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Archive-name: sci/nonlinear-faq Posting-Frequency: monthly This is version 1.0.4 (December 1995) of the Frequently Asked Questions document for the newsgroup sci.nonlinear. This document can also be found in html format as: <http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html> <http://amath.colorado.edu/appm/faculty/jdm/faq.html> <http://www.cis.ohio-state.edu/hypertext/faq/usenet/sci/ nonlinear-faq/faq.html> and in Microsoft Word format as: <ftp://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.hqx> and in text form as: <ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/nonlinear-faq> What's New: Question [17] Added URL for quantum chaos Question [20] Added new book about nonlinear circuits Question [21] A new question: simple experimental demonstrations. Please send suggestions! Question [25] Soliton WWW site added. Question [28] Updates to bibliography sites, Electronic Texts, Conference Announcements Question [29] Addional Software sites added, and updated some outdated links This FAQ is maintained by Jim Meiss <jdm@boulder.colorado.edu>. Copyright (c) 1995 by James D. Meiss, all rights reserved. This FAQ may be posted to any USENET newsgroup, on-line service, or BBS as long as it is posted in its entirety and includes this copyright statement. This FAQ may not be distributed for financial gain. This FAQ may not be included in commercial collections or compilations without express permission from the author. Table of Contents [1] What is nonlinear? [2] What is nonlinear science? [3] What is a dynamical system? [4] What is phase space? [5] What is a degree of freedom? [6] What is a map? [7] How are maps related to flows (differential equations)? [8] What is chaos? [9] What is sensitive dependence on initial conditions? [10] What are Lyapunov exponents? [11] What is Generic? [12] What is the minimum phase space dimension for chaos? [13] What are complex systems? [14] What are fractals? [15] What do fractals have to do with chaos? [16] What are topological and fractal dimension? [17] What is quantum chaos? [18] How do I know if my data is deterministic? [19] What is the control of chaos? [20] How can I build a chaotic circuit? [21] What are simple experiments that I can do to demonstrate chaos? [22] What is targeting? [23] What is time series analysis? [24] Is there chaos in the stock market? [25] What are solitons? [26] What should I read to learn more? [27] What technical journals have nonlinear science articles? [28] What are net sites for nonlinear science materials? [29] What nonlinear science software is available? [30] Acknowledgments ********** [1] What is nonlinear? In geometry, linearity refers to Euclidean objects: lines, planes, (flat) three dimensional space, etc.--these objects appear the same no matter how we examine them. A nonlinear object, a sphere for example, looks different on different scales--when looked at closely enough it looks like a plane, and from a far enough distance it looks like a point. In algebra, we define linearity in terms of functions which have the property f(x+y) = f(x)+f(y) and f(ax) = af(x). Nonlinear is defined as the negation of linear. This means that the result f may be out of proportion to the input x or y. The result may be more than linear, as when a diode begins to pass current; or less than linear, as when finite resources limit Malthusian population growth. Thus the fundamental simplifying tools of linear analysis are no longer available: for example, for a linear system, if we have two zeros, f(x) = 0 and f(y) = 0, then we automatically have a third zero f(x+y) = 0 (in fact there are infinitely many zeros as well, since linearity implies that f(ax+by) = 0 for any a and b). This is called the principle of superposition--it gives many solutions from a few. For nonlinear systems, each solution much be fought for (generally) with unvarying ardor! ********** [2] What is nonlinear science? Stanislaw Ulam reportedly said (something like) "Calling a science 'nonlinear' is like calling zoology 'the study of non-human animals'. So why do we have a name that appears to be merely a negative? Firstly, linearity is rather special, and no model of a real system is truly linear (you might protest that quantum mechanics is an exception, however this is at the expense of infinite dimensionality which is just as bad or worse-- and 'any' finite dimensional nonlinear model can be turned into an infinite dimensional linear one). Some things are profitably studied as linear approximations to the real models--for example the fact that Hooke's law, the linear law of elasticity (strain is proportional to stress) is approximately valid for a pendulum of small amplitude implies that its period is approximately independent of amplitude (i.e. Period(Amplitude) = Period(2xAmplitude)). However, as the amplitude gets large the period gets longer, a fundamental effect of nonlinearity in the pendulum equations. 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 ([Q3]see [Q3]). 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 has a whole. Nonlinear science has applications to a wide variety of fields, from mathematics, physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects--that it brings researchers from many disciplines together with a common language. ********** [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 coin toss has two consequents with equal probability for each initial state). Most of nonlinear science--and everything in this FAQ--deals with deterministic systems. A dynamical system can have discrete or continuous time. The discrete case is defined by a map, z_1 = f(z_0), that gives the state z_1 resulting from the initial state z_0 at the next time value. The continuous case is defined by a "flow", z(t) = \phi_t(z_0), which gives the state at time t, given that the state was z_0 at time 0. A smooth flow can be differentiated w.r.t. time to give a differential equation, dz/dt = F(z). In this case we call F(z) a "vector field," it gives a vector pointing in the direction of the velocity at every point in phase