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\sbasedon0\snext2 text;}{\s3\sb200\sa200\keep\tqc\tx4680\tx8640 \f20 \sbasedon2\snext3 equation;}{\s4 \f20 \sbasedon0\snext4 numbered list;}{\s5\tx180 \f20 \sbasedon0\snext5 bullet list;}{\s6 \f20 \sbasedon0\snext6 glossary;}{\s7 \v\f20\uldb
\sbasedon0\snext7 reference;}{\s8 \outl\v\f20 \sbasedon0\snext8 mark;}{\s9 \f4\fs18 \sbasedon0\snext9 pre;}{\s10\brdrb\brdrs \f20 \sbasedon0\snext10 hr;}{\s11\tx540\tqc\tx3680 \f20 \sbasedon5\snext11 bullet list 1;}{\s12\tx720\tx1440\tx7920 \f20
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{\author James D. Meiss}{\keywords chaos, dynamical systems}}\margl1440\margr1440\widowctrl\ftnbj \sectd \sbknone\pgnrestart\linemod0\cols1\endnhere\titlepg {\header \pard\plain \s244\qr\pvpg\phpg\posx10041\posy1167\absw576\tqc\tx4320\tqr\tx8640 \f20 {
\chpgn }\par
\pard \s244\tqc\tx4320\tqr\tx8640 \par
}{\footer \pard\plain \s243\tqc\tx4320\tqr\tx8640 \f20 Sci.nonlinear FAQ, version 2.0\tab \tab Sept 2003\par
\pard\plain \f20 \'a9 J.D. Meiss\par
}{\headerf \pard\plain \s244\tqc\tx4320\tqr\tx8640 \f20 {\fs48 \tab Frequently Asked Questions \par
\tab about Nonlinear Science\par
}{\fs36 J.D. Meiss\tab \tab Sept 2003\par
}\pard\plain \f20 \par
\par
\tab \par
}{\footerf \pard\plain \s243\tqc\tx4320\tqr\tx8640 \f20 Sci.nonlinear FAQ, version 2.0\tab \tab \'a9 J.D. Meiss\par
}\pard\plain \s9 \f4\fs18 {\v From: jdm@boulder.colorado.edu (James Meiss)\par
Newsgroups: sci.nonlinear,sci.answers,news.answers\par
Subject: Nonlinear Science FAQ\par
Followup-To: poster\par
Approved: news-answers-request@MIT.EDU\par
Summary: Frequently asked questions about Nonlinear Science,\par
Chaos, and Dynamical Systems\par
\par
Archive-name: sci/nonlinear-faq\par
Posting-Frequency: monthly\par
\par
}\pard\plain \s255\sb240\tx540 \b\f21 [1]\tab About Sci.nonlinear FAQ\par
\pard\plain \f20 \par
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\par
\par
\pard\plain \s5\tx180 \f20 \tab Html format from:\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://amath.colorado.edu/faculty/jdm/faq.html} Colorado,\par
\tab {\cf5 http://www-chaos.engr.utk.edu/faq.html} Tennessee,\par
\tab {\cf5 http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html} England, \par
\tab {\cf5 http://www.sci.usq.edu.au/mirror/sci.nonlinear.faq/faq.html} Australia,\par
\tab {\cf5 http://www.faqs.org/faqs/sci/nonlinear-faq/} Hypertext FAQ Archive\par
\pard\plain \s5\tx180 \f20 \tab Or in other formats: \par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.pdf} PDF Format,\par
\tab {\cf5 http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.rtf} RTF Format,\par
\tab {\cf5 http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.tex} old version in TeX,\par
\tab {\cf5 http://www.faqs.org/ftp/faqs/sci/nonlinear-faq} the FAQ's site, text version\par
\tab {\cf5 ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/nonlinear-faq} text format.\par
\pard\plain \f20 \par
\par
This FAQ is maintained by Jim Meiss {\cf5 jdm@boulder.colorado.edu}.\par
\par
\par
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 fi
nancial gain. This FAQ may not be included in commercial collections or compilations without express permission from the author. \par
\par
\pard\plain \s254\sb120 \b\f21 [1.1] What's New?\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab \tab Fixed lots of broken and outdated links. A few sites seem to be gone, and some new sites appeared.\par
\par
\tab To some extent this FAQ is now been superseded by the Dynamical Systems site run by SIAM. See {\cf5 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.\par
\pard\plain \f20 {\cf5 \par
}\par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s233\sb240\tx540\tx720\tqr\tx9360 \b\f20 [1]\tab About Sci.nonlinear FAQ\par
\pard\plain \s232\tqr\tx9360 \i\f20 [1.1] What's New?\par
\pard\plain \s233\sb240\tx540\tx720\tqr\tx9360 \b\f20 [2]\tab Basic Theory\par
\pard\plain \s232\tqr\tx9360 \i\f20 [2.1] What is nonlinear?\par
[2.2] What is nonlinear science?\par
[2.3] What is a dynamical system?\par
[2.4] What is phase space?\par
[2.5] What is a degree of freedom?\par
[2.6] What is a map?\par
[2.7] How are maps related to flows (differential equations)?\par
[2.8] What is an attractor?\par
[2.9] What is chaos?\par
[2.10] What is sensitive dependence on initial conditions?\par
[2.11] What are Lyapunov exponents?\par
[2.12] What is a Strange Attractor?\par
[2.13] Can computers simulate chaos?\par
[2.14] What is generic?\par
[2.15] What is the minimum phase space dimension for chaos?\par
\pard\plain \s233\sb240\tx540\tx720\tqr\tx9360 \b\f20 [3]\tab Applications and Advanced Theory\par
\pard\plain \s232\tqr\tx9360 \i\f20 [3.1] What are complex systems?\par
[3.2] What are fractals?\par
[3.3] What do fractals have to do with chaos?\par
[3.4] What are topological and fractal dimension?\par
[3.5] What is a Cantor set?\par
[3.6] What is quantum chaos?\par
[3.7] How do I know if my data are deterministic?\par
