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- From: James Meiss <jdm@boulder.colorado.edu>
- Newsgroups: sci.nonlinear,sci.answers,news.answers
- Subject: Nonlinear Science FAQ
- Followup-To: poster
- Date: Wed, 15 Oct 2003 15:58:56 -0600
- Organization: University of Colorado
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- Summary: Frequently asked questions about Nonlinear Science,
- Chaos, and Dynamical Systems
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- Archive-name: sci/nonlinear-faq
- Posting-Frequency: annually
- Xref: senator-bedfellow.mit.edu sci.nonlinear:13615 sci.answers:15559 news.answers:259875
-
- 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
-
- Html format from:
- http://amath.colorado.edu/faculty/jdm/faq.html Colorado,
- http://www-chaos.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/nonlinear-faq/ 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/nonlinear-faq the FAQ's site
- ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/nonlinear-faq text format.
-
-
- This FAQ is maintained by Jim Meiss jdm@boulder.colorado.edu.
-
-
- Copyright (c) 1995-2003 by James D. Meiss, all rights reserved. This FAQ may
- be posted to any USENET newsgroup, on-line service, or BBS as long as it is
- posted in its entirety and includes this copyright statement. This FAQ may
- 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- Fixed lots of broken and outdated links.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- To some extent this FAQ is now been superseded by the Dynamical Systems site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 spatio-temporal chaos?
- [3.16] What are cellular automata?
- [3.17] What is a Bifurcation?
- [3.18] What is a Hamiltonian Chaos?
- [4] To Learn More
- [4.1] What should I read to learn more?
- [4.2] What technical journals have nonlinear science articles?
- [4.3] What are net sites for nonlinear science materials?
- [5] Computational Resources
- [5.1] What are general computational resources?
- [5.2] Where can I find specialized programs for nonlinear science?
- [6] Acknowledgments
-
-
- [2] Basic Theory
- [2.1] What is nonlinear?
-
- In geometry, linearity refers to Euclidean objects: lines, planes, (flat)
- three-dimensional space, etc.--these objects appear the same no matter how we
- examine them. A nonlinear object, a sphere for example, looks different on
- different scales--when looked at closely enough it looks like a plane, and
- from a far enough distance it looks like a point.
-
- In algebra, we define linearity in terms of functions that have the property
- f(x+y) = f(x)+f(y) and f(ax) = af(x). Nonlinear is defined as the negation of
- linear. This means that the result f may be out of proportion to the input x
- or y. The result may be more than linear, as when a diode begins to pass
- current; or less than linear, as when finite resources limit Malthusian
- population growth. Thus the fundamental simplifying tools of linear analysis
- are no longer available: for example, for a linear system, if we have two
- zeros, f(x) = 0 and f(y) = 0, then we automatically have a third zero f(x+y) =
- 0 (in fact there are infinitely many zeros as well, since linearity implies
- that f(ax+by) = 0 for any a and b). This is called the principle of
- superposition--it gives many solutions from a few. For nonlinear systems, each
- solution must be fought for (generally) with unvarying ardor!
-
-
- [2.2] What is nonlinear science?
-
- Stanislaw Ulam reportedly said (something like) "Calling a science 'nonlinear'
- is like calling zoology 'the study of non-human animals'. So why do we have a
- name that appears to be merely a negative?
-
- Firstly, linearity is rather special, and no model of a real system is truly
- linear. Some things are profitably studied as linear approximations to the
- real models--for example the fact that Hooke's law, the linear law of
- elasticity (strain is proportional to stress) is approximately valid for a
- pendulum of small amplitude implies that its period is approximately
- independent of amplitude. However, as the amplitude gets large the period gets
- longer, a fundamental effect of nonlinearity in the pendulum equations (see
- http://monet.physik.unibas.ch/~elmer/pendulum/upend.htm and [3.10]).
-
- (You might protest that quantum mechanics is the fundamental theory and that
- it is linear! However this is at the expense of infinite dimensionality which
- is just as bad or worse--and 'any' finite dimensional nonlinear model can be
- turned into an infinite dimensional linear one--e.g. a map x' = f(x) is
- equivalent to the linear integral equation often called the Perron-Frobenius
- equation
- p'(x) = integral [ p(y) \delta(x-f(y)) dy ])
- Here p(x) is a density, which could be interpreted as the probability of
- finding oneself at the point x, and the Dirac-delta function effectively moves
- the points according to the map f to give the new density. So even a nonlinear
- map is equivalent to a linear operator.)
-
- Secondly, nonlinear systems have been shown to exhibit surprising and complex
- effects that would never be anticipated by a scientist trained only in linear
- techniques. Prominent examples of these include bifurcation, chaos, and
- solitons. Nonlinearity has its most profound effects on dynamical systems (see
- [2.3]).
-
- Further, while we can enumerate the linear objects, nonlinear ones are
- nondenumerable, and as of yet mostly unclassified. We currently have no
- general techniques (and very few special ones) for telling whether a
- particular nonlinear system will exhibit the complexity of chaos, or the
- simplicity of order. Thus since we cannot yet subdivide nonlinear science into
- proper subfields, it exists as a whole.
-
- Nonlinear science has applications to a wide variety of fields, from
- mathematics, physics, biology, and chemistry, to engineering, economics, and
- medicine. This is one of its most exciting aspects--that it brings researchers
- from many disciplines together with a common language.
-
-
- [2.3] What is a dynamical system?
-
- A dynamical system consists of an abstract phase space or state space, whose
- coordinates describe the dynamical state at any instant; and a dynamical rule
- which specifies the immediate future trend of all state variables, given only
- the present values of those same state variables. Mathematically, a dynamical
- system is described by an initial value problem.
-
- Dynamical systems are "deterministic" if there is a unique consequent to every
- state, and "stochastic" or "random" if there is more than one consequent
- chosen from some probability distribution (the "perfect" coin toss has two
- consequents with equal probability for each initial state). Most of nonlinear
- science--and everything in this FAQ--deals with deterministic systems.
-
- A dynamical system can have discrete or continuous time. The discrete case is
- defined by a map, z_1 = f(z_0), that gives the state z_1 resulting from the
- initial state z_0 at the next time value. The continuous case is defined by a
- "flow", z(t) = \phi_t(z_0), which gives the state at time t, given that the
- state was z_0 at time 0. A smooth flow can be differentiated w.r.t. time to
- give a differential equation, dz/dt = F(z). In this case we call F(z) a
- "vector field," it gives a vector pointing in the direction of the velocity at
- every point in phase space.
-
-
- [2.4] What is phase space?
-
- Phase space is the collection of possible states of a dynamical system. A
- phase space can be finite (e.g. for the ideal coin toss, we have two states
- heads and tails), countably infinite (e.g. state variables are integers), or
- uncountably infinite (e.g. state variables are real numbers). Implicit in the
- notion is that a particular state in phase space specifies the system
- completely; it is all we need to know about the system to have complete
- knowledge of the immediate future. Thus the phase space of the planar pendulum
- is two-dimensional, consisting of the position (angle) and velocity. According
- to Newton, specification of these two variables uniquely determines the
- subsequent motion of the pendulum.
-
- Note that if we have a non-autonomous system, where the map or vector field
- depends explicitly on time (e.g. a model for plant growth depending on solar
- flux), then according to our definition of phase space, we must include time
- as a phase space coordinate--since one must specify a specific time (e.g. 3PM
- on Tuesday) to know the subsequent motion. Thus dz/dt = F(z,t) is a dynamical
- system on the phase space consisting of (z,t), with the addition of the new
- dynamics dt/dt = 1.
-
- The path in phase space traced out by a solution of an initial value problem
- is called an orbit or trajectory of the dynamical system. If the state
- variables take real values in a continuum, the orbit of a continuous-time
- system is a curve, while the orbit of a discrete-time system is a sequence of
- points.
-
-
- [2.5] What is a degree of freedom?
-
- The notion of "degrees of freedom" as it is used for
- http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Hamiltonian systems means
- one canonical conjugate pair, a configuration, q, and its conjugate momentum
- p. Hamiltonian systems (sometimes mistakenly identified with the notion of
- conservative systems) always have such pairs of variables, and so the phase
- space is even dimensional.
-
- In the study of dissipative systems the term "degree of freedom" is often used
- differently, to mean a single coordinate dimension of the phase space. This
- can lead to confusion, and it is advisable to check which meaning of the term
- is intended in a particular context.
-
- Those with a physics background generally prefer to stick with the Hamiltonian
- definition of the term "degree of freedom." For a more general system the
- proper term is "order" which is equal to the dimension of the phase space.
-
- Note that a dynamical system with N d.o.f. Hamiltonian nominally moves in a
- 2N dimensional phase space. However, if H(q,p) is time independent, then
- energy is conserved, and therefore the motion is really on a 2N-1 dimensional
- energy surface, H(q,p) = E. Thus e.g. the planar, circular restricted 3 body
- problem is 2 d.o.f., and motion is on the 3D energy surface of constant
- "Jacobi constant." It can be reduced to a 2D area preserving map by PoincarΘ
- section (see [2.6]).
-
- If the Hamiltonian is time dependent, then we generally say it has an
- additional 1/2 degree of freedom, since this adds one dimension to the phase
- space. (i.e. 1 1/2 d.o.f. means three variables, q, p and t, and energy is no
- longer conserved).
-
-
- [2.6] What is a map?
-
- A map is simply a function, f, on the phase space that gives the next state,
- f(z) (the image), of the system given its current state, z. (Often you will
- find the notation z' = f(z), where the prime means the next point, not the
- derivative.)
-
- Now a function must have a single value for each state, but there could be
- several different states that give rise to the same image. Maps that allow
- every state in the phase space to be accessed (onto) and which have precisely
- one pre-image for each state (one-to-one) are invertible. If in addition the
- map and its inverse are continuous (with respect to the phase space coordinate
- z), then it is called a homeomorphism. A homeomorphism that has at least one
- continuous derivative (w.r.t. z) and a continuously differentiable inverse is
- a diffeomorphism.
-
- Iteration of a map means repeatedly applying the map to the consequents of the
- previous application. Thus we get a sequence
- n
- z = f(z ) = f(f(z )...) = f (z )
- n n-1 n-2 0
-
- This sequence is the orbit or trajectory of the dynamical system with initial
- condition z_0.
-
-
- [2.7] How are maps related to flows (differential equations)?
-
- Every differential equation gives rise to a map, the time one map, defined by
- advancing the flow one unit of time. This map may or may not be useful. If the
- differential equation contains a term or terms periodic in time, then the time
- T map (where T is the period) is very useful--it is an example of a PoincarΘ
- section. The time T map in a system with periodic terms is also called a
- stroboscopic map, since we are effectively looking at the location in phase
- space with a stroboscope tuned to the period T. This map is useful because it
- permits us to dispense with time as a phase space coordinate: the remaining
- coordinates describe the state completely so long as we agree to consider the
- same instant within every period.
-
- In autonomous systems (no time-dependent terms in the equations), it may also
- be possible to define a PoincarΘ section and again reduce the phase space
- dimension by one. Here the PoincarΘ section is defined not by a fixed time
- interval, but by successive times when an orbit crosses a fixed surface in
- phase space. (Surface here means a manifold of dimension one less than the
- phase space dimension).
-
- However, not every flow has a global PoincarΘ section (e.g. any flow with an
- equilibrium point), which would need to be transverse to every possible orbit.
-
- Maps arising from stroboscopic sampling or PoincarΘ section of a flow are
- necessarily invertible, because the flow has a unique solution through any
- point in phase space--the solution is unique both forward and backward in
- time. However, noninvertible maps can be relevant to differential equations:
- PoincarΘ maps are sometimes very well approximated by noninvertible maps. For
- example, the Henon map (x,y) -> (-y-a+x^2,bx) with small |b| is close to the
- logistic map, x -> -a+x^2.
