sections, bifurcation diagrams. System: Unix, X windows. Available: ftp://cam.cornell.edu/pub/dstool/ Dynamical Software Pro Analyze nonlinear dynamics and chaos. Includes ODEs, delay differential equations, discrete maps, numerical integration, time series embedding, etc. System: DOS. Microsoft Fortran compiler for user defined equations. Available: SciTech http://www.scitechint.com/ Dynamics: Numerical Explorations. A book + disk by H. Nusse, and J.Yorke. A hands on approach to learning the concepts and the many aspects in computing relevant quantities in chaos System: PCcompatible computer or Xwindows system on Unix computers Available: $$ http://www.springerny.com/detail.tpl?isbn=0387982647 Dynamics Solver Dynamics Solver solve numerically both initialvalue 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: PCcompatible 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 2D vector graphics in many file formats, and for doing vector graphics animations. The package also contains several commandline 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 reactiondiffusion 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, 10811089 (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 threedimensional 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 onedimensional (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 nonlinear 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 usercontributed 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, 116, (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/macmath.html Available: $$ SpringerVerlag http://www.springer ny.com/detail.tpl?isbn=0387941355 Madonna Solves Differential and Difference Equations. Runs STELLA. Has a parser with a control language. By Robert Macey and George Oster at Berkeley System: Macintosh or Windows 95 or later Available : $$ 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: PCcompatible 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, MSDOS. Available: http://www.physik.tudarmstadt.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/ 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 1D4D differential equationsmany not chaoticand 1D3D difference equations. System: PCcompatible Available: SpringerVerlag 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/ 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: $$ http://www.hpsinc.com/edu/stella/stella.htm Time Series Tools An extensive list of Unix tools for Time Series analysis System: Unix For more info: http://chuchi.df.uba.ar/guille/TS/tools/tools.html (Link down??) Time Series Analysis from Darmstadt Four prgrams Time Series analysis and Dimension calculation from the Institute of Applied Physics at Darmstadt. System: OS2 or Solaris/Linux/Win9X/NT + Fortran source For more info: http://www.physik.tudarmstadt.de/nlp/distribution.html Time Series Analysis from Kennel The program mkball finds the minimum embedding dimension using the false strands enhancement of the false neighbors algorithm of Kennel & Abarbanel. System: any C compiler Available: ftp://lyapunov.ucsd.edu/pub/nonlinear/mbkall.tar.gz TISEAN Time Series Analysis Agorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing. By Rainer Hegger, Holger Kantz and Thomas Schreiber System: C, C++ and Fortran Codes for Unix, Available: http://www.mpipksdresden.mpg.de/~tisean/ Tufillaro's Programs From the book Nonlinear Dynamics and Chaos by Tufillaro, Abbot and Reilly (1992) (for a sample section see http://www.drchaos.net/drchaos/Book/node1.html). A collection of programs for the Macintosh. System: Macintosh Available: http://www.drchaos.net/drchaos/bb.html Unified Life Models (ULM) ULM, by Stephane Legendre, is a program to study population dynamics and more generally, discrete dynamical systems. It models any species life cycle graph (matrix models) inter and intraspecific competition (non linear systems), environmental stochasticity, demographic stochasticity (branching processes), and metapopulations, migrations (coupled systems). System: PC/Windows 3.X Available: from http://www.snv.jussieu.fr Virtual Laboratory Simulations of 2D active media by the Complex Systems Group at the Max Planck Inst. in Berlin. System: Requires PVWave by Visual Numerics $$http://www.vni.com/products/wave/ Available: $$ http://w3.rzberlin.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 Xwindows systems (for systems like Lorenz, Rossler). Plot, rotate in 3d, Poincaré sections, etc. By Thomas P. Witelski System: Xwindows, Unix, SunOS 4 binary Available: http://www.alumni.caltech.edu/~witelski/xphased.html XPPAut Differential equations and maps for xwindows systems. Links to Auto for bifurcation analysis. By Bard Ermentrout System: Xwindows, Binaries for many unix systems Available : ftp://ftp.math.pitt.edu/pub/bardware/tut/ XSpiral Simulate pattern formation in 2D excitable media (in particular 2 models, one of them the FitzHughNagumo). By Flavio Fenton. System: Xwindows Available : (Missing??) [6] Acknowledgments Alan Champneys a.r.champneys@bristol.ac.uk Jim Crutchfield chaos@gojira.Berkeley.EDU S. H. Doole Stuart.Doole@Bristol.ac.uk David Elliot delliott@isr.umd.edu Fred Klingener klingener@BrockEng.com Matt Kennel kennel@msr.epm.ornl.gov Jose Korneluk jose.korneluk@sfwmd.gov Wayne Hayes wayne@cs.toronto.edu Justin Lipton JML@basil.eng.monash.edu.au Ronnie Mainieri ronnie@cnls.lanl.gov Zhen Mei meizhen@mathematik.unimarburg.de Gerard Middleton middleto@mcmail.CIS.McMaster.CA Andy de Paoli andrea.depaoli@mail.esrin.esa.it Lou Pecora pecora@zoltar.nrl.navy.mil Pavel Pokorny pokornp@tiger.vscht.cz, Leon Poon lpoon@Glue.umd.edu Hawley Rising rising@crl.com, Michael Rosenstein MTR1a@aol.com Harold Ruhl hjr@connix.com Troy Shinbrot shinbrot@bart.chemeng.nwu.edu Viorel Stancu vstancu@sb.tuiasi.ro Jaroslav Stark j.stark@ucl.ac.uk Bruce Stewart bstewart@bnlux1.bnl.gov Richard Tasgal tasgal@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 jdm@boulder.colorado.edu User Contributions:Comment about this article, ask questions, or add new information about this topic:
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