[3.8] What is the control of chaos?\par
[3.9] How can I build a chaotic circuit?\par
[3.10] What are simple experiments to demonstrate chaos?\par
[3.11] What is targeting?\par
[3.12] What is time series analysis?\par
[3.13] Is there chaos in the stock market?\par
[3.14] What are solitons?\par
[3.15] What is spatio-temporal chaos?\par
[3.16] What are cellular automata?\par
[3.17] What is a Bifurcation?\par
[3.18] What is a Hamiltonian Chaos?\par
\pard\plain \s233\sb240\tx540\tx720\tqr\tx9360 \b\f20 [4]\tab To Learn More\par
\pard\plain \s232\tqr\tx9360 \i\f20 [4.1] What should I read to learn more?\par
[4.2] What technical journals have nonlinear science articles?\par
[4.3] What are net sites for nonlinear science materials?\par
\pard\plain \s233\sb240\tx540\tx720\tqr\tx9360 \b\f20 [5]\tab Computational Resources\par
\pard\plain \s232\tqr\tx9360 \i\f20 [5.1] What are general computational resources?\par
[5.2] Where can I find specialized programs for nonlinear science?\par
\pard\plain \s233\sb240\tx540\tqr\tx9360 \b\f20 [6] Acknowledgments\par
\pard\plain \s10\brdrb\brdrs \f20 \par
\pard\plain \s255\sb240\tx540 \b\f21 \page [2]\tab Basic Theory\par
\pard\plain \s254\sb120 \b\f21 [2.1] What is nonlinear?\par
\pard\plain \f20 \par
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. \par
\par
In algebra, we define linearity in terms of functions that have the property {\i f}({\i x}+{\i y}) = {\i f(x})+{\i f}({\i y}) and {\i f}({\i ax}) = {\i af}({\i x}). Nonlinear is defined as the negation of linear. This means that the result {\i f}
may be out of proportion to the input {\i x} or {\i 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, {\i f}({\i x}) = 0 and {\i f}({\i y}) = 0, then we automatically have a third zero {\i f}({\i x}+{\i y}) = 0 (in fact there are infinitely many zeros as well, since linearity implies that {\i f}({\i ax}+{\i by}
) = 0 for any {\i a} and {\i 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! \par
\par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s254\sb120 \b\f21 [2.2] What is nonlinear science?\par
\pard\plain \f20 \par
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? \par
\par
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 amplit
ude gets large the period gets longer, a fundamental effect of nonlinearity in the pendulum equations (see {\cf5 http://monet.physik.unibas.ch/~elmer/pendulum/upend.htm} and {\cf5 [3.10]}).\par
\par
(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 {\i x'} = {\i f}({\i x}) is equivalent to the linear integral equation often called the Perron-Frobenius equation\par
\pard \qc {\i p'}({\i x}) = integral [ {\i p}({\i y}) \\delta({\i x}-{\i f}({\i y})) {\i dy} ]) \par
\pard Here {\i p}({\i x}) is a density, which could be interpreted as the probability of finding oneself at the point {\i x}, and the Dirac-delta function effectively moves the points according to the map {\i f}
to give the new density. So even a nonlinear map is equivalent to a linear operator.)\par
\par
Secondly, nonlinear systems have been shown to exhibit surprising and complex effects that would never be anticipated by a scientist train
ed only in linear techniques. Prominent examples of these include bifurcation, chaos, and solitons. Nonlinearity has its most profound effects on dynamical systems (see {\cf5 [2.3]}).\par
\par
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. \par
\par
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.\par
\par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s254\sb120 \b\f21 [2.3] What is a dynamical system?\par
\pard\plain \f20 \par
A dynamical system consists of an abstract phase space or state space, whose coor
dinates 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. \par
\par
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 cons
equents with equal probability for each initial state). Most of nonlinear science--and everything in this FAQ--deals with deterministic systems.\par
\par
A dynamical system can have discrete or continuous time. The discrete case is defined by a map, {\i z}_1 = {\i f}({\i z}_0), that gives the state {\i z}_1 resulting from the initial state {\i z}
_0 at the next time value. The continuous case is defined by a "flow", {\i z}({\i t}) = \\{\i phi}_{\i t}({\i z}_0), which gives the state at time {\i t}, given that the state was z_0 at time 0. A smooth flow can be differentiat
ed w.r.t. time to give a differential equation, {\i dz/dt} = {\i F}({\i z}). In this case we call {\i F}({\i z}) a "vector field," it gives a vector pointing in the direction of the velocity at every point in phase space.\par
\par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s254\sb120 \b\f21 [2.4] What is phase space?\par
\pard\plain \f20 \par
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 infi
nite (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 sp
ace 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.\par
\par