-
- It is often (though not always) possible to go backwards, from an invertible
- map to a differential equation having the map as its PoincarΘ map. This is
- called a suspension of the map. One can also do this procedure approximately
- for maps that are close to the identity, giving a flow that approximates the
- map to some order. This is extremely useful in bifurcation theory.
-
- Note that any numerical solution procedure for a differential initial value
- problem which uses discrete time steps in the approximation is effectively a
- map. This is not a trivial observation; it helps explain for example why a
- continuous-time system which should not exhibit chaos may have numerical
- solutions which do--see [2.15].
-
-
- [2.8] What is an attractor?
-
- Informally an attractor is simply a state into which a system settles (thus
- dissipation is needed). Thus in the long term, a dissipative dynamical system
- may settle into an attractor.
- Interestingly enough, there is still some controversy in the mathematics
- community as to an appropriate definition of this term. Most people adopt the
- definition
- Attractor: A set in the phase space that has a neighborhood in which every
- point stays nearby and approaches the attractor as time goes to infinity.
- Thus imagine a ball rolling inside of a bowl. If we start the ball at a point
- in the bowl with a velocity too small to reach the edge of the bowl, then
- eventually the ball will settle down to the bottom of the bowl with zero
- velocity: thus this equilibrium point is an attractor. The neighborhood of
- points that eventually approach the attractor is the basin of attraction for
- the attractor. In our example the basin is the set of all configurations
- corresponding to the ball in the bowl, and for each such point all small
- enough velocities (it is a set in the four dimensional phase space [2.4]).
- Attractors can be simple, as the previous example. Another example of an
- attractor is a limit cycle, which is a periodic orbit that is attracting
- (limit cycles can also be repelling). More surprisingly, attractors can be
- chaotic (see [2.9]) and/or strange (see [2.12]).
- The boundary of a basin of attraction is often a very interesting object
- since it distinguishes between different types of motion. Typically a basin
- boundary is a saddle orbit, or such an orbit and its stable manifold. A crisis
- is the change in an attractor when its basin boundary is destroyed.
- An alternative definition of attractor is sometimes used because there
- are systems that have sets that attract most, but not all, initial conditions
- in their neighborhood (such phenomena is sometimes called riddling of the
- basin). Thus, Milnor defines an attractor as a set for which a positive
- measure (probability, if you like) of initial conditions in a neighborhood are
- asymptotic to the set.
-
-
- [2.9] What is chaos?
-
- It has been said that "Chaos is a name for any order that produces confusion
- in our minds." (George Santayana, thanks to Fred Klingener for finding this).
- However, the mathematical definition is, roughly speaking,
- Chaos: effectively unpredictable long time behavior arising in a deterministic
- dynamical system because of sensitivity to initial conditions.
- It must be emphasized that a deterministic dynamical system is perfectly
- predictable given perfect knowledge of the initial condition, and is in
- practice always predictable in the short term. The key to long-term
- unpredictability is a property known as sensitivity to (or sensitive
- dependence on) initial conditions.
-
- For a dynamical system to be chaotic it must have a 'large' set of initial
- conditions which are highly unstable. No matter how precisely you measure the
- initial condition in these systems, your prediction of its subsequent motion
- goes radically wrong after a short time. Typically (see [2.14] for one
- definition of 'typical'), the predictability horizon grows only
- logarithmically with the precision of measurement (for positive Lyapunov
- exponents, see [2.11]). Thus for each increase in precision by a factor of 10,
- say, you may only be able to predict two more time units (measured in units of
- the Lyapunov time, i.e. the inverse of the Lyapunov exponent).
-
- More precisely: A map f is chaotic on a compact invariant set S if
- (i) f is transitive on S (there is a point x whose orbit is dense in S), and
- (ii) f exhibits sensitive dependence on S (see [2.10]).
- To these two requirements #DevaneyDevaney adds the requirement that periodic
- points are dense in S, but this doesn't seem to be really in the spirit of the
- notion, and is probably better treated as a theorem (very difficult and very
- important), and not part of the definition.
-
- Usually we would like the set S to be a large set. It is too much to hope for
- except in special examples that S be the entire phase space. If the dynamical
- system is dissipative then we hope that S is an attractor (see [2.8]) with a
- large basin. However, this need not be the case--we can have a chaotic saddle,
- an orbit that has some unstable directions as well as stable directions.
-
- As a consequence of long-term unpredictability, time series from chaotic
- systems may appear irregular and disorderly. However, chaos is definitely not
- (as the name might suggest) complete disorder; it is disorder in a
- deterministic dynamical system, which is always predictable for short times.
-
- The notion of chaos seems to conflict with that attributed to Laplace: given
- precise knowledge of the initial conditions, it should be possible to predict
- the future of the universe. However, Laplace's dictum is certainly true for
- any deterministic system, recall [2.3]. The main consequence of chaotic motion
- is that given imperfect knowledge, the predictability horizon in a
- deterministic system is much shorter than one might expect, due to the
- exponential growth of errors. The belief that small errors should have small
- consequences was perhaps engendered by the success of Newton's mechanics
- applied to planetary motions. Though these happen to be regular on human
- historic time scales, they are chaotic on the 5 million year time scale (see
- e.g. "Newton's Clock", by Ivars Peterson (1993 W.H. Freeman).
-
-
- [2.10] What is sensitive dependence on initial conditions?
-
- Consider a boulder precariously perched on the top of an ideal hill. The
- slightest push will cause the boulder to roll down one side of the hill or the
- other: the subsequent behavior depends sensitively on the direction of the
- push--and the push can be arbitrarily small. Of course, it is of great
- importance to you which direction the boulder will go if you are standing at
- the bottom of the hill on one side or the other!
-
- Sensitive dependence is the equivalent behavior for every initial condition--
- every point in the phase space is effectively perched on the top of a hill.
-
- More precisely a set S exhibits sensitive dependence if there is an r such
- that for any epsilon > 0 and for each x in S, there is a y such that |x - y| <
- epsilon, and |x_n - y_n| > r for some n > 0. Then there is a fixed distance r
- (say 1), such that no matter how precisely one specifies an initial state
- there are nearby states that eventually get a distance r away.
-
- Note: sensitive dependence does not require exponential growth of
- perturbations (positive Lyapunov exponent), but this is typical (see [2.14])
- for chaotic systems. Note also that we most definitely do not require ALL
- nearby initial points diverge--generically [2.14] this does not happen--some
- nearby points may converge. (We may modify our hilltop analogy slightly and
- say that every point in phase space acts like a high mountain pass.) Finally,
- the words "initial conditions" are a bit misleading: a typical small
- disturbance introduced at any time will grow similarly. Think of "initial" as
- meaning "a time when a disturbance or error is introduced," not necessarily
- time zero.
-
-
- [2.11] What are Lyapunov exponents?
- (Thanks to Ronnie Mainieri & Fred Klingener for contributing to this answer)
-
- The hardest thing to get right about Lyapunov exponents is the spelling of
- Lyapunov, which you will variously find as Liapunov, Lyapunof and even
- Liapunoff. Of course Lyapunov is really spelled in the Cyrillic alphabet:
- (Lambda)(backwards r)(pi)(Y)(H)(0)(B). Now that there is an ANSI standard of
- transliteration for Cyrillic, we expect all references to converge on the
- version Lyapunov.
-
- Lyapunov was born in Russia in 6 June 1857. He was greatly influenced by
- Chebyshev and was a student with Markov. He was also a passionate man:
- Lyapunov shot himself the day his wife died. He died 3 Nov. 1918, three days
- later. According to the request on a note he left, Lyapunov was buried with
- his wife. [biographical data from a biography by A. T. Grigorian].
-
- Lyapunov left us with more than just a simple note. He left a collection of
- papers on the equilibrium shape of rotating liquids, on probability, and on
- the stability of low-dimensional dynamical systems. It was from his
- dissertation that the notion of Lyapunov exponent emerged. Lyapunov was
- interested in showing how to discover if a solution to a dynamical system is
- stable or not for all times. The usual method of studying stability, i.e.
- linear stability, was not good enough, because if you waited long enough the
- small errors due to linearization would pile up and make the approximation
- invalid. Lyapunov developed concepts (now called Lyapunov Stability) to
- overcome these difficulties.
-
- Lyapunov exponents measure the rate at which nearby orbits converge or
- diverge. There are as many Lyapunov exponents as there are dimensions in the
- state space of the system, but the largest is usually the most important.
- Roughly speaking the (maximal) Lyapunov exponent is the time constant, lambda,
- in the expression for the distance between two nearby orbits, exp(lambda *
- t).á If lambda is negative, then the orbits converge in time, and the
- dynamical system is insensitive to initial conditions.á However, if lambda is
- positive, then the distance between nearby orbits grows exponentially in time,
- and the system exhibits sensitive dependence on initial conditions.
-
- There are basically two ways to compute Lyapunov exponents. In one way one
- chooses two nearby points, evolves them in time, measuring the growth rate of
- the distance between them. This is useful when one has a time series, but has
- the disadvantage that the growth rate is really not a local effect as the
- points separate. A better way is to measure the growth rate of tangent vectors
- to a given orbit.
-
- More precisely, consider a map f in an m dimensional phase space, and its
- derivative matrix Df(x). Let v be a tangent vector at the point x. Then we
- define a function
- 1 n
- L(x,v) = lim --- ln |( Df (x)v )|
- n -> oo n
- Now the Multiplicative Ergodic Theorem of Oseledec states that this limit
- exists for almost all points x and all tangent vectors v. There are at most m
- distinct values of L as we let v range over the tangent space. These are the
- Lyapunov exponents at x.
-
- For more information on computing the exponents see
-
- Wolf, A., J. B. Swift, et al. (1985). "Determining Lyapunov Exponents from a
- Time Series." Physica D 16: 285-317.
- Eckmann, J.-P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from
- time series." Phys. Rev. A 34: 4971-4979.
-
-
- [2.12] What is a Strange Attractor?
- Before Chaos (BC?), the only known attractors (see [2.8]) were fixed
- points, periodic orbits (limit cycles), and invariant tori (quasiperiodic
- orbits). In fact the famous PoincarΘ-Bendixson theorem states that for a pair
- of first order differential equations, only fixed points and limit cycles can
- occur (there is no chaos in 2D flows).
- In a famous paper in 1963, Ed Lorenz discovered that simple systems of
- three differential equations can have complicated attractors. The Lorenz
- attractor (with its butterfly wings reminding us of sensitive dependence (see
- [2.10])) is the "icon" of chaos
- http://kong.apmaths.uwo.ca/~bfraser/version1/lorenzintro.html. Lorenz showed
- that his attractor was chaotic, since it exhibited sensitive dependence.
- Moreover, his attractor is also "strange," which means that it is a fractal
- (see [3.2]).
- The term strange attractor was introduced by Ruelle and Takens in 1970
- in their discussion of a scenario for the onset of turbulence in fluid flow.
- They noted that when periodic motion goes unstable (with three or more modes),
- the typical (see [2.14]) result will be a geometrically strange object.
- Unfortunately, the term strange attractor is often used for any chaotic
- attractor. However, the term should be reserved for attractors that are
- "geometrically" strange, e.g. fractal. One can have chaotic attractors that
- are not strange (a trivial example would be to take a system like the cat map,
- which has the whole plane as a chaotic set, and add a third dimension which is
- simply contracting onto the plane). There are also strange, nonchaotic
- attractors (see Grebogi, C., et al. (1984). "Strange Attractors that are not
- Chaotic." Physica D 13: 261-268).
-
-
- [2.13] Can computers simulate chaos?