space. ********** [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 coin toss, we have two states heads and tails), countably infinite (e.g. state variables are integers), or uncountably infinite (e.g. state variables are real numbers). Implicit in the notion is that a particular state in phase space specifies the system completely; it is all we need to know about the system to have complete knowledge of the immediate future. Thus the phase space of the planar pendulum is two dimensional, consisting of the position (angle) and velocity. According to Newton, specification of these two variables uniquely determines the subsequent motion of the pendulum. Note that if we have a non-autonomous system, where the map or vector field depends explicitly on time (e.g. a model for plant growth depending on solar flux), then according to our definition of phase space, we must include time as a phase space coordinate--since one must specify a specific time (e.g. 3PM on Tuesday) to know the subsequent motion. Thus dz/dt = F(z,t) is a dynamical system on the phase space consisting of (z,t), with the addition the new dynamical equation dt/dt = 1. The path in phase space traced out by a solution of an initial value problem is called an orbit or trajectory of the dynamical system. If the state variables take real values in a continuum, the orbit of a continuous-time system is a curve, while the orbit of a discrete-time system is a sequence of points. ********** [5] What is a degree of freedom? The notion of "degrees of freedom" as it is used for 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 the 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 Hamiltonian H(q,p) with N d.o.f. nominally moves in a 2N dimensional phase space. However, energy is conserved, and therefore the motion is really on a 2N-1 dimensional energy surface, H(q,p) = E. Thus e.g. the planar, circular restricted 3 body problem is 2 d.o.f., and motion is on the 3D energy surface of constant "Jacobi constant." It can be reduced to a 2D area preserving map by Poincare section (see Q6]). 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). ********** [6] What is a map? A map is simply a function, f, on the phase space that gives the next state, f(z), (the image) of the system given its current state, z. (Often you will find the notation z' = f(z), where the prime means the next point, not the derivative.) Now a function must have a single value for each state, but there could be several different states that give rise to the same image. Maps that allow every state in the phase space to be accessed (onto) and which have precisely one pre-image for each state (one-to-one) are invertible. If in addition the map and its inverse are continuous (with respect to the phase space coordinate z), then it is called a homeomorphism. A homeomorphism that has at least one continuous derivative (w.r.t. z) and a continuously differentiable inverse is a diffeomorphism. Iteration of a map means repeatedly applying the map to the consequents of the previous application. Thus we get a sequence n z = f(z ) = f(f(z ).... = f (z ) n n-1 n-2 0 This sequence is the orbit or trajectory of the dynamical system with initial condition z_0. ********** [7] How are maps related to flows (differential equations)? Every differential equation gives rise to a map, the time one map, defined by advancing the flow one unit of time. This map may or may not be useful. If the differential equation contains a term or terms periodic in time, then the time T map (where T is the period) is very useful--it is an example of a Poincare section. The time T map in a system with periodic terms is also called a stroboscopic map, since we are effectively looking at the location in phase space with a stroboscope tuned to the period T. This map is useful because it permits us to dispense with time as a phase space coordinate: the remaining coordinates describe the state completely so long as we agree to consider the same instant within every period. In autonomous systems (no time-dependent terms in the equations), it may also be possible to define a Poincare section and again reduce the phase space dimension by one. Here the Poincare 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 Poincare 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 Poincare section of a flow are necessarily invertible, because the flow has a unique solution through any point in phase space--the solution is unique both forward and backward in time. However, noninvertible maps can be relevant to differential equations: Poincare maps are sometimes very well approximated by noninvertible maps. For example, the Henon map (x,y) -> (-y-a+x^2,bx) with small |b| is close to the logistic map, x -> -a+x^2. It is often (though not always) possible to go backwards, from an invertible map to a differential equation having the map as its Poincare map. This is called a suspension of the map. One can also do this procedure approximately for maps that are close to the identity, giving a flow that approximates the map to some order. This is extremely useful in bifurcation theory. Note that any numerical solution procedure for a differential initial value problem which uses discrete time steps in the approximation is effectively a map. This is not a trivial observation; it helps explain for example why a continuous-time system which should not exhibit chaos may have numerical solutions which do--[Q12]see [Q12]. ********** [8] What is chaos? Roughly speaking, chaos is 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 further is in practice always predictable in the short term. The key to long- term unpredictability is a property known as sensitivity to (or sensitive dependence on) initial conditions. For a dynamical system to be chaotic it must have a 'large' set of initial conditions which are highly unstable. No matter how precisely you measure the initial condition in these systems, your prediction of its subsequent motion goes radically wrong after a short time. Typically (see [Q20] for one definition of 'typical'), the predictability horizon grows only logarithmically with the precision of measurement (for positive Lyapunov exponents, see [Q10]). Thus for each increase in precision by a factor of 10, say, you may only be able to predict two more time units. 