Note that if we have a non-autonomous syst
em, 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 {\i dz/dt }= {\i F}({\i z},{\i t}) is a dynamical system on the phase space consisting of ({\i z},{\i t}), with the addition of the new dynamics {\i dt/dt }= 1.\par
\par
The path in phase space traced out by a solution of an initial value pr
oblem 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.\par
\par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s254\sb120 \b\f21 [2.5] What is a degree of freedom?\par
\pard\plain \f20 \par
The notion of "degrees of freedom" as it is used for {\v\uldb http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/}{\uldb Hamiltonian }systems means one canonical conjugate pair, a configuration, {\i q}, and its conjugate momentum {\i 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.\par
\par
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 partic
ular context.\par
\par
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.\par
\par
Note that a dynamical system with N d.o.f. Hamiltonian nominally moves in a 2{\i N} dimensional phase space. However, if {\i H}({\i q,p}) is time independent, then energy is conserved, and therefore the motion is really on a 2{\i N}
-1 dimensional energy surface, {\i H}({\i q},{\i p}) = {\i E}. Thus e.g. the planar, circular restricted 3 body problem is 2 d.o.f., and m
otion is on the 3D energy surface of constant "Jacobi constant." It can be reduced to a 2D area preserving map by Poincar\'8e section (see {\cf5 [2.6]}).\par
\par
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).\par
\par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s254\sb120 \b\f21 [2.6] What is a map?\par
\pard\plain \f20 \par
A map is simply a function, {\i f}, on the phase space that gives the next state, {\i f}({\i z}) (the image), of the system given its current state, {\i z}. (Often you will find the notation {\i z}' = {\i f}({\i z}
), where the prime means the next point, not the derivative.)\par
\par
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 ar
e 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.\par
\par
Iteration of a map means repeatedly applying the map to the consequents of the previous application. Thus we get a sequence\par
\pard\plain \s9 \f4\fs18 n\par
z = f(z ) = f(f(z )...) = f (z )\par
n n-1 n-2 0\par
\pard\plain \f20 \par
This sequence is the orbit or trajectory of the dynamical system with initial condition z_0.\par
\par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s254\sb120 \b\f21 [2.7] How are maps related to flows (differential equations)?\par
\pard\plain \f20 \par
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 {\i T} map (where {
\i T} is the period) is very useful--it is an example of a Poincar\'8e section. The time {\i 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 spac
e coordinate: the remaining coordinates describe the state completely so long as we agree to consider the same instant within every period.\par
\par
In autonomous systems (no time-dependent terms in the equations), it may also be possible to define a Poincar\'8e section and again reduce the phase space dimension by one. Here the Poincar\'8e
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).\par
\par
However, not every flow has a global Poincar\'8e section (e.g. any flow with an equilibrium point), which would need to be transverse to every possible orbit.\par
\par
Maps arising from stroboscopic sampling or Poincar\'8e
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: Poincar\'8e maps are sometimes very well approximated by noninvertible maps. For example, the Henon map ({\i x},{\i y}) -> (-{\i y}-{\i a}+{\i x}^2,{\i bx}) with small |{\i b}| is close to the logistic map, {\i x} -> -{\i a}+{\i x}
^2.\par
\par
It is often (though not always) possible to go backwards, from an invertible map to a differential equation having the map as its Poincar\'8e
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.\par
\par
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 observa
tion; it helps explain for example why a continuous-time system which should not exhibit chaos may have numerical solutions which do--see {\cf5 [2.15]}.\par
\par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s254\sb120 \b\f21 [2.8] What is an attractor?\par
\pard\plain \f20 \par
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.\par
\tab Interestingly enough, there is still some controversy in the mathematics community as to an appropriate definition of this term. Most people adopt the definition\par
\pard\plain \s6 \f20 {\i 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.\par
\pard\plain \f20
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 {\i 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 {\cf5 [2.4]}).\par
\tab 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 {\cf5 [2.9]}
) and/or strange (see {\cf5 [2.12]}).\par
\tab 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 {\i crisis }