-
- Strictly speaking, chaos cannot occur on computers because they deal with
- finite sets of numbers. Thus the initial condition is always precisely known,
- and computer experiments are perfectly predictable, in principle. In
- particular because of the finite size, every trajectory computed will
- eventually have to repeat (an thus be eventually periodic). On the other hand,
- computers can effectively simulate chaotic behavior for quite long times (just
- so long as the discreteness is not noticeable). In particular if one uses
- floating point numbers in double precision to iterate a map on the unit
- square, then there are about 10^28 different points in the phase space, and
- one would expect the "typical" chaotic orbit to have a period of about 10^14
- (this square root of the number of points estimate is given by Rannou for
- random diffeomorphisms and does not really apply to floating point operations,
- but nonetheless the period should be a big number). See, e.g.,
-
- Earn, D. J. D. and S. Tremaine, "Exact Numerical Studies of Hamiltonian
- Maps: Iterating without Roundoff Error," Physica D 56, 1-22 (1992).
- Binder, P. M. and R. V. Jensen, "Simulating Chaotic Behavior with Finite
- State Machines," Phys. Rev. 34A, 4460-3 (1986).
- Rannou, F., "Numerical Study of Discrete Plane Area-Preserving Mappings,"
- Astron. and Astrophys. 31, 289-301 (1974).
-
-
- [2.14] What is generic?
- (Thanks to Hawley Rising for contributing to this answer)
-
- Generic in dynamical systems is intended to convey "usual" or, more properly,
- "observable". Roughly speaking, a property is generic over a class if any
- system in the class can be modified ever so slightly (perturbed), into one
- with that property.
-
- The formal definition is done in the language of topology: Consider the class
- to be a space of systems, and suppose it has a topology (some notion of a
- neighborhood, or an open set). A subset of this space is dense if its closure
- (the subset plus the limits of all sequences in the subset) is the whole
- space. It is open and dense if it is also an open set (union of
- neighborhoods). A set is countable if it can be put into 1-1 correspondence
- with the counting numbers. A countable intersection of open dense sets is the
- intersection of a countable number of open dense sets. If all such
- intersections in a space are also dense, then the space is called a Baire
- space, which basically means it is big enough. If we have such a Baire space
- of dynamical systems, and there is a property which is true on a countable
- intersection of open dense sets, then that property is generic.
-
- If all this sounds too complicated, think of it as a precise way of defining a
- set which is near every system in the collection (dense), which isn't too big
- (need not have any "regions" where the property is true for every system).
- Generic is much weaker than "almost everywhere" (occurs with probability 1),
- in fact, it is possible to have generic properties which occur with
- probability zero. But it is as strong a property as one can define
- topologically, without having to have a property hold true in a region, or
- talking about measure (probability), which isn't a topological property (a
- property preserved by a continuous function).
-
-
- [2.15] What is the minimum phase space dimension for chaos?
-
- This is a slightly confusing topic, since the answer depends on the type of
- system considered. First consider a flow (or system of differential
- equations). In this case the PoincarΘ-Bendixson theorem tells us that there is
- no chaos in one or two-dimensional phase spaces. Chaos is possible in three-
- dimensional flows--standard examples such as the Lorenz equations are indeed
- three-dimensional, and there are mathematical 3D flows that are provably
- chaotic (e.g. the 'solenoid').
-
- Note: if the flow is non-autonomous then time is a phase space coordinate, so
- a system with two physical variables + time becomes three-dimensional, and
- chaos is possible (i.e. Forced second-order oscillators do exhibit chaos.)
-
- For maps, it is possible to have chaos in one dimension, but only if the map
- is not invertible. A prominent example is the Logistic map
- x' = f(x) = rx(1-x).
- This is provably chaotic for r = 4, and many other values of r as well (see
- e.g. #DevaneyDevaney). Note that every point x < f(1/2) has two preimages, so
- this map is not invertible.
-
- For homeomorphisms, we must have at least two-dimensional phase space for
- chaos. This is equivalent to the flow result, since a three-dimensional flow
- gives rise to a two-dimensional homeomorphism by PoincarΘ section (see [2.7]).
-
- Note that a numerical algorithm for a differential equation is a map, because
- time on the computer is necessarily discrete. Thus numerical solutions of two
- and even one dimensional systems of ordinary differential equations may
- exhibit chaos. Usually this results from choosing the size of the time step
- too large. For example Euler discretization of the Logistic differential
- equation, dx/dt = rx(1-x), is equivalent to the logistic map. See e.g. S.
- Ushiki, "Central difference scheme and chaos," Physica 4D (1982) 407-424.
-
-
-
- [3] Applications and Advanced Theory
- [3.1] What are complex systems?
- (Thanks to Troy Shinbrot for contributing to this answer)
-
- Complex systems are spatially and/or temporally extended nonlinear systems
- characterized by collective properties associated with the system as a whole--
- and that are different from the characteristic behaviors of the constituent
- parts.
-
- While, chaos is the study of how simple systems can generate complicated
- behavior, complexity is the study of how complicated systems can generate
- simple behavior. An example of complexity is the synchronization of biological
- systems ranging from fireflies to neurons (e.g. Matthews, PC, Mirollo, RE &
- Strogatz, SH "Dynamics of a large system of coupled nonlinear oscillators,"
- Physica 52D (1991) 293-331). In these problems, many individual systems
- conspire to produce a single collective rhythm.
-
- The notion of complex systems has received lots of popular press, but it is
- not really clear as of yet if there is a "theory" about a "concept". We are
- withholding judgment. See
-
- http://www.calresco.org/index.htm The Complexity & Artificial Life Web Site
- http://www.calresco.org/sos/sosfaq.htm The self-organized systems FAQ
-
-
- [3.2] What are fractals?
-
- One way to define "fractal" is as a negation: a fractal is a set that does not
- look like a Euclidean object (point, line, plane, etc.) no matter how closely
- you look at it. Imagine focusing in on a smooth curve (imagine a piece of
- string in space)--if you look at any piece of it closely enough it eventually
- looks like a straight line (ignoring the fact that for a real piece of string
- it will soon look like a cylinder and eventually you will see the fibers, then
- the atoms, etc.). A fractal, like the Koch Snowflake, which is topologically
- one dimensional, never looks like a straight line, no matter how closely you
- look. There are indentations, like bays in a coastline; look closer and the
- bays have inlets, closer still the inlets have subinlets, and so on. Simple
- examples of fractals include Cantor sets (see [3.5], Sierpinski curves, the
- Mandelbrot set and (almost surely) the Lorenz attractor (see [2.12]).
- Fractals also approximately describe many real-world objects, such as clouds
- (see http://makeashorterlink.com/?Z50D42C16) mountains, turbulence,
- coastlines, roots and branches of trees and veins and lungs of animals.
-
- "Fractal" is a term which has undergone refinement of definition by a lot of
- people, but was first coined by B. Mandelbrot,
- http://physics.hallym.ac.kr/reference/physicist/Mandelbrot.html, and defined
- as a set with fractional (non-integer) dimension (Hausdorff dimension, see
- [3.4]). Mandelbrot defines a fractal in the following way:
-
- A geometric figure or natural object is said to be fractal if it
- combines the following characteristics: (a) its parts have the same
- form or structure as the whole, except that they are at a different
- scale and may be slightly deformed; (b) its form is extremely irregular,
- or extremely interrupted or fragmented, and remains so, whatever the scale
- of examination; (c) it contains "distinct elements" whose scales are very
- varied and cover a large range." (Les Objets Fractales 1989, p.154)
-
- See the extensive FAQ from sci.fractals at
- <ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq
-
-
- [3.3] What do fractals have to do with chaos?
-
- Often chaotic dynamical systems exhibit fractal structures in phase space.
- However, there is no direct relation. There are chaotic systems that have
- nonfractal limit sets (e.g. Arnold's cat map) and fractal structures that can
- arise in nonchaotic dynamics (see e.g. Grebogi, C., et al. (1984). "Strange
- Attractors that are not Chaotic." Physica 13D: 261-268.)
-
-
- [3.4] What are topological and fractal dimension?
-
- See the fractal FAQ:
- ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq
- or the site
- http://pro.wanadoo.fr/quatuor/mathematics.htm
-
-
- [3.5] What is a Cantor set?
- (Thanks to Pavel Pokorny for contributing to this answer)
-
- A Cantor set is a surprising set of points that is both infinite (uncountably
- so, see [2.14]) and yet diffuse. It is a simple example of a fractal, and
- occurs, for example as the strange repellor in the logistic map (see [2.15])
- when r>4. The standard example of a Cantor set is the "middle thirds" set
- constructed on the interval between 0 and 1. First, remove the middle third.
- Two intervals remain, each one of length one third. From each remaining
- interval remove the middle third. Repeat the last step infinitely many times.
- What remains is a Cantor set.
-
- More generally (and abstrusely) a Cantor set is defined topologically as a
- nonempty, compact set which is perfect (every point is a limit point) and
- totally disconnected (every pair of points in the set are contained in
- disjoint covering neighborhoods).
-
- See also
- http://www.shu.edu/html/teaching/math/reals/topo/defs/cantor.html
- http://personal.bgsu.edu/~carother/cantor/Cantor1.html
- http://mizar.uwb.edu.pl/JFM/Vol7/cantor_1.html
-
- Georg Ferdinand Ludwig Philipp Cantor was born 3 March 1845 in St Petersburg,
- Russia, and died 6 Jan 1918 in Halle, Germany. To learn more about him see:
- http://turnbull.dcs.st-and.ac.uk/history/Mathematicians/Cantor.html
- http://www.shu.edu/html/teaching/math/reals/history/cantor.html
-
- To read more about the Cantor function (a function that is continuous,
- differentiable, increasing, non-constant, with a derivative that is zero
- everywhere except on a set with length zero) see
- http://www.shu.edu/projects/reals/cont/fp_cantr.html
-
-
- [3.6] What is quantum chaos?
- (Thanks to Leon Poon for contributing to this answer)
-
- According to the correspondence principle, there is a limit where classical
- behavior as described by Hamilton's equations becomes similar, in some
- suitable sense, to quantum behavior as described by the appropriate wave
- equation. Formally, one can take this limit to be h -> 0, where h is Planck's
- constant; alternatively, one can look at successively higher energy levels.
- Such limits are referred to as "semiclassical". It has been found that the
- semiclassical limit can be highly nontrivial when the classical problem is
- chaotic. The study of how quantum systems, whose classical counterparts are
- chaotic, behave in the semiclassical limit has been called quantum chaos. More
- generally, these considerations also apply to elliptic partial differential
- equations that are physically unrelated to quantum considerations. For
- example, the same questions arise in relating classical waves to their
- corresponding ray equations. Among recent results in quantum chaos is a
- prediction relating the chaos in the classical problem to the statistics of
- energy-level spacings in the semiclassical quantum regime.
-
- Classical chaos can be used to analyze such ostensibly quantum systems as the
- hydrogen atom, where classical predictions of microwave ionization thresholds
- agree with experiments. See Koch, P. M. and K. A. H. van Leeuwen (1995).
- "Importance of Resonances in Microwave Ionization of Excited Hydrogen Atoms."
- Physics Reports 255: 289-403.
-
- See also:
- http://sagar.physics.neu.edu/qchaos/qc.html Quantum Chaos
- http://www.mpipks-dresden.mpg.de/~noeckel/microlasers.html Microlaser
- Cavities
-
-
-
- [3.7] How do I know if my data are deterministic?
- (Thanks to Justin Lipton for contributing to this answer)
-
- How can I tell if my data is deterministic? This is a very tricky problem. It
- is difficult because in practice no time series consists of pure 'signal.'
- There will always be some form of corrupting noise, even if it is present as
- round-off or truncation error or as a result of finite arithmetic or
- quantization. Thus any real time series, even if mostly deterministic, will be
- a stochastic processes
-
- All methods for distinguishing deterministic and stochastic processes rely on
- the fact that a deterministic system will always evolve in the same way from a
- given starting point. Thus given a time series that we are testing for
- determinism we
- (1) pick a test state
- (2) search the time series for a similar or 'nearby' state and
- (3) compare their respective time evolution.