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 [Q9]). To these two requirements Devaney 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 with a large basin. However, this need not be the case--we can have a chaotic saddle, an orbit that has some unstable directions as well as stable directions. As a consequence of long-term unpredictability, time series from chaotic systems may appear irregular and disorderly. However, chaos is definitely not (as the name might suggest) complete disorder; it is disorder in a deterministic dynamical system, which is always predictable for short times. The possibility of a predictability horizon in a deterministic system came as something of a shock to mathematicians and physicists, long used to a notion attributed to Laplace that, given precise knowledge of the initial conditions, it should be possible to predict the future of the universe. This mistaken faith in predictability was engendered by the success of Newton's mechanics applied to planetary motions, which happen to be regular on human historic time scales, but chaotic on the 5 million year time scale (see e.g. "Newton's Clock", by Ivars Peterson (1993 W.H. Freeman) . ********** [9] What is sensitive dependence on initial conditions? Consider a boulder precariously perched on the top of an ideal hill. The slightest push will cause the boulder to roll down one side of the hill or the other: the subsequent behavior depends sensitively on the direction of the push--and the push can be arbitrarily small. If you are standing at the bottom of the hill on one side, then you would dearly like to know which direction the boulder will fall. 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. That is 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 Q[20]) for chaotic systems. Note also that we most definitely do not require ALL nearby initial points diverge--generically [Q20] this does not happen--some nearby points may converge. (We may modify our hilltop analogy slightly and say the 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. ********** [10] What are Lyapunov exponents? The hardest thing to get right about Lyapunov exponents is the spelling of Lyapunov, which you will variously find as Liapunov, Lyapunof and even Liapunoff. Of course Lyapunov is really spelled in the Cyrillic alphabet: (Lambda)(backwards r)(pi)(Y)(H)(0)(B). Now that there is an ANSI standard of transliteration for Cyrillic, we expect all references to converge on the version Lyapunov. Lyapunov was born in Russia in 6 June 1857. He was greatly influenced by Chebyshev and was a student with Markov. He was also a passionate man: Lyapunov shot himself the day his wife died. He died 3 Nov. 1918, three days later. According to the request on a note he left, Lyapunov was buried with his wife. [biographical data from a biography by A. T.Grigorian]. Lyapunov left us with more than just a simple note. He left a collection of papers on the equilibrium shape of rotating liquids, on probability, and on the stability of low-dimensional dynamical systems. It was from his dissertation that the notion of Lyapunov exponent emerged. Lyapunov was interested in showing how to discover if a solution to a dynamical system is stable or not for all times. The usual method of studying stability --- linearizing around the solution --- was not good enough. If you waited long enough the small errors due to linearization would pile up and make the approximation invalid. Lyapunov developed concepts to overcome these difficulties. Lyapunov exponents measure the rate of divergence of nearby orbits. Roughly speaking the (maximal) Lyapunov exponent is the time average logarithmic growth rate of the distance between two nearby orbits. Positive Lyapunov exponents indicate sensitive dependence on initial conditions, since the distance then grows (on average in time and locally in phase space) exponentially in time. There are basically two ways to compute Lyapunov exponents. In one way one chooses two nearby points, evolves them in time, measuring the growth rate of the distance between them. This is useful when one has a time series, but has the disadvantage that the growth rate is really not a local effect as the points separate. A better way is to measure the growth rate of tangent vectors to a given orbit. More precisely, consider a map f in an m dimensional phase space, and its derivative matrix Df(x). Let v be a tangent vector at the point x. Then we define a function 1 n L(x,v) = lim --- ln |( Df (x)v )| n -> oo n Now the Multiplicative Ergodic Theorem of Oseledec states that this limit exists for almost all points x and all tangent vectors v. There are at most m distinct values of L as we let v range over the tangent space. These are the Lyapunov exponents at x. For more information on computing the exponents see Wolf, A., J. B. Swift, et al. (1985). "Determining Lyapunov Exponents from a Time Series." Physica D 16: 285-317. Eckmann, J.-P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from time series." Phys. Rev. A 34: 4971-4979. ********** [11] What is Generic? Generic in dynamical systems is intended to convey "usual" or, more properly, "observable". Roughly speaking, a property is generic over a class if any system in the class can be modified ever so slightly (perturbed), into one with that property. The formal definition is done in the language of topology: Consider the class to be a space of systems, and suppose it has a topology (some notion of a neighborhood, or an open set). A subset of this space is *dense* if its closure (the subset plus the limits of all sequences in the subset) is the whole space. It is *open and dense* if it is also an open set (union of neighborhoods). A set is *countable* if it can be put into 1-1 correspondence with the counting numbers. A *countable intersection of open dense sets* is the intersection of a countable number of open dense sets. If all such intersections in a space are also dense, then the space is called a *Baire* space, which basically means its 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, them 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 (needn't 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). ********** [12] 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 Poincare-Bendixson theorem tells us that there is no chaos in one or two dimensional phase spaces. Chaos is possible in three dimensional flows--standard examples such as the Lorenz equations are indeed three dimensional, and there are mathematical 3D flows that are provably chaotic (e.g. the 'solenoid'). Note: if the flow is non-autonomous then time is a phase space coordinate, so a system with two physical variables + time becomes three dimensional, and chaos is possible (i.e. Forced second-order oscillators do exhibit chaos.) For maps, it is possible to have chaos in one dimension, but only if the map is not invertible. A prominent example is the Logistic map x' = f(x) = rx(1- x). This is provably chaotic for r = 4, and many other values of r as well (see e.g. Devaney). Note that every point has two preimages, except for the image of the critical point x=1/2, so this map is not invertible. For homeomorphisms, we must have at least two dimensional phase space for chaos. This is equivalent to the flow result, since a three dimensional flow gives rise to a two dimensional homeomorphism by Poincare section (see [Q6]). Note that a numerical algorithm for a differential equation is a map, because time on the computer is necessarily discrete. Thus numerical solutions of two and even one dimensional systems of ordinary differential equations may exhibit chaos. Usually this results from choosing the size of the time step too large. For example Euler discretization of the Logistic differential equation, dx/dt = rx(1-x), is equivalent to the logistic map. See e.g. S. Ushiki, Central difference scheme and chaos, Physica D, vol. 4 (1982) 407-424. ********** [13] What are complex systems? A complex system, as I understand it, is a system with many inequivalent degrees of freedom. 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 D _52_ (1991) 293-331). In these problems, many individual systems conspire to produce a single collective rhythm. The notion of complex systems has received lots of popular press, but it is not really clear as of yet if there is a "theory" about a "concept". We are withholding judgement. ********** [14] 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. "Fractal" is a term which has undergone refinement of definition by a lot of people, but was first coined by B. Mandelbrot and defined as a set with fractional (non-integer) dimension (Hausdorff dimension, see [Q16]). While this definition has a lot of drawbacks, note that it says nothing about self- similarity--even though the most commonly known fractals are indeed self- similar. See the extensive FAQ from sci.fractals at <ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq> <http://www.cis.ohio-state.edu/hypertext/faq/usenet/fractal-faq/faq.html> ********** [15] What do fractals have to do with chaos? Often chaotic dynamical systems exhibit fractal structures in phase space. However, there is no direct relation. There are chaotic systems that have nonfractal limit sets (e.g. Arnold's cat map) and fractal structures that can arise in nonchaotic dynamics (see e.g. Grebogi, C., et al. (1984). "Strange Attractors that are not Chaotic." Physica 13D: 261-268.) ********** [16] What are topological and fractal dimension? See the fractal FAQ: <ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq> <http://www.cis.ohio-state.edu/hypertext/faq/usenet/fractal-faq/faq.html> ********** [17] What is quantum chaos? 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, etc. 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 acoustic waves to their corresponding ray equations. Among recent results in quantum chaos is a prediction relating the chaos in the classical problem to the statistics of energy-level spacings in the semiclassical quantum regime. Classical chaos can be used to analyze such ostensibly quantum systems as the hydrogen atom, where classical predictions of microwave ionization thresholds agree with experiments. See Koch, P. M. and K. A. H. van Leeuwen (1995). "Importance of Resonances in Microwave Ionization of Excited Hydrogen Atoms." Physics Reports 255: 289-403. See: <http://sagar.cas.neu.edu/qchaos/qc.html> Quantum Chaos Home Page ********** [18] How do I know if my data is deterministic? 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 s system one typically relies on phase space embedding methods, see [23]. Typically one chooses an embedding dimension, and investigates the propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you can increase the dimension to obtain a deterministic looking error, then you are done. Though it may sound simple it is not really! One complication is that as the dimension increases the search for a nearby state requires a lot more computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) to find a suitably close candidate. If the embedding dimension (number of measures per state) is chosen too small (less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the dimension too large--the method will work. Practically, anything approaching about 10 dimensions is considered so large that a stochastic description is probably more suitable and convenient anyway. See e.g., Sugihara, G. and R. M. May (1990). "Nonlinear Forcasting as a Way of Distinguishing Chaos from Measurement Error in Time Series." Nature 344: 734-740. ********** [19] What is the control of chaos? Control of chaos has come to mean the two things: (1) stabilization of unstable periodic orbits, (2) use of recurrence to allow stabilization to be applied locally. Thus term "control of chaos" is somewhat of a misnomer--but the name has stuck. The ideas for controlling chaos originated in the work of Hubler followed by the Maryland Group. Hubler, A. W. (1989). "Adaptive Control of Chaotic Systems." Helv. Phys. Acta 62: 343-346). Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review Letters 64(11): 1196-1199. <http://www-chaos.umd.edu/publications/abstracts.html#prl64.1196> The idea that chaotic systems can in fact be controlled may be counterintuitive -- after all they are unpredictable in the long term. Nevertheless, numerous theorists have independently developed methods which can be applied to chaotic systems, and many experimentalists have demonstrated that physical chaotic systems respond well to both simple and sophisticated control strategies. Applications have been proposed in such diverse areas of research as communications, electronics, physiology, epidemiology, fluid mechanics and chemistry. The great bulk of this work has been restricted to low-dimensional systems; more recently, a few researchers have proposed control techniques for application to high- or infinite-dimensional systems. The literature on the subject of the control of chaos is quite voluminous; nevertheless several reviews of the literature are available, including: Shinbrot, T. Ott, E., Grebogi, C. & Yorke, J.A., "Using Small Perturbations to Control Chaos," Nature, 363 (1993) 411-7. Shinbrot, T., "Chaos: Unpredictable yet Controllable?" Nonlin. Sciences Today, 3:2 (1993) 1-8. Shinbrot, T., "Progress in the Control of Chaos," Advance in Physics (in press). Ditto, WL & Pecora, LM "Mastering Chaos," Scientific American (Aug. 1993), 78-84. Chen, G. & Dong, X, "From Chaos to Order -- Perspectives and Methodologies in Controlling Chaotic Nonlinear Dynamical Systems," Int. J. Bif. & Chaos 3 (1993) 1363-1409. It is generically quite difficult to control high dimensional systems; an alternative approach is to use control to reduce the dimension before applying one of the above techniques. This approach is in its infancy; see: Auerbach, D., Ott, E., Grebogi, C., and Yorke, J.A. "Controlling Chaos in High Dimensional Systems," Phys. Rev. Lett. 69 (1992) 3479-82 <http://www-chaos.umd.edu/publications/abstracts.html#prl69.3479> ********** [20] How can I build a chaotic circuit? There are many different physical systems which display chaos, dripping faucets, water wheels, oscillating magnetic ribbons etc. but the most simple systems which can be easily implemented are chaotic circuits. In fact an electronic circuit was one of the first demonstrations of chaos which showed that chaos is not just a mathematical abstraction. Leon Chua designed the circuit 1983. The circuit he designed, now known as Chua's circuit, consists of a piecewise linear resistor as its nonlinearity (making analysis very easy) plus two capacitors, one resistor and one inductor--the circuit is unforced (autonomous). In fact the chaotic aspects (bifurcation values, Lyapunov exponents, various dimensions etc.) of this circuit have been extensively studied in the literature both experimentally and theoretically. It is extremely easy to build and presents beautiful attractors (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: Kennedy M. P., "Robust OP Amp Realization of Chua's Circuit", Frequenz, vol. 46, no. 3-4, 1992. Madan, R. A., "Chua's Circuit: A paradigm for chaos", ed. R. A. Madan, Singapore: World Scientific, 1993. Pecora, L. and Carroll, T. "Nonlinear Dynamics in Circuits", Singapore: World Scientific, 1995. ******** [21] What are simple experiments that I can do 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://www.ibm.com/Stretch/EOS/chaos.html>. My favorite double pendulum consists of two identical planar pendula, so that you can demonstrate sensitive dependence [Q9]. One of the simplest chemical systems that shows chaos is the Belousov- Zhabotinsky reacation.The book by Strogatz [Q26] has a good introduction to this subject, see also <http://taylor.mc.duke.edu/~rubin/BZ/BZexplain.html> for some more information. The Chaotic waterwheel, while not so simple to build, is an exact realization of Lorenz famous equaions. This is nicely discussed in Strogatz book [Q26] as well. Chua's nonlinear curcuit is also a good example. See [Q20] above. ******** [22] What is targeting? To direct trajectories in chaotic systems, one can generically apply small perturbations; see: Shinbrot, T. Ott, E., Grebogi, C. & Yorke, J.A., "Using Small Perturbations to Control Chaos," Nature, 363 (1993) 411-7). We are still awaiting a good answer to this question. ********** [23] What is time series analysis? 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 dynamicists 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 in three dimensional space by measuring only one of its coordinates, say x(t), and plotting the delay coordinates (x(t), x(t+h), x(t+2h)) for a fixed h. It is important to emphasize that the idea of using derivatives or delay coordinates in time series modeling is nothing new. It goes back at least to the work of Yule, who in 1927 used an autoregressive (AR) model to make a predictive model for the sunspot cycle. AR models are nothing more than delay coordinates used with a linear model. Delays, derivatives, principal components, and a variety of other methods of reconstruction have been widely used in time series analysis since the early 50's, and are described in several hundred books. The new aspects raised by dynamical systems theory are (i) the implied geometric view of temporal behavior and (ii) the existence of "geometric invariants", such as dimension and Lyapunov exponents. The central question was not whether delay coordinates are useful for time series analysis, but rather whether reconstruction methods preserve the geometry and the geometric invariants of dynamical systems. (Packard, Crutchfield, Farmer & Shaw) G.U. Yule, Phil. Trans. R. Soc. London A 226 (1927) p. 267. N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, "Geometry from a time series", Phys. Rev. Lett. vol. 