-
- Define the error as the difference between the time evolution of the 'test'
- state and the time evolution of the nearby state. A deterministic system will
- have an error that either remains small (stable, regular solution) or increase
- exponentially with time (chaotic solution). A stochastic system will have a
- randomly distributed error.
-
- Essentially all measures of determinism taken from time series rely upon
- finding the closest states to a given 'test' state (i.e., correlation
- dimension, Lyapunov exponents, etc.). To define the state of a system one
- typically relies on phase space embedding methods, see [3.14].
-
- Typically one chooses an embedding dimension, and investigates the propagation
- of the error between two nearby states. If the error looks random, one
- increases the dimension. If you can increase the dimension to obtain a
- deterministic looking error, then you are done. Though it may sound simple it
- is not really! One complication is that as the dimension increases the search
- for a nearby state requires a lot more computation time and a lot of data (the
- amount of data required increases exponentially with embedding dimension) to
- find a suitably close candidate. If the embedding dimension (number of
- measures per state) is chosen too small (less than the 'true' value)
- deterministic data can appear to be random but in theory there is no problem
- choosing the dimension too large--the method will work. Practically, anything
- approaching about 10 dimensions is considered so large that a stochastic
- description is probably more suitable and convenient anyway.
-
- See e.g.,
- Sugihara, G. and R. M. May (1990). "Nonlinear Forecasting as a Way of
- Distinguishing Chaos from Measurement Error in Time Series." Nature
- 344: 734-740.
-
-
- [3.8] What is the control of chaos?
-
- Control of chaos has come to mean the two things:
- stabilization of unstable periodic orbits,
- use of recurrence to allow stabilization to be applied locally.
- Thus term "control of chaos" is somewhat of a misnomer--but the name has
- stuck. The ideas for controlling chaos originated in the work of Hubler
- followed by the Maryland Group.
-
- Hubler, A. W. (1989). "Adaptive Control of Chaotic Systems." Helv. Phys.
- Acta 62: 343-346.
- Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review
- Letters 64(11): 1196-1199. http://www-
- chaos.umd.edu/publications/abstracts.html#prl64.1196
-
- The idea that chaotic systems can in fact be controlled may be
- counterintuitive--after all they are unpredictable in the long term.
- Nevertheless, numerous theorists have independently developed methods which
- can be applied to chaotic systems, and many experimentalists have demonstrated
- that physical chaotic systems respond well to both simple and sophisticated
- control strategies. Applications have been proposed in such diverse areas of
- research as communications, electronics, physiology, epidemiology, fluid
- mechanics and chemistry.
-
- The great bulk of this work has been restricted to low-dimensional systems;
- more recently, a few researchers have proposed control techniques for
- application to high- or infinite-dimensional systems. The literature on the
- subject of the control of chaos is quite voluminous; nevertheless several
- reviews of the literature are available, including:
-
- Shinbrot, T. Ott, E., Grebogi, C. & Yorke, J.A., "Using Small Perturbations
- to Control Chaos," Nature, 363 (1993) 411-7.
- Shinbrot, T., "Chaos: Unpredictable yet Controllable?" Nonlin. Sciences
- Today, 3:2 (1993) 1-8.
- Shinbrot, T., "Progress in the Control of Chaos," Advance in Physics (in
- press).
- Ditto, WL & Pecora, LM "Mastering Chaos," Scientific American (Aug. 1993),
- 78-84.
- Chen, G. & Dong, X, "From Chaos to Order -- Perspectives and Methodologies
- in Controlling Chaotic Nonlinear Dynamical Systems," Int. J. Bif. & Chaos 3
- (1993) 1363-1409.
-
- It is generically quite difficult to control high dimensional systems; an
- alternative approach is to use control to reduce the dimension before applying
- one of the above techniques. This approach is in its infancy; see:
-
- Auerbach, D., Ott, E., Grebogi, C., and Yorke, J.A. "Controlling Chaos in
- High Dimensional Systems," Phys. Rev. Lett. 69 (1992) 3479-82
- http://www-chaos.umd.edu/publications/abstracts.html#prl69.3479
-
-
- [3.9] How can I build a chaotic circuit?
- (Thanks to Justin Lipton and Jose Korneluk for contributing to this answer)
-
- There are many different physical systems which display chaos, dripping
- faucets, water wheels, oscillating magnetic ribbons etc. but the most simple
- systems which can be easily implemented are chaotic circuits. In fact an
- electronic circuit was one of the first demonstrations of chaos which showed
- that chaos is not just a mathematical abstraction. Leon Chua designed the
- circuit 1983.
-
- The circuit he designed, now known as Chua's circuit, consists of a piecewise
- linear resistor as its nonlinearity (making analysis very easy) plus two
- capacitors, one resistor and one inductor--the circuit is unforced
- (autonomous). In fact the chaotic aspects (bifurcation values, Lyapunov
- exponents, various dimensions etc.) of this circuit have been extensively
- studied in the literature both experimentally and theoretically. It is
- extremely easy to build and presents beautiful attractors (see [2.8]) (the
- most famous known as the double scroll attractor) that can be displayed on a
- CRO.
-
- For more information on building such a circuit try: see
-
- http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html Chua's Circuit Applet
-
- References
- Matsumoto T. and Chua L.O. and Komuro M. "Birth and Death of the Double
- Scroll" Physica D24 97-124, 1987.
- Kennedy M. P., "Robust OP Amp Realization of Chua's Circuit", Frequenz
- 46, no. 3-4, 1992
- Madan, R. A., Chua's Circuit: A paradigm for chaos, ed. R. A. Madan,
- Singapore: World Scientific, 1993.
- Pecora, L. and Carroll, T. Nonlinear Dynamics in Circuits, Singapore:
- World Scientific, 1995.
- Nonlinear Dynamics of Electronic Systems, Proceedings of the Workshop
- NDES 1993, A.C.Davies and W.Schwartz, eds., World Scientific, 1994,
- ISBN 981-02-1769-2.
- Parker, T.S., and L.O.Chua, Practical Numerical Algorithms for Chaotic
- Systems, Springer-Verlag, 1989, ISBN's: 0-387-96689-7
- and 3-540-96689-7.
-
-
- [3.10] What are simple experiments to demonstrate chaos?
-
- There are many "chaos toys" on the market. Most consist of some sort of
- pendulum that is forced by an electromagnet. One can of course build a simple
- double pendulum to observe beautiful chaotic behavior see
- http://quasar.mathstat.uottawa.ca/~selinger/lagrange/doublependulum.html
- Experimental Pendulum Designs
- http://www.maths.tcd.ie/~plynch/SwingingSpring/doublependulum.html Java
- Applet
- http://monet.physik.unibas.ch/~elmer/pendulum/ Java Applets Pendulum Lab
-
- My favorite double pendulum consists of two identical planar pendula, so that
- you can demonstrate sensitive dependence [2.10], for a Java applet simulation
- see http://www.cs.mu.oz.au/~mkwan/pendulum/pendulum.html. Another cute toy is
- the "Space Circle" that you can find in many airport gift shops. This is
- discussed in the article:
-
- A. Wolf & T. Bessoir, Diagnosing Chaos in the Space Circle, Physica 50D,
- 1991.
-
- One of the simplest chemical systems that shows chaos is the Belousov-
- Zhabotinsky reaction. The book by Strogatz [4.1] has a good introduction to
- this subject,. For the recipe see
- http://www.ux.his.no/~ruoff/BZ_Phenomenology.html. Chemical chaos is modeled
- (in a generic sense) by the "Brusselator" system of differential equations.
- See
-
- Nicolis, Gregoire & Prigogine, (1989) Exploring Complexity: An
- Introduction W. H. Freeman
-
- The Chaotic waterwheel, while not so simple to build, is an exact realization
- of Lorenz famous equations. This is nicely discussed in Strogatz book [4.1] as
- well.
-
- Billiard tables can exhibit chaotic motion, see
- http://www.maa.org/mathland/mathland_3_3.html, though it might be hard to see
- this next time you are in a bar, since a rectangular table is not chaotic!
-
-
- [3.11] What is targeting?
- (Thanks to Serdar Iplikτi for contributing to this answer)
-
- Targeting is the task of steering a chaotic system from any initial point to
- the target, which can be either an unstable equilibrium point or an unstable
- periodic orbit, in the shortest possible time, by applying relatively small
- perturbations. In order to effectively control chaos, [3.8] a targeting
- strategy is important. See:
-
- Kostelich, E., C. Grebogi, E. Ott, and J. A. Yorke, "Higher
- Dimensional Targeting," Phys Rev. E,. 47, , 305-310 (1993).
- Barreto, E., E. Kostelich, C. Grebogi, E. Ott, and J. A. Yorke, "Efficient
- Switching Between Controlled Unstable Periodic Orbits in Higher
- Dimensional Chaotic Systems," Phys Rev E, 51, 4169-4172 (1995).
-
- One application of targeting is to control a spacecraft's trajectory so that
- one can find low energy orbits from one planet to another. Recently targeting
- techniques have been used in the design of trajectories to asteroids and even
- of a grand tour of the planets. For example,
-
- E. Bollt and J. D. Meiss, "Targeting Chaotic Orbits to the Moon
- Through Recurrence," Phys. Lett. A 204, 373-378 (1995).
- http://www.cds.caltech.edu/~marsden/software/spacecraft_orbits.html
-
-
- [3.12] What is time series analysis?
- (Thanks to Jim Crutchfield for contributing to this answer)
-
- This is the application of dynamical systems techniques to a data series,
- usually obtained by "measuring" the value of a single observable as a function
- of time. The major tool in a dynamicist's toolkit is "delay coordinate
- embedding" which creates a phase space portrait from a single data series. It
- seems remarkable at first, but one can reconstruct a picture equivalent
- (topologically) to the full Lorenz attractor (see [2.12])in three-dimensional
- space by measuring only one of its coordinates, say x(t), and plotting the
- delay coordinates (x(t), x(t+h), x(t+2h)) for a fixed h.
-
- It is important to emphasize that the idea of using derivatives or delay
- coordinates in time series modeling is nothing new. It goes back at least to
- the work of Yule, who in 1927 used an autoregressive (AR) model to make a
- predictive model for the sunspot cycle. AR models are nothing more than delay
- coordinates used with a linear model. Delays, derivatives, principal
- components, and a variety of other methods of reconstruction have been widely
- used in time series analysis since the early 50's, and are described in
- several hundred books. The new aspects raised by dynamical systems theory are
- (i) the implied geometric view of temporal behavior and (ii) the existence of
- "geometric invariants", such as dimension and Lyapunov exponents. The central
- question was not whether delay coordinates are useful for time series
- analysis, but rather whether reconstruction methods preserve the geometry and
- the geometric invariants of dynamical systems. (Packard, Crutchfield, Farmer &
- Shaw)
-
- G.U. Yule, Phil. Trans. R. Soc. London A 226 (1927) p. 267.
- N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, "Geometry
- from a time series", Phys. Rev. Lett. 45, no. 9 (1980) 712.
- F. Takens, "Detecting strange attractors in fluid turbulence", in: Dynamical
- Systems and Turbulence, eds. D. Rand and L.-S. Young
- (Springer, Berlin, 1981)
- Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.Sh.T.
- "The analysis of observed chaotic data in physical systems",
- Rev. Modern Physics 65 (1993) 1331-1392.
- D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics,
- Springer-Verlag http://www.cnd.mcgill.ca/books_understanding.html
- E. Peters (1994) Fractal Market Analysis : Applying Chaos Theory to
- Investment and Economics, Wiley
- http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471585246.html
-
-
- [3.13] Is there chaos in the stock market?
- (Thanks to Bruce Stewart for Contributions to this answer)
-
- In order to address this question, we must first agree what we mean by chaos,
- see [2.9].