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. of Modern Physics 65 (1993) 1331-1392. D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics, Springer-Verlag ********** [24] Is there chaos in the stock market? In order to address this question, we must first agree what we mean by chaos, see [Q8]. In dynamical systems theory, chaos means irregular fluctuations in a deterministic system (see [Q3] and [Q18]). This means the system behaves irregularly because of its own internal logic, not because of random forces acting from outside. Of course if you define your dynamical system to be the socio-economic behavior of the entire planet, nothing acts randomly from outside (except perhaps the occasional meteor), so you have a dynamical system. But its dimension (number of state variables--see [Q4]) is vast, and there is no hope of exploiting the determinism. This is high-dimensional chaos, which might just as well be truly random behavior. In this sense, the stock market is chaotic, but who cares? To be useful, economic chaos would have to involve some kind of collective behavior which can be fully described by a small number of variables. In the lingo, the system would have to be self-organizing, resulting in low- dimensional chaos. If this turns out to be true, then you can exploit the low- dimensional chaos to make short-term predictions. The problem is to identify the state variable 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. 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. There are a number of people and groups trying to find these patterns. Some of these groups are known to outsiders, because they include prominent researchers in the field of chaos; we have no idea whether they are succeeding or not. If you know chaos theory and would like to make yourself a slave to the rhythms of market trading, you can probably find a major 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. ********** [25] What are solitons? Consider this frequently asked question: The Fourier transform can simplify the evolution of linear differential equations; is there a counterpart which similarly simplifies nonlinear equations? The answer is No. Nonlinear equations are qualitatively more complex than linear equations, and a procedure which gives the dynamics as simply as for linear equations must contain a mistake. There are, however, exceptions to any rule. Certain nonlinear differential equations can be fully solved by, e.g., the "inverse scattering method." Examples are the Korteweg-de Vries, nonlinear Schrodinger, and sine-Gordon equations. In these cases the real space maps, in a rather abstract way, to an inverse space, which is comprised of continuous and discrete parts and evolves linearly in time. The continuous part typically corresponds to radiation and the discrete parts to stable solitary waves, i.e. pulses, which are called solitons. The linear evolution of the inverse space means that solitons will emerge virtually unaffected from interactions with anything, giving them great stability. More broadly, there is a wide variety of systems which support stable solitary waves through a balance of dispersion and nonlinearity. Though these systems may not be integrable as above, in many cases they are close to systems which are, and the solitary waves may share many of the stability properties of true solitons, especially that of surviving interactions with other solitary waves (mostly) unscathed. It is widely accepted to call these solitary waves solitons, albeit with qualifications. Why solitons? Solitons are simply a fundamental nonlinear wave phenomenon. Many very basic linear systems with the addition of the simplest possible or first order nonlinearity support solitons; this universality means that solitons will arise in many important physical situations. Optical fibers can support solitons, which because of their great stability are an ideal medium for transmitting information. In a few years long distance telephone communications will likely be carried via solitons. The soliton literature is by now vast. Two books which contain clear discussions of solitons as well as references to original papers are Alan C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, Penn. (1985) Mark J. Ablowitz, Solitons, nonlinear evolution equations and inverse scattering, Cambridge (1991). See the Soliton Home page: <http://www.ma.hw.ac.uk/solitons/> ********** [26] What should I read to learn more? Popularizations 1. Gleick, J. (1987). Chaos, the Making of a New Science. London, Heinemann. 2. Stewart, I. (1989). Does God Play Dice? Cambridge, Blackwell. 3. Devaney, R. L. (1990). Chaos, Fractals, and Dynamics : Computer Experiments in Mathematics. Menlo Park, Addison-Wesley Pub. Co. 4. Lorenz, E., (1994) The Essence of Chaos, University of Washington Press. Introductory Texts 1. Percival, I. C. and D. Richard (1982). Introduction to Dynamics. Cambridge, Cambridge Univ. Press. <http://www.cup.org/Titles/28/0521281490.html> 2. Devaney, R. L. (1986). An Introduction to Chaotic Dynamical Systems. Menlo Park, Benjamin/Cummings. <http://www.aw.com/he/Math/MathCategories/ABP/devaney13046.html> 3. Baker, G. L. and J. P. Gollub (1990). Chaotic Dynamics. Cambridge, Cambridge Univ. Press. <http://www.cup.org/Titles/38/052138897X.html> 4. Tufillaro, N., T. Abbott, et al. (1992). An Experimental Approach to Nonlinear Dynamics and Chaos. Redwood City, Addison-Wesley. <http://www.aw.com/he/Math/MathCategories/ABP/tufillaro55441.html> 5. Jurgens, H., H.-O. Peitgen, et al. (1993). Chaos and Fractals: New Frontiers of Science. New York, Springer Verlag. <http://www.springer-ny.com> 6. Glendinning, P. (1994). Stability, Instability and Chaos. Cambridge, Cambridge Univ Press. <http://www.cup.org/Titles/415/0521415535.html> 7. Strogatz, S. (1994). Nonlinear Dynamics and Chaos. Reading, Addison- Wesley. <http://www.aw.com/he/Math/MathCategories/Chaos/strogatz54344.html> 8. Moon, F. C. (1992). Chaotic and Fractal Dynamics. New York, John Wiley. <gopher://gopher.infor.com:6000/0exec%3A-v%20a%20R9469895-9471436-/ .text/Main%3A/.bin/aview> 9. Turcotte, Donald L. (1992). Fractals and Chaos in Geology and Geophysics, Cambridge Univ. Press. <http://www.wiley.com> 10. Ott, Edward (1993). Chaos in Dynamical Systems. Cambridge, Cambridge University Press. <http://www.cup.org/Titles/43/0521432154.html> 11. D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics, Springer-Verlag New York. <http://www.cnd.mcgill.ca/Understanding/> Introductory Articles 1. May, R. M. (1986). "When Two and Two Do Not Make Four." Proc. Royal Soc. B228: 241. 2. Berry, M. V. (1981). "Regularity and Chaos in Classical Mechanics, Illustrated by Three Deformations of a Circular Billiard." Eur. J. Phys. 2: 91-102. 3. Crawford, J. D. (1991). "Introduction to Bifurcation Theory." Reviews of Modern Physics 63(4): 991-1038. 4. Shinbrot, T., C. Grebogi, et al. (1992). "Chaos in a Double Pendulum." Am. J. Phys 60: 491-499. 5. David Ruelle. (1980). "Strange Attractors," The Mathematical Intelligencer 2: 126-37. ********** [27] What technical journals have nonlinear science articles? Physica D The premier journal in Nonlinear Dynamics Nonlinearity Good mix, with a mathematical bias Chaos AIP Journal, with a good physical bent Physics Letters A Has a good nonlinear science section Physical Review E Lots of Physics articles with nonlinear emphasis Ergodic Theory and Rigorous mathematics, and careful work Dynamical Systems J Differential Equations A premier journal, but very mathematical J Dynamics and Diff. Eq. Good, more focused version of the above J Dynamics and Stability Focused on Eng. applications. New editorial of Systems board--stay tuned. J Statistical Physics Used to contain seminal dynamical systems papers SIAM Journals Only the odd dynamical systems paper J Fluid Mechanics Some expt. papers, e.g. transition to turbulence Nonlinear Dynamics Haven't read enough to form an opinion J Nonlinear Science a newer journal--haven't read enough yet. Nonlinear Science Today News of the week see: <http://www.springer-ny.com/nst> International J of lots of color pictures, variable quality. Bifurcation and Chaos Chaos Solitons and Fractals Variable quality, some good applications Communications in Math Phys an occasional paper on dynamics Nonlinear Processes in New, variable quality...may be improving Geophysics ********** [28] What are net sites for nonlinear science materials? Bibliography <http://www.uni-mainz.de/FB/Physik/Chaos/chaosbib.html> <ftp://ftp.uni-mainz.de/pub/chaos/chaosbib/> <http://t13.lanl.gov/ronnie/cabinet.html> <http://www-chaos.umd.edu/publications/references.html> <http://www-chaos.umd.edu/~msanjuan/biblio.html> Preprint Archives <http://cnls-www.lanl.gov/nbt/intro.html> Los Alamos Preprint Server <http://xyz.lanl.gov/> Nonlinear Science Eprint Server <http://www.ma.utexas.edu/mp_arc/mp_arc-home.html> Math-Physics Archive <http://e-math.ams.org/web/preprints/preprints-home.html> AMS Preprint Conference Announcements <http://t13.lanl.gov/~nxt/meet.html> <http://www.nonlin.tu-muenchen.de/chaos/termine.html> <http://xxx.lanl.gov/Announce/Conference/> <http://www.math.psu.edu/weiss/conf.html> 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 Electronic Journals <http://www.springer-ny.com/nst/> Nonlinear Science Today <http://www.santafe.edu/sfi/Complexity> The Complexity Journal <http://www.csu.edu.au/ci/ci.html> Complexity International Journal Electronic Texts <http://www.lib.rmit.edu.au/fractals/exploring.html> Exploring Chaos & Fractals <http://www.nbi.dk/~predrag/QCcourse/> Cvitanovic's Lecture Notes <http://www.students.uiuc.edu/~ag-ho/chaos/chaos.html> Chaos Intro Institutes and Academic Programs <http://www.physics.mcgill.ca/physics-services/physics_complex.html> <http://www.physics.mcgill.ca/physics-services/physics_complex2.html> Extensive List of Physics Groups in Nonlinear Phenonmena <http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/institutes.html> Extensive List of Nonlinear Groups Who is Who in Nonlinear Dynamics <http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/wiw.html> Nonlinear Lists <http://cnls-www.lanl.gov/nbt/sites.html> Extensive List of Nonlinear <http://www.ar.com/ger/sci.nonlinear.html> URLs from Sci.nonlinear <http://www.industrialstreet.com/chaos/metalink.htm#SCIENCE> Chaos URLs Time Series sites <http://cnls-www.lanl.gov/nbt/intro.html> Dynamics and Time Series <http://chuchi.df.uba.ar/series.html> time series http://chuchi.df.uba.ar/tools/tools.html <ftp://ftp.cs.colorado.edu/pub/Time-Series/TSWelcome.html> Santa Fe Time Series Competition Chaos Sites <http://ucmp1.berkeley.edu/henon.html> Expt. henon attractor <http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html> All about Feigenbaum Constants <http://members.aol.com/MTRw3/w3/sw/sw00.html> Mike Rosenstein's Chaos Page. <http://www.prairienet.org/business/ptech/full/chaostry.html> Chaos Network <gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/lorenz.gif> Lorenz Attractor Complexity Sites <http://life.anu.edu.au/complex_systems/complex.html> Complex Sytems <http://www.cc.duth.gr/~mboudour/nonlin.html> Complexity Home Page Fractals Sites <ftp://spanky.triumf.ca/fractals/> The Spanky Fractal DataBase <http://sprott.physics.wisc.edu/fractals.htm> Sprott's Fractal Gallery <http://www-syntim.inria.fr/fractales/> Groupe Fractales <http://acacia.ens.fr:8080/home/massimin/quat/f_gal.ang.html> 3D Fractals <http://www.cnam.fr/fractals.html> Fractal Gallery> <http://homepage.seas.upenn.edu/~lau/fractal.html> <http://homepage.seas.upenn.edu/~rajiyer/math480.html> Course on Fractal Geometry ********** [29] What nonlinear science software is available? General Resources "Guide to Available Mathematical Software" maintained by NIST: <http://gams.cam.nist.gov/> "Mathematics Archives Software" <http://archives.math.utk.edu/software.html> dstool Free software from Guckenheimer's group at Cornell; DSTool has lots of examples of chaotic systems, Poincare' sections, bifurcation diagrams. System: Unix, X windows. Available by anonymous ftp: <ftp://macomb.tn.cornell.edu/pub/dstool/> AUTO Bifurcation/Continuation Software (THE standard). AUTO94 with a GUI requires X and Motif to be present. There is also a command line version AUTO86 The softare is transported as a compressed, encoded file called auto.tar.Z.uu. You should describe your UNIX server in the email. System: versions to run under X windows--SUN or sgi Available: send email to doedel@cs.concordia.ca Chaos Visual simulation in two- and three-dimensional phase space; based on visual algorithms rather than canned numerical algorithms; well-suited for educational use; comes with tutorial exercises. System: Silicon Graphics workstations, IBM