-
- In dynamical systems theory, chaos means irregular fluctuations in a
- deterministic system (see [2.3] and [3.7]). This means the system behaves
- irregularly because of its own internal logic, not because of random forces
- acting from outside. Of course, if you define your dynamical system to be the
- socio-economic behavior of the entire planet, nothing acts randomly from
- outside (except perhaps the occasional meteor), so you have a dynamical
- system. But its dimension (number of state variables--see [2.4]) is vast, and
- there is no hope of exploiting the determinism. This is high-dimensional
- chaos, which might just as well be truly random behavior. In this sense, the
- stock market is chaotic, but who cares?
-
- To be useful, economic chaos would have to involve some kind of collective
- behavior which can be fully described by a small number of variables. In the
- lingo, the system would have to be self-organizing, resulting in low-
- dimensional chaos. If this turns out to be true, then you can exploit the low-
- dimensional chaos to make short-term predictions. The problem is to identify
- the state variables which characterize the collective modes. Furthermore,
- having limited the number of state variables, many events now become external
- to the system, that is, the system is operating in a changing environment,
- which makes the problem of system identification very difficult.
-
- If there were such collective modes of fluctuation, market players would
- probably know about them; economic theory says that if many people recognized
- these patterns, the actions they would take to exploit them would quickly
- nullify the patterns. Market participants would probably not need to know
- chaos theory for this to happen. Therefore if these patterns exist, they must
- be hard to recognize because they do not emerge clearly from the sea of noise
- caused by individual actions; or the patterns last only a very short time
- following some upset to the markets; or both.
-
- A number of people and groups have tried to find these patterns. So far the
- published results are negative. There are also commercial ventures involving
- prominent researchers in the field of chaos; we have no idea how well they are
- succeeding, or indeed whether they are looking for low-dimensional chaos. In
- fact it seems unlikely that markets remain stationary long enough to identify
- a chaotic attractor (see [2.12]). If you know chaos theory and would like to
- devote yourself to the rhythms of market trading, you might find a trading
- firm which will give you a chance to try your ideas. But don't expect them to
- give you a share of any profits you may make for them :-) !
-
- In short, anyone who tells you about the secrets of chaos in the stock market
- doesn't know anything useful, and anyone who knows will not tell. It's an
- interesting question, but you're unlikely to find the answer.
-
- On the other hand, one might ask a more general question: is market behavior
- adequately described by linear models, or are there signs of nonlinearity in
- financial market data? Here the prospect is more favorable. Time series
- analysis (see [3.14]) has been applied these tests to financial data; the
- results often indicate that nonlinear structure is present. See e.g. the book
- by Brock, Hsieh, LeBaron, "Nonlinear Dynamics, Chaos, and Instability", MIT
- Press, 1991; and an update by B. LeBaron, "Chaos and nonlinear forecastability
- in economics and finance," Philosophical Transactions of the Royal Society,
- Series A, vol 348, Sept 1994, pp 397-404. This approach does not provide a
- formula for making money, but it is stimulating some rethinking of economic
- modeling. A book by Richard M. Goodwin, "Chaotic Economic Dynamics," Oxford
- UP, 1990, begins to explore the implications for business cycles.
-
-
- [3.14] What are solitons?
-
- The process of obtaining a solution of a linear (constant coefficient)
- differential equations is simplified by the Fourier transform (it converts
- such an equation to an algebraic equation, and we all know that algebra is
- easier than calculus!); is there a counterpart which similarly simplifies
- nonlinear equations? The answer is No. Nonlinear equations are qualitatively
- more complex than linear equations, and a procedure which gives the dynamics
- as simply as for linear equations must contain a mistake. There are, however,
- exceptions to any rule.
-
- Certain nonlinear differential equations can be fully solved by, e.g., the
- "inverse scattering method." Examples are the Korteweg-de Vries, nonlinear
- Schrodinger, and sine-Gordon equations. In these cases the real space maps, in
- a rather abstract way, to an inverse space, which is comprised of continuous
- and discrete parts and evolves linearly in time. The continuous part typically
- corresponds to radiation and the discrete parts to stable solitary waves, i.e.
- pulses, which are called solitons. The linear evolution of the inverse space
- means that solitons will emerge virtually unaffected from interactions with
- anything, giving them great stability.
-
- More broadly, there is a wide variety of systems which support stable solitary
- waves through a balance of dispersion and nonlinearity. Though these systems
- may not be integrable as above, in many cases they are close to systems which
- are, and the solitary waves may share many of the stability properties of true
- solitons, especially that of surviving interactions with other solitary waves
- (mostly) unscathed. It is widely accepted to call these solitary waves
- solitons, albeit with qualifications.
-
- Why solitons? Solitons are simply a fundamental nonlinear wave phenomenon.
- Many very basic linear systems with the addition of the simplest possible or
- first order nonlinearity support solitons; this universality means that
- solitons will arise in many important physical situations. Optical fibers can
- support solitons, which because of their great stability are an ideal medium
- for transmitting information. In a few years long distance telephone
- communications will likely be carried via solitons.
-
- The soliton literature is by now vast. Two books which contain clear
- discussions of solitons as well as references to original papers are
- A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia,
- Penn. (1985)
- M.J. Ablowitz and P.A.Clarkson, Solitons, nonlinear evolution equations and
- inverse
- scattering, Cambridge (1991).
- http://www.cup.org/titles/catalogue.asp?isbn=0521387302
- See http://www.ma.hw.ac.uk/solitons/
-
-
- [3.15] What is spatio-temporal chaos?
-
- Spatio-temporal chaos occurs when system of coupled dynamical systems
- gives rise to dynamical behavior that exhibits both spatial disorder (as in
- rapid decay of spatial correlations) and temporal disorder (as in nonzero
- Lyapunov exponents). This is an extremely active, and rather unsettled area of
- research. For an introduction see:
- Cross, M. C. and P. C. Hohenberg (1993). "Pattern Formation outside of
- Equilibrium." Rev. Mod. Phys. 65: 851-1112.
- http://www.cmp.caltech.edu/~mcc/st_chaos.html Spatio-Temporal Chaos
-
- An interesting application which exhibits pattern formation and spatio-
- temporal chaos is to excitable media in biological or chemical systems. See
-
- Chaos, Solitions and Fractals 5 #3&4 (1995) Nonlinear Phenomena in Excitable
- Physiological System, http://www.elsevier.nl/locate/chaos
- http://ojps.aip.org/journal_cgi/dbt?KEY=CHAOEH&Volume=8&Issue=1
- Chaos focus issue on Fibrillation
-
-
- [3.16] What are cellular automata?
- (Thanks to Pavel Pokorny for Contributions to this answer)
-
- A Cellular automaton (CA) is a dynamical system with discrete time (like
- a map, see [2.6]), discrete state space and discrete geometrical space (like
- an ODE), see [2.7]). Thus they can be represented by a state s(i,j) for
- spatial state i, at time j, where s is taken from some finite set. The update
- rule is that the new state is some function of the old states, s(i,j+1) =
- f(s). The following table shows the distinctions between PDE's, ODE's, coupled
- map lattices (CML) and CA in taking time, state space or geometrical space
- either continuous (C) of discrete (D):
- time state space geometrical space
- PDE C C C
- ODE C C D
- CML D C D
- CA D D D
-
- Perhaps the most famous CA is Conway's game "life." This CA evolves
- according to a deterministic rule which gives the state of a site in the next
- generation as a function of the states of neighboring sites in the present
- generation. This rule is applied to all sites.
-
- For further reading see
-
- S. Wolfram (1986) Theory and Application of Cellular Automata, World
- Scientific Singapore.
- Physica 10D (1984)--the entire volume
-
- Some programs that do CA, as well as more generally "artificial life" are
- available at
- http://www.alife.org/links.html
- http://www.kasprzyk.demon.co.uk/www/ALHome.html
-
-
- [3.17] What is a Bifurcation?
- (Thanks to Zhen Mei for Contributions to this answer)
-
- A bifurcation is a qualitative change in dynamics upon a small variation in
- the parameters of a system.
-
- Many dynamical systems depend on parameters, e.g. Reynolds number, catalyst
- density, temperature, etc. Normally a gradually variation of a parameter in
- the system corresponds to the gradual variation of the solutions of the
- problem. However, there exists a large number of problems for which the number
- of solutions changes abruptly and the structure of solution manifolds varies
- dramatically when a parameter passes through some critical values. For
- example, the abrupt buckling of a stab when the stress is increased beyond a
- critical value, the onset of convection and turbulence when the flow
- parameters are changed, the formation of patterns in certain PDE's, etc. This
- kind of phenomena is called bifurcation, i.e. a qualitative change in the
- behavior of solutions of a dynamics system, a partial differential equation or
- a delay differential equation.
-
- Bifurcation theory is a method for studying how solutions of a nonlinear
- problem and their stability change as the parameters varies. The onset of
- chaos is often studied by bifurcation theory. For example, in certain
- parameterized families of one dimensional maps, chaos occurs by infinitely
- many period doubling bifurcations
- (See http://www.stud.ntnu.no/~berland/math/feigenbaum/)
-
- There are a number of well constructed computer tools for studying
- bifurcations. In particular see [5.2] for AUTO and DStool.
-
-
- [3.18] What is a Hamiltonian Chaos?
-
- The transition to chaos for a Hamiltonian (conservative) system is somewhat
- different than that for a dissipative system (recall [2.5]). In an integrable
- (nonchaotic) Hamiltonian system, the motion is "quasiperiodic", that is motion
- that is oscillatory, but involves more than one independent frequency (see
- also [2.12]). Geometrically the orbits move on tori, i.e. the mathematical
- generalization of a donut. Examples of integrable Hamiltonian systems include
- harmonic oscillators (simple mass on a spring, or systems of coupled linear
- springs), the pendulum, certain special tops (for example the Euler and
- Lagrange tops), and the Kepler motion of one planet around the sun.
-
- It was expected that a typical perturbation of an integrable Hamiltonian
- system would lead to "ergodic" motion, a weak version of chaos in which all of
- phase space is covered, but the Lyapunov exponents [2.11] are not necessarily
- positive. That this was not true was rather surprisingly discovered by one of
- the first computer experiments in dynamics, that of Fermi, Pasta and Ulam.
- They showed that trajectories in nonintegrable system may also be surprisingly
- stable. Mathematically this was shown to be the case by the celebrated theorem
- of Kolmogorov Arnold and Moser (KAM), first proposed by Kolmogorov in 1954.
- The KAM theorem is rather technical, but in essence says that many of the
- quasiperiodic motions are preserved under perturbations. These orbits fill out
- what are called KAM tori.
-
- An amazing extension of this result was started with the work of John Greene
- in 1968. He showed that if one continues to perturb a KAM torus, it reaches a
- stage where the nearby phase space [2.4] becomes self-similar (has fractal
- structure [3.2]). At this point the torus is "critical," and any increase in
- the perturbation destroys it. In a remarkable sequence of papers, Aubry and
- Mather showed that there are still quasiperiodic orbits that exist beyond this
- point, but instead of tori they cover cantor sets [3.5]. Percival actually
- discovered these for an example in 1979 and named them "cantori."
- Mathematicians tend to call them "Aubry-Mather" sets. These play an important
- role in limiting the rate of transport through chaotic regions.
-
- Thus, the transition to chaos in Hamiltonian systems can be thought of as the
- destruction of invariant tori, and the creation of cantori. Chirikov was the
- first to realize that this transition to "global chaos" was an important
- physical phenomena. Local chaos also occurs in Hamiltonian systems (in the
- regions between the KAM tori), and is caused by the intersection of stable and
- unstable manifolds in what PoincarΘ called the "homoclinic trellis."
-
- To learn more: See the introductory article by Berry, the text by Percival and
- Richards and the collection of articles on Hamiltonian systems by MacKay and
- Meiss [4.1]. There are a number of excellent advanced texts on Hamiltonian
- dynamics, some of which are listed in [4.1], but we also mention
-
- Meyer, K. R. and G. R. Hall (1992), Introduction to Hamiltonian dynamical
- systems and the N-body problem (New York, Springer-Verlag).
-
-
-
- [4] To Learn More
- [4.1] What should I read to learn more?
- Popularizations
- 1 Gleick, J. (1987). Chaos, the Making of a New Science.
- London, Heinemann. http://www.around.com/chaos.html
- 2 Stewart, I. (1989). Does God Play Dice? Cambridge, Blackwell.
- http://www.amazon.com/exec/obidos/ASIN/1557861064
- 3 Devaney, R. L. (1990). Chaos, Fractals, and Dynamics: Computer
- Experiments in Mathematics. Menlo Park, Addison-Wesley
- http://www.amazon.com/exec/obidos/ASIN/1878310097
- 4 Lorenz, E., (1994) The Essence of Chaos, Univ. of Washington Press.
- http://www.amazon.com/exec/obidos/ASIN/0295975148
- 5 Schroeder, M. (1991) Fractals, Chaos, Power: Minutes from an infinite paradise
- W. H. Freeman New York:
- Introductory Texts
- 1 Abraham, R. H. and C. D. Shaw (1992) Dynamics: The Geometry of
- Behavior, 2nd ed. Redwood City, Addison-Wesley.
- 2 Baker, G. L. and J. P. Gollub (1990). Chaotic Dynamics.
- Cambridge, Cambridge Univ. Press.
- http://www.cup.org/titles/catalogue.asp?isbn=0521471060
- 3 DevaneyDevaney, R. L. (1986). An Introduction to Chaotic Dynamical
- Systems. Menlo Park, Benjamin/Cummings.
- http://math.bu.edu/people/bob/books.html
- 4 Kaplan, D. and L. Glass (1995). Understanding Nonlinear Dynamics,
- Springer-Verlag New York. http://www.cnd.mcgill.ca/books_understanding.html
- 5 Glendinning, P. (1994). Stability, Instability and Chaos.
- Cambridge, Cambridge Univ Press.
- http://www.cup.org/Titles/415/0521415535.html
- 6 Jurgens, H., H.-O. Peitgen, et al. (1993). Chaos and Fractals: New
- Frontiers of Science. New York, Springer Verlag.
- http://www.springer-ny.com/detail.tpl?isbn=0387979034
- 7 Moon, F. C. (1992). Chaotic and Fractal Dynamics. New York, John Wiley.
- http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471545716.html
- 8 Percival, I. C. and D. Richard (1982). Introduction to Dynamics. Cambridge,
- Cambridge Univ. Press.
- http://www.cup.org/titles/catalogue.asp?isbn=0521281490
- 9 Scott, A. (1999). NONLINEAR SCIENCE: Emergence and Dynamics of
- Coherent Structures, Oxford http://www4.oup.co.uk/isbn/0-19-850107-2
- http://www.imm.dtu.dk/documents/users/acs/BOOK1.html
- 10 Smith, P (1998) Explaining Chaos, Cambridge
- http://us.cambridge.org/titles/catalogue.asp?isbn=0521477476
- 11 Strogatz, S. (1994). Nonlinear Dynamics and Chaos. Reading,
- Addison-Wesley
- http://www.perseusbooksgroup.com/perseus-cgi-bin/display/0-7382-0453-6
- 12 Thompson, J. M. T. and H. B. Stewart (1986) Nonlinear Dynamics and
- Chaos. Chichester, John Wiley and Sons.
- http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471876844.html
- 13 Tufillaro, N., T. Abbott, et al. (1992). An Experimental Approach
- to Nonlinear Dynamics and Chaos. Redwood City, Addison-Wesley.
- http://www.amazon.com/exec/obidos/ASIN/0201554410/
- 14 Turcotte, Donald L. (1992). Fractals and Chaos in Geology and
- Geophysics, Cambridge Univ. Press.
- http://www.cup.org/titles/catalogue.asp?isbn=0521567335
-
- Introductory Articles
- 1 May, R. M. (1986). "When Two and Two Do Not Make Four."
- Proc. Royal Soc. B228: 241.
- 2 Berry, M. V. (1981). "Regularity and Chaos in Classical Mechanics,
- Illustrated by Three Deformations of a Circular Billiard."
- Eur. J. Phys. 2: 91-102.
- 3 Crawford, J. D. (1991). "Introduction to Bifurcation Theory."
- Reviews of Modern Physics 63(4): 991-1038.
- 3 Shinbrot, T., C. Grebogi, et al. (1992). "Chaos in a Double Pendulum."
- Am. J. Phys 60: 491-499.
- 5 David Ruelle. (1980). "Strange Attractors,"
- The Mathematical Intelligencer 2: 126-37.
-
- Advanced Texts
- 1 Arnold, V. I. (1978). Mathematical Methods of Classical Mechanics.
- New York, Springer.
- http://www.springer-ny.com/detail.tpl?isbn=038796890
- 2 Arrowsmith, D. K. and C. M. Place (1990), An Introduction to Dynamical Systems.
- Cambridge, Cambridge University Press.
- http://us.cambridge.org/titles/catalogue.asp?isbn=0521316502
- 3 Guckenheimer, J. and P. Holmes (1983), Nonlinear Oscillations, Dynamical
- Systems, and Bifurcation of Vector Fields, Springer-Verlag New York.
- 4 Kantz, H., and T. Schreiber (1997). Nonlinear time series analysis.
- Cambridge, Cambridge University Press
- http://www.mpipks-dresden.mpg.de/~schreibe/myrefs/book.html
- 5 Katok, A. and B. Hasselblatt (1995), Introduction to the Modern
- Theory of Dynamical Systems, Cambridge, Cambridge Univ. Press.
- http://titles.cambridge.org/catalogue.asp?isbn=0521575575
- 6 Hilborn, R. (1994), Chaos and Nonlinear Dyanamics: an Introduction for
- Scientists and Engineers, Oxford Univesity Press.
- http://www4.oup.co.uk/isbn/0-19-850723-2
- 7 Lichtenberg, A.J. and M. A. Lieberman (1983), Regular and Chaotic Motion,
- Springer-Verlag, New York .
- 8 Lind, D. and Marcus, B. (1995) An Introduction to Symbolic Dynamics and
- Coding, Cambridge University Press, Cambridge
- http://www.math.washington.edu/SymbolicDynamics/
- 9 MacKay, R.S and J.D. Meiss (eds) (1987), Hamiltonian Dynamical Systems
- A reprint selection, , Adam Hilger, Bristol
- 10 Nayfeh, A.H. and B. Balachandran (1995), Applied Nonlinear Dynamics:
- Analytical, Computational and Experimental Methods
- John Wiley& Sons Inc., New York
- http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471593486.html
- 11 Ott, E. (1993). Chaos in Dynamical Systems. Cambridge University Press,
- Cambridge. http://us.cambridge.org/titles/catalogue.asp?isbn=0521010845
- 12 L.E. Reichl, (1992), The Transition to Chaos, in Conservative and
- Classical Systems: Quantum Manifestations Springer-Verlag, New York
- 13 Robinson, C. (1999), Dynamical Systems: Stability, Symbolic
- Dynamics, and Chaos, 2nd Edition, Boca Raton, CRC Press.
- http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=8495
- 14 Ruelle, D. (1989), Elements of Differentiable Dynamics and Bifurcation
- Theory, Academic Press Inc.
- 15 Tabor, M. (1989), Chaos and Integrability in Nonlinear Dynamics:
- an Introduction, Wiley, New York.
- http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471827282.html
- 16 Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems
- and Chaos, Springer-Verlag New York.
- 17 Wiggins, S. (1988), Global Bifurcations and Chaos, Springer-Verlag New
- York.
-
-
-
- [4.2] What technical journals have nonlinear science articles?
-
- Physica D The premier journal in Applied Nonlinear Dynamics
- Nonlinearity Good mix, with a mathematical bias
- Chaos AIP Journal, with a good physical bent
- SIAM J. of Dynamical Systems Online Journal with multimedia
- http://www.siam.org/journals/siads/siads.htm
- Chaos Solitons and Fractals Low quality, some good applications
- Communications in Math Phys an occasional paper on dynamics
- Comm. in Nonlinear Sci. New Elsevier journal
- and Num. Sim. http://www.elsevier.com/locate/cnsns
- Ergodic Theory and Rigorous mathematics, and careful work
- Dynamical Systems
- International J of lots of color pictures, variable quality.
- Bifurcation and Chaos
- J Differential Equations A premier journal, but very mathematical
- J Dynamics and Diff. Eq. Good, more focused version of the above
- J Dynamics and Stability Focused on Eng. applications. New editorial
- of Systems board--stay tuned.
- J Fluid Mechanics Some expt. papers, e.g. transition to turbulence
- J Nonlinear Science a newer journal--haven't read enough yet.
- J Statistical Physics Used to contain seminal dynamical systems papers
- Nonlinear Dynamics Haven't read enough to form an opinion
- Nonlinear Science Today Weekly News: http://www.springer-ny.com/nst/
- Nonlinear Processes in New, variable quality...may be improving
- Geophysics
- Physics Letters A Has a good nonlinear science section
- Physical Review E Lots of Physics articles with nonlinear emphasis
- Regular and Chaotic Dynamics Russian Journal http://web.uni.udm.ru/~rcd/
-
-
-
- [4.3] What are net sites for nonlinear science materials?