RISC workstations with GL Available by anonymous ftp: <http://msg.das.bnl.gov/~bstewart/software.html> Xphased Phase Plane plotter for x-windows systems System: X-windows, Unix, SunOS 4 binary Available by anonymous ftp: <http://www.ama.caltech.edu/~tpw/xphased.html> 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 by anonymous ftp: <ftp://amath.colorado.edu/pub/dynamics/programs/> Lyapunov Exponents and Time Series Based on Alan Wolf's algorithm, see[Q10], but a more efficient version. System: Comes as C source, Fortran source, PC executable, etc Available by anonymous ftp: <http://www.users.interport.net/~wolf/> Lyapunov Exponents Keith Briggs Fortran codes for Lyapunov exponents System: any with a Fortran compiler Available by anonymous ftp: <http:www.pd.uwa.edu.au/Keith/homepage.html> MTRChaos MTRCHAOS and MTRLYAP compute correlation dimension and largest Lyapunov exponents, delay portraits. By Mike Rosenstein. System: PC-compatible computer running DOS 3.1 or higher, 640K RAM, and EGA display. VGA & coprocessor recommended Available by anonymous ftp: <ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/> Chaos Plot ChaosPlot is a simple program which plots the chaotic behavior of a damped, driven anharmonic oscillator. System: Macintosh Available from: <ftp://archives.math.utk.edu/software/mac/diffEquations/ ChaosPlot/ChaosPlot.sea.hqx> MatLab Chaos A collection of routines from the Mathworks folks for generating diagrams which illustrate chaotic behavior associated with the logistic equation. System: Requires MatLab. Available by anonymous ftp: <ftp://ftp.mathworks.com/pub/contrib/misc/chaos/> 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 by anonymous ftp: <ftp://ftp.inria.fr/INRIA/Projects/Meta2/Scilab> Cubic Oscillator Explorer The CUBIC OSCILLATOR EXPLORER is a Macintosh application which allows interactive exploration of the chaotic processes of the Cubic Oscillator, commonly known as Duffing's System. System: Macintosh Available from WWW FRACTAL MUSIC PROJECT at: <http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic/> Dynamics: Numerical Explorations. Nusse, Helena E. and J.E. Yorke, 1994. book + diskette. A hands on approach to learning the concepts and the many aspects in computing relevant quantities in chaos System: PC-compatible computer or X-windows system on Unix computers Available: Springer-Verlag PHASER Kocak, H., 1989. Differential and Difference Equations through Computer Experiments: with a supplementary diskette comtaining PHASER: An Animator/Simulator for Dynamical Systems emonstrates a large number of 1D- 4D differential equations--many not chaotic--and 1D-3D difference equations. System: PC-compatible computer + ??? Available: Springer-Verlag The Academic Software Library: Chaos Simulations Bessoir, T., and A. Wolf, 1990. Demonstrates logistic map, Lyapunov exponents, billiards in a stadium, sensitive dependence, n-body gravitational motion. Available: The Academic Software Library, (800) 955-TASL. $70. Chaos Data Analyser A PC program for analyzing time series. By Sprott, J.C. and G. Rowlands. Available: The Academic Software Library, (800) 955-TASL. $70. For more information see: <http://sprott.physics.wisc.edu/cda.htm> Chaos Demonstrations A PC program for demonstrating chaos, fractals, cellular automata, and related nonlinear phenomena. By J. C. Sprott and G. Rowlands. System: IBM PC or compatible with at least 512K of memory. Available: The Academic Software Library, (800) 955-TASL. $70. Chaotic Dynamics Workbench Performs interactive numerical experiments on systems modeled by ordinary differential equations, including: four versions of driven Duffing oscillators, pendulum, Lorenz, driven Van der Pol osc., driven Brusselator, and the Henon-Heils system. By R. Rollins. System: IBM PC or compatible, 512 KB memory. Available: The Academic Software Library, (800) 955-TASL, $70. Chaos A Program Collection for the PC by Korsch, H.J. and H-J. Jodl, 1994, A book/disk combo that gives a hands-on, computer experiment approach to learning nonlinear dynamics. Some of the modules cover billiard systems, double pendulum, Duffing oscillator, 1D iterative maps, an "electronic chaos-generator", the Mandelbrot set, and ODEs. System: IBM PC or compatible. Available: Springer-Verlag 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). Quality is uneven, and expected Macintosh features (color, resizeable windows) are not always supported (in version 9.0). System: Macintosh See: <http://archives.math.utk.edu/cgibin/fife.test/mkTxtPage.pl?/ ftp/software/mac/calculus/MacMath/MacMath.abstract> Available: Springer-Verlag Tufillaro's Programs From the book Nonlinear Dynamics and Chaos by Tufillaro, Abbot and Reilly (1992). A collection of programs for the Macintosh. System: Macintosh Available: Addison-Wesley For more info see: <http://cnls-www.lanl.gov/nbt/qm.html> <http://cnls-www.lanl.gov/nbt/bb.html> Applied Chaos Tools Software package for time series analysis based on the UCSD group's, work. This package is a companion for Abarbanel's book "Analysis of Observed Chaotic Data", Springer-Verlag. System: Unix, and soon Windows 95 For more info see: <http://pm.znet.com/apchaos/csp.html> ********** [30] Acknowledgments Thanks to Hawley Rising <mailto://rising@crl.com>, Bruce Stewart <mailto://bstewart@bnlux1.bnl.gov> Alan Champneys <mailto://a.r.champneys@bristol.ac.uk> Michael Rosenstein <mailto://MTR1a@aol.com> Troy Shinbrot <mailto://shinbrot@bart.chem-eng.nwu.edu> Matt Kennel <mailto://kennel@msr.epm.ornl.gov> Lou Pecora <mailto://pecora@zoltar.nrl.navy.mil> Richard Tasgal <mailto://tasgal@math.tau.ac.il> Wayne Hayes <mailto://wayne@cs.toronto.edu> S. H. Doole <mailto://Stuart.Doole@Bristol.ac.uk> Pavel Pokorny <mailto://pokornp@tiger.vscht.cz>, Gerard Middleton <mailto://middleto@mcmail.CIS.McMaster.CA> Ronnie Mainieri <mailto://ronnie@cnls.lanl.gov> Leon Poon <mailto://lpoon@Glue.umd.edu> Justin Lipton <mailto://JML@basil.eng.monash.edu.au> Anyone else who would like to contribute, please do! Send me your comments: Jim Meiss <mailto://jdm@boulder.colorado.edu> -- James Meiss Program in Applied Math jdm@boulder.colorado.edu