-
- Bibliography
- http://www.uni-mainz.de/FB/Physik/Chaos/chaosbib.html Mainz http site
- ftp://ftp.uni-mainz.de/pub/chaos/chaosbib/ Mainz ftp site
- http://www-chaos.umd.edu/publications/searchbib.html Seach the Mainz Site
- http://www-chaos.umd.edu/publications/references.html Maryland
- http://www.cpm.mmu.ac.uk/~bruce/combib/ Complexity Bibliography
- http://www.mth.uea.ac.uk/~h720/research/ Ergodic Theory and Dynamical Systems
- http://www.drchaos.net/drchaos/intro.html Nonlinear Dynamics Resources (pdf file)
- http://www.nonlin.tu-muenchen.de/chaos/Projects/miguelbib Sanjuan's Bibliography
-
- Preprint Archives
- http://www.math.sunysb.edu/dynamics/preprints/ StonyBrook
- http://cnls.lanl.gov/People/nbt/intro.html Los Alamos Preprint Server
- http://xxx.lanl.gov/ Nonlinear Science Eprint Server
- http://www.ma.utexas.edu/mp_arc/mp_arc-home.html Math-Physics Archive
- http://www.ams.org/global-preprints/ AMS Preprint Servers List
-
- Conference Announcements
- http://at.yorku.ca/amca/conferen.htm Mathematics Conference List
- http://www.math.sunysb.edu/dynamics/conferences/conferences.html
-
- StonyBrook List
- http://www.nonlin.tu-muenchen.de/chaos/termine.html Munich List
- http://xxx.lanl.gov/Announce/Conference/ Los Alamos List
- http://www.tam.uiuc.edu/Events/conferences.html Theoretical & Applied Mechanics
- http://www.siam.org/meetings/ds99/index.htm SIAM Dynamical Systems 1999
-
- Newsletters
- gopher://gopher.siam.org:70/11/siag/ds SIAM Dynamical Systems Group
- http://www.amsta.leeds.ac.uk/Applied/news.dir/ UK Nonlinear News
-
- Education Sites
- http://math.bu.edu/DYSYS/ Devaney's Dynamical Systems Project
-
- Electronic Journals
- http://www.springer-ny.com/nst/ Nonlinear Science Today
- http://www3.interscience.wiley.com/cgi-bin/jtoc?ID=38804 Complexity
- http://journal-ci.csse.monash.edu.au/ Complexity International Journal
-
- Electronic Texts
- http://cnls.lanl.gov/People/nbt//Book/node1.html An experimental approach
- to nonlinear dynamics and chaos
- http://www.nbi.dk/~predrag/QCcourse/ Lecture Notes on Periodic Orbits
- http://hypertextbook.com/chaos/ The Chaos HyperTextBook
-
- Institutes and Academic Programs
- http://physicsweb.org/resources/dsearch.phtml Physics Institutes
- http://ip-service.com/WiW/institutes.html Nonlinear Groups
- http://www-chaos.engr.utk.edu/related.html Research Groups in Chaos
-
- Java Applets Sites
- http://physics.hallym.ac.kr/education/TIPTOP/VLAB/about.html Virtual Laboratory
- http://monet.physik.unibas.ch/~elmer/pendulum/ Java Pendulum
- http://kogs-www.informatik.uni-hamburg.de/~wiemker/applets/fastfrac/fastfrac.html
- Java Fractal Explorer
- http://www.apmaths.uwo.ca/~bfraser/index.html B. Fraser╣s Nonlinear Lab
- http://www.cmp.caltech.edu/~mcc/Chaos_Course/ Mike Cross' Demos
-
- Who is Who in Nonlinear Dynamics
- http://www.chaos-gruppe.de/wiw/wiw.html Munich List
- http://www.math.sunysb.edu/dynamics/people/list.html Stonybrook List
-
- Lists of Nonlinear sites
- http://makeashorterlink.com/?C58C23C16 Netscape╣s List
- http://cnls.lanl.gov/People/nbt/sites.html Tufillaro's List
- http://cires.colorado.edu/people/peckham.scott/chaos.html Peckham's List
- http://members.tripod.com/~IgorIvanov/physics/nonlinear.html Physics Encyclopedia
- http://www.maths.ex.ac.uk/~hinke/dss/index.html Osinga's Software List
-
- Dynamical Systems
- http://www.math.sunysb.edu/dynamics/ Dynamical Systems Home Page
- http://www.math.psu.edu/gunesch/entropy.html Entropy and Dynamics
-
- Chaos sites
- http://www.industrialstreet.net/chaosmetalink/ Chaos Metalink
- http://bofh.priv.at/ifs/ Iterated Function Systems Playground
- http://www.xahlee.org/PageTwo_dir/more.html Xah Lee's dynamics and Fractals pages
- http://acl2.physics.gatech.edu/tutorial/outline.htm Tutorial on Control of Chaos
- http://www.mathsoft.com/mathresources/constants/wellknown/article/0,,2090,00.html
- All about Feigenbaum Constants
- http://www.stud.ntnu.no/~berland/math/feigenbaum/ The Feigenbaum Fractal
- http://members.aol.com/MTRw3/index.html Mike Rosenstein's Chaos Page.
- http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/cspls.html Chaos in Psychology
- http://www.eie.polyu.edu.hk/~cktse/NSR/ Movies and Demonstrations
-
- Time Series
- http://www.drchaos.net/drchaos/refs.html Dynamics and Time Series
- http://astro.uni-tuebingen.de/groups/time/ Time series Analysis
- http://www-personal.buseco.monash.edu.au/~hyndman/TSDL/index.htm
- Time Series Data Library
-
- Complex Systems Sites
- http://www.math.upatras.gr/~mboudour/nonlin.html Complexity Home Page
- http://www.calresco.org/ The Complexity & Artificial Life Web Site
- http://www.physionet.org/ Complexity and Physiology Site
-
- Fractals Sites
- http://forum.swarthmore.edu/advanced/robertd/index.html#frac A Fractal Gallery
- http://spanky.triumf.ca/www/welcome1.html The Spanky Fractal DataBase
- http://sprott.physics.wisc.edu/fractals.htm Sprott's Fractal Gallery
- http://fractales.inria.fr/ Projet Fractales
- http://force.stwing.upenn.edu/~lau/fractal.html Lau's Fractal Stuff
- http://skal.planet-d.net/quat/f_gal.html 3D Fractals
- http://www.cnam.fr/fractals.html Fractal Gallery
- http://www.fractaldomains.com/ Fractal Domains Gallery
- http://home1.swipnet.se/~w-17723/fracpro.html Fractal Programs
- http://xahlee.org/PageTwo_dir/MathPrograms_dir/mathPrograms.html#Fractals
- Fractal Programs
-
-
-
- [5] Computational Resources
-
- [5.1] What are general computational resources?
- CAIN Europe Archives
- http://www.can.nl/education/material/software.html Software Area
- FAQ guide to packages from sci.math.num-analysis
- ftp://rtfm.mit.edu/pub/usenet/news.answers/num-analysis/faq/part1
- NIST Guide to Available Mathematical Software
- http://gams.cam.nist.gov/
- Mathematics Archives Software
- http://archives.math.utk.edu/software.html
- Matpack, C++ numerical methods and data analysis library
- http://www.matpack.de/
- Numerical Recipes Home Page
- http://www.nr.com/
-
- [5.2] Where can I find specialized programs for nonlinear
- science?
-
- The Academic Software Library:
- Chaos Simulations
- Bessoir, T., and A. Wolf, 1990. Demonstrates logistic map, Lyapunov exponents,
- billiards in a stadium, sensitive dependence, n-body gravitational motion.
- Chaos Data Analyser
- A PC program for analyzing time series. By Sprott, J.C. and G. Rowlands.
- For more info:http://sprott.physics.wisc.edu/cda.htm
- Chaos Demonstrations
- A PC program for demonstrating chaos, fractals, cellular automata, and related
- nonlinear phenomena. By J. C. Sprott and G. Rowlands.
- System: IBM PC or compatible with at least 512K of memory.
- Available: The Academic Software Library, (800) 955-TASL. $70.
- Chaotic Dynamics Workbench
- Performs interactive numerical experiments on systems modeled by ordinary
- differential equations, including: four versions of driven Duffing
- oscillators, pendulum, Lorenz, driven Van der Pol osc., driven Brusselator,
- and the Henon-Heils system. By R. Rollins.
- System: IBM PC or compatible, 512 KB memory.
- Available: The Academic Software Library, (800) 955-TASL, $70
-
- Applied Chaos Tools
- Software package for time series analysis based on the UCSD group's, work.
- This package is a companion for Abarbanel's book Analysis of Observed Chaotic
- Data, Springer-Verlag.
- System: Unix-Motif, Windows 95/NT
- For more info see: http://www.zweb.com/apnonlin/csp.html
-
- AUTO
- Bifurcation/Continuation Software (THE standard). The latest version is
- AUTO97. The GUI requires X and Motif to be present. There is also a command
- line version AUTO86. The software is transported as a compressed file called
- auto.tar.Z.
- System: versions to run under X windows--SUN or sgi or LINUX
- Available: anonymous ftp from ftp://ftp.cs.concordia.ca/pub/doedel/auto
-
- BZphase
- Models Belousov- Zhabotinsky reaction based on the scheme of Ruoff and Noyes.
- The dynamics ranges from simple quasisinusoidal oscillations to quasiperiodic,
- bursting, complex periodic and chaotic.
- System: DOS 6 and higher + PMODE/W DOS Extender. Also openGL version
- Available: http://members.tripod.com/~RedAndr/BZPhase.htm
-
- Chaos
- Visual simulation in two- and three-dimensional phase space; based on visual
- algorithms rather than canned numerical algorithms; well-suited for
- educational use; comes with tutorial exercises. By Bruce Stewart
- System: Silicon Graphics workstations, IBM RISC workstations with GL
- Available: http://msg.das.bnl.gov/~bstewart/software.html
-
- Chaos
- A Program Collection for the PC by Korsch, H.J. and H-J. Jodl, 1994, A
- book/disk combo that gives a hands-on, computer experiment approach to
- learning nonlinear dynamics. Some of the modules cover billiard systems,
- double pendulum, Duffing oscillator, 1D iterative maps, an "electronic chaos-
- generator", the Mandelbrot set, and ODEs.
- System: IBM PC or compatible.
- Available: $$http://www.springer-ny.com/catalog/np/updates/0-387-57457-3.html
-
- CHAOS II
- Chaos Programs to go with Baker, G. L. and J. P. Gollub (1990) Chaotic
- Dynamics. Cambridge, Cambridge Univ.
- http://www.cup.org/titles/catalogue.asp?isbn=0521471060
- System: IBM, 512K memory, CGA or EGA graphics, True Basic
- For more info: contact Gregory Baker, P.O. Box 278 ,Bryn Athyn, PA, 19009
-
- Chaos Analyser
- Programs to Time delay embedding, Attractor (3d) viewing and animation,
- PoincarΘ sections, Mutual information, Singular Value Decomposition embedding,
- Full Lyapunov spectra (with noise cancellation), Local SVD analysis (for
- determining the systems dimension). By Mike Banbrook.
- System: Unix, X windows
- For more info: http://www.ee.ed.ac.uk/~mb/analysis_progs.html
-
- Chaos Cookbook
- These programs go with J. Pritchard's book, The Chaos Cookbook System:
- Programs written in Visual Basic & Turbo Pascal
- Available: $$http://www.amazon.com/exec/obidos/ASIN/0750617772
-
- Chaos Plot
- ChaosPlot is a simple program which plots the chaotic behavior of a damped,
- driven anharmonic oscillator.
- System: Macintosh
- For more info:
- http://archives.math.utk.edu/software/mac/diffEquations/.directory.html
-
- Cubic Oscillator Explorer
- The CUBIC OSCILLATOR EXPLORER is a Macintosh application which allows
- interactive exploration of the chaotic processes of the Cubic Oscillator,
- i.e..Duffing's equation.
- System: Macintosh + Digidesign DSPácard, Digisystem init 2.6 and (optional)
- MIDI Manager
- Available: (Missing??) Fractal Music
-
- DataPlore
- Signal and time series analysis package. Contains standard facilities for
- signal processing as well as advanced features like wavelet techniques and
- methods of nonlinear dynamics.
- Systems: MS Windows, Linux, SUN Solaris 2.6
- Available: $$http://www.datan.de/dataplore/
-
- dstool
- Free software from Guckenheimer's group at Cornell; DSTool has lots of
- examples of chaotic systems, PoincarΘ sections, bifurcation diagrams.
- System: Unix, X windows.
- Available: ftp://cam.cornell.edu/pub/dstool/
-
- Dynamical Software Pro
- Analyze non-linear dynamics and chaos. Includes ODEs, delay differential
- equations, discrete maps, numerical integration, time series embedding, etc.
- System: DOS. Microsoft Fortran compiler for user defined equations.
- Available: SciTech http://www.scitechint.com/
-
- Dynamics: Numerical Explorations.
- A book + disk by H. Nusse, and J.Yorke. A hands on approach to learning the
- concepts and the many aspects in computing relevant quantities in chaos
- System: PC-compatible computer or X-windows system on Unix computers
- Available: $$ http://www.springer-ny.com/detail.tpl?isbn=0387982647
-
- Dynamics Solver
- Dynamics Solver solve numerically both initial-value problems and boundary-
- value problems for continuous and discrete dynamical systems.
- System: Windows 3.1 or Windows 95/98/NT
- Available: http://tp.lc.ehu.es/jma/ds/ds.html
-
- DynaSys
- Phase plane portraits of 2D ODEs by Etienne Dupuis
- System: Windows 95/98
- Available: (Missing??)
-
- FD3
- A program to estimate fractal dimensions of a set. By DiFalco/Sarraille
- System: C source code, suitable for compiling for use on a Unix or DOS
- platform.
- Available: ftp://ftp.cs.csustan.edu/pub/fd3/
-
- FracGen
- FracGen is a freeware program to create fractal images using Iterated
- Function Systems. A tutorial is provided with the program. By Patrick Bangert
- System: PC-compatible computer, Windows 3.1
- Available: http://212.201.48.1/pbangert/site/fracgen.html
-
- Fractal Domains
- Generates of Mandelbrot and Julia sets. By Dennis C. De Mars
- System: Power Macintosh
- Available: http://www.fractaldomains.com/
-
- Fractal Explorer
- Generates Mandelbrot and Newton's method fractals. By Peter Stone
- System: Power Macintosh
- Available: http://usrwww.mpx.com.au/~peterstone/index.html
-
- GNU Plotutils
- The GNU plotutils package contains C/C++ function library for exporting 2-D
- vector graphics in many file formats, and for doing vector graphics
- animations. The package also contains several command-line programs for
- plotting scientific data, such as GNU graph, which is based on libplot, and
- ODE integration software.
- System: GNU/Linux, FreeBSD, and Unix systems.
- Available: http://www.gnu.org/software/plotutils/plotutils.html
-
- Ilya
- A program to visually study a reaction-diffusion model based on the
- Brusselator from Future Skills Software, Herber Sauro.
- System: Requires Windows 95, at least 256 colours
- Available : http://www.fssc.demon.co.uk/rdiffusion/ilya.htm
-
- INSITE
- (It's a Nonlinear Systems Investigative Toolkit for Everyone) is a collection
- for the simulation and characterization of dynamical systems, with an emphasis
- on chaotic systems. Companion software for T.S. Parker and L.O. Chua (1989)
- Practical Numerical Algorithms for Chaotic Systems Springer Verlag. See their
- paper "INSITE A Software Toolkit for the Analysis of Nonlinear Dynamical
- Systems," Proc. of the IEEE, 75, 1081-1089 (1987).
- System: C codes in Unix Tar or DOS format (later requires QuickWindowC
- or MetaWINDOW/Plus 3.7C. and MS C compiler 5.1)
- Available: INSITE SOFTWARE, p.o. Box 9662, Berkeley, CA , U.S.A.
-
- Institut fur ComputerGraphik
- A collection of programs for developing advanced visualization techniques in
- the field of three-dimensional dynamical systems. By L÷ffelmann H., Gr÷ller E.
- System: various, requires AVS
- Available: http://www.cg.tuwien.ac.at/research/vis/dynsys/
-
- KAOS1D
- A tool for studying one-dimensional (1D) discrete dynamical systems. Does
- bifurcation diagrams, etc. for a number of maps
- System: PC compatible computer, DOS, VGA graphics
- Available: http://www.if.ufrgs.br/~arenzon/jsoftw.html
-
- LOCBIF
- An interactive tool for bifurcation analysis of non-linear ordinary
- differential equations ODE's and maps. By Khibnik, Nikolaev, Kuznetsov and V.
- Levitin
- System: Now part of XPP (See below)
- Available: http://www.math.pitt.edu/~bard/classes/wppdoc/locbif.html
-
- Lyapunov Exponents
- Keith Briggs Fortran codes for Lyapunov exponents
- System: any with a Fortran compiler
- Available: http://more.btexact.com/people/briggsk2/
-
- Lyapunov Exponents and Time Series
- Based on Alan Wolf's algorithm, see [2.11], but a more efficient version.
- System: Comes as C source, Fortran source, PC executable, etc
- Available: http://www.cooper.edu/engineering/physics/wolf/ (Seems to be
- missing?)
-
- Lyapunov Exponents and Time Series
- Michael Banbrook's C codes for Lyapunov exponents & time series analysis
- System: Sun with X windows.
- Available: http://www.see.ed.ac.uk/~mb/analysis_progs.html
-
- Lyapunov Exponents Toolbox (LET)
- A user-contributed MATLAB toolbox that provides a graphical user interface
- for users to determine the full sets of Lyapunov exponents and Lyapunov
- dimensions of discrete and continuous chaotic systems.
- System: MATLAB 5
- Available: ftp://ftp.mathworks.com/pub/contrib/v5/misc/let
-
- Lyapunov.m
- A Matlab program based on the QR Method , by von Bremen, Udwadia, and
- Proskurowski, Physica D, vol. 101, 1-16, (1997)
- System: Matlab
- Available: http://www.usc.edu/dept/engineering/mecheng/DynCon/
-
- Macintosh Dynamics Programs
- Lists available at: http://hypertextbook.com/chaos/92.shtml
- and http://www.xahlee.org/PageTwo_dir/MathPrograms_dir/mathPrograms.html
-
- MacMath
- Comes on a disk with the book MacMath, by Hubbard and West. A collection of
- programs for dynamical systems (1 & 2 D maps, 1 to 3D flows). Version 9.2 is
- the current version, but West is working on a much improved update.
- System: Macintosh
- For more info: http://www.math.hmc.edu/codee/solvers/mac-math.html
- Available: $$ Springer-Verlag http://www.springer-
- ny.com/detail.tpl?isbn=0387941355
-
- Madonna
- Solves Differential and Difference Equations. Runs STELLA. Has a parser with a
- control language. By Robert Macey and George Oster at Berkeley
- System: Macintosh or Windows 95 or later
- Available : $$ http://www.berkeleymadonna.com/
-
- MatLab Chaos
- A collection of routines for generate diagrams which illustrate chaotic
- behavior associated with the logistic equation.
- System: Requires MatLab.
- Available : ftp://ftp.mathworks.com/pub/contrib/misc/chaos/
-
- MTRChaos
- MTRCHAOS and MTRLYAP compute correlation dimension and largest Lyapunov
- exponents, delay portraits. By Mike Rosenstein.
- System: PC-compatible computer running DOS 3.1 or higher, 640K RAM, and EGA
- display. VGA & coprocessor recommended
- Available: ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/
-
- Nonlinear Dynamics Toolbox
- Josh Reiss' NDT includes routines for the analysis of chaotic data, such as
- power spectral analyses, determination of the Lyapunov spectrum, mutual
- information function, prediction, noise reduction, and dimensional analysis.
- System: Windows 95, 98, or NT
- Available : Missing??
-
- NLD Toolbox
- This toolbox has many of the standard dynamical systems, By Jeff Brush
- System: PC, MS-DOS.
- Available: http://www.physik.tu-darmstadt.de/nlp/nldtools/nldtools.html
-
- ODECalc
- A program for integrating boundary value and initial value Problems for up to
- 9th order ODEs. By Optimal Designs.
- System: PC 386+, DOS 3.3+, 16 bit arch.
- Available : ftp://ftp.mecheng.asme.org/pub/EDU_TOOL/Ode200.exe
-
- PHASER
- Kocak, H., 1989. Differential and Difference Equations through Computer
- Experiments: with a supplementary diskette containing PHASER: An
- Animator/Simulator for Dynamical Systems. Demonstrates a large number of 1D-4D
- differential equations--many not chaotic--and 1D-3D difference equations.
- System: PC-compatible
- Available: Springer-Verlag http://www.springer-
- ny.com/detail.tpl?isbn=0387142029
-
- PhysioToolkit
- Software for physiologic signal processing and analysis, detection of
- physiologically significant events using both classical techniques and novel
- methods based on statistical physics and nonlinear dynamics
- System: Unix
- Available: http://www.physionet.org/physiotools/
-
- Recurrence Quantification Analysis
- Recurrence plots give a visual indication of deterministic behavior in complex
- time series. The program, by Webber and Zbilut creates the plots and
- quantifies the determinism with five measures.
- System: DOS executable
- Available:http://homepages.luc.edu/~cwebber/
-
- SciLab
- A simulation program similar in intent to MatLab. It's primarily designed for
- systems/signals work, and is large. From INRIA in France.
- System: Unix, X Windows, 20 Meg Disk space.
- Available : ftp://ftp.inria.fr/INRIA/Projects/Meta2/Scilab
-
- StdMap
- Iterates Area Preserving Maps, by J. D. Meiss. Iterates 8 different maps. It
- will find periodic orbits, cantori, stable and unstable manifolds, and allows
- you to iterate curves.
- System: Macintosh
- Available: http://amath.colorado.edu/faculty/jdm/stdmap.html
-
- STELLA
- Simulates dynamics for Biological and Social systems modelling. Uses a
- building block metaphor constructing models.
- System: Macintosh and Windows PC
- Available: $$ http://www.hps-inc.com/edu/stella/stella.htm
-
- Time Series Tools
- An extensive list of Unix tools for Time Series analysis
- System: Unix
- For more info: http://chuchi.df.uba.ar/guille/TS/tools/tools.html (Link
- down??)
-
- Time Series Analysis from Darmstadt
- Four prgrams Time Series analysis and Dimension calculation from the Institute
- of Applied Physics at Darmstadt.
- System: OS2 or Solaris/Linux/Win9X/NT + Fortran source
- For more info: http://www.physik.tu-darmstadt.de/nlp/distribution.html
-
- Time Series Analysis from Kennel
- The program mkball finds the minimum embedding dimension using the false
- strands enhancement of the false neighbors algorithm of Kennel & Abarbanel.
- System: any C compiler
- Available: ftp://lyapunov.ucsd.edu/pub/nonlinear/mbkall.tar.gz
-
- TISEAN Time Series Analysis
- Agorithms for data representation, prediction, noise reduction, dimension and
- Lyapunov estimation, and nonlinearity testing. By Rainer Hegger, Holger Kantz
- and Thomas Schreiber
- System: C, C++ and Fortran Codes for Unix,
- Available: http://www.mpipks-dresden.mpg.de/~tisean/
-
- Tufillaro's Programs
- From the book Nonlinear Dynamics and Chaos by Tufillaro, Abbot and Reilly
- (1992) (for a sample section see
- http://www.drchaos.net/drchaos/Book/node1.html). A collection of programs for
- the Macintosh.
- System: Macintosh
- Available: http://www.drchaos.net/drchaos/bb.html
-
- Unified Life Models (ULM)
- ULM, by Stephane Legendre, is a program to study population dynamics and more
- generally, discrete dynamical systems. It models any species life cycle graph
- (matrix models) inter- and intra-specific competition (non linear systems),
- environmental stochasticity, demographic stochasticity (branching processes),
- and metapopulations, migrations (coupled systems).
- System: PC/Windows 3.X
- Available: from http://www.snv.jussieu.fr
-
- Virtual Laboratory
- Simulations of 2D active media by the Complex Systems Group at the Max Planck
- Inst. in Berlin.
- System: Requires PV-Wave by Visual Numerics
- $$http://www.vni.com/products/wave/
- Available: $$ http://w3.rz-berlin.mpg.de/~mik/oertzen/vlm/m_contents.htm
-
- VRA (Visual Recurrence Analysis)
- VRA is a software to display and Study the recurrence plots, first described
- by Eckmann, Oliffson Kamphorst And Ruelle in 1987. With RP, one can
- graphically detect hidden patterns and structural changes in data or see
- similarities in patterns across the time series under study. By Eugene Kononov
- Stystem: Windows 95
- Available: http://pweb.netcom.com/~eugenek/download.html
-
- Xphased
- Phase 3D plane program for X-windows systems (for systems like Lorenz,
- Rossler). Plot, rotate in 3-d, PoincarΘ sections, etc. By Thomas P. Witelski
- System: X-windows, Unix, SunOS 4 binary
- Available: http://www.alumni.caltech.edu/~witelski/xphased.html
-
- XPP-Aut
- Differential equations and maps for x-windows systems. Links to Auto for
- bifurcation analysis. By Bard Ermentrout
- System: X-windows, Binaries for many unix systems
- Available : ftp://ftp.math.pitt.edu/pub/bardware/tut/start.html
-
- XSpiral
- Simulate pattern formation in 2-D excitable media (in particular 2 models, one
- of them the FitzHugh-Nagumo). By Flavio Fenton.
- System: X-windows
- Available : (Missing??)
-
-
-
-
- [6] Acknowledgments
-
- Alan Champneys 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.uni-marburg.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.chem-eng.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
-