is the change in an attractor when its basin boundary is destroyed.\par
\tab 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 neighbor
hood (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.\par
\par
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\pard\plain \s254\sb120 \b\f21 [2.9] What is chaos?\par
\pard\plain \f20 \par
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, \par
\pard\plain \s6 \f20 {\i Chaos}: effectively unpredictable long time behavior arising in a deterministic dynamical system because of sensitivity to initial conditions.\par
\pard\plain \f20
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 sen
sitivity to (or sensitive dependence on) initial conditions.\par
\par
For a dynamical system to be {\i 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 {\cf5 [2.14]}
for one definition of 'typical'), the predictability horizon grows only logarithmically with the precision of measurement (for positive Lyapunov exponents, see {\cf5 [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).\par
\par
More precisely: A map f is {\i chaotic} on a compact invariant set S if\par
\pard\plain \s5\tx180 \f20 \tab (i) f is transitive on S (there is a point x whose orbit is dense in S), and\par
\tab (ii) f exhibits sensitive dependence on S (see {\cf5 [2.10]}).\par
\pard\plain \f20 To these two requirements {\v\uldb #Devaney}{\uldb Devaney} adds the requirement that periodic points are dense in S, but this doesn't seem to be really in the spirit of the notion, and is probably better treated as a theorem (very
difficult and very important), and not part of the definition.\par
\par
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 {\cf5 [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.\par
\par
As a consequence of long-term unpredictability, time series fr
om 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.\par
\par
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 {\ul deterministic}
system, recall {\cf5 [2.3]}. The mai
n 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 consequen
ces 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).\par
\par
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\pard\plain \s254\sb120 \b\f21 [2.10] What is sensitive dependence on initial conditions?\par
\pard\plain \f20 \par
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 arb
itrarily 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!\par
\par
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.\par
\par
More precisely a set {\i S} exhibits sensitive dependence if there is an {\i r} such that for any {\i epsilon} > 0 and for each {\i x} in {\i S}, there is a {\i y} such that |{\i x} - {\i y}| < {\i epsilon}, and |{\i x}_{\i n} - {\i y}_{\i n}| > {\i r}
for some {\i n} > 0. Then there is a fixed distance {\i r }(say 1), such that no matter how precisely one specifies an initial state there are nearby states that eventually get a distance {\i r} away.\par
\par
Note: sensitive dependence does not require exponential growth of perturbations (positive Lyapunov exponent), but this is typical (see {\cf5 [2.14]}
) for chaotic systems. Note also that we most definitely do not require ALL nearby initial points diverge--generically {\cf5 [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 "ini
tial 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.\par
\par
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\pard\plain \s254\sb120 \b\f21 [2.11] What are Lyapunov exponents?\par
\pard\plain \f20 (Thanks to Ronnie Mainieri & Fred Klingener for contributing to this answer)\par
\par
The hardest thing to get right about Lyapunov exponents is the spelling of Lyapunov, which you will variously find as Liapunov, Lyapun
of 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.
\par
\par
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].\par
\par
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 Lyap
unov 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 u
sual 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 Stabil
ity) to overcome these difficulties.\par
\par
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 importa
nt. Roughly speaking the (maximal) Lyapunov exponent is the time constant, lambda, in the expression for the distance between two nearby orbits, exp({\i lambda }* {\i t)}.\~
If lambda is negative, then the orbits converge in time, and the dynamical system is insensitive to initial conditions.\~ However, if lambda is positive, then the distance between nearby orbits grows exponentially in time, and the system exhibits sensiti
ve dependence on initial conditions.\par
\par
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 se
parate. A better way is to measure the growth rate of tangent vectors to a given orbit.\par
\par
More precisely, consider a map{\i f} in an{\i m} dimensional phase space, and its derivative matrix {\i Df}({\i x}). Let {\i v }be a tangent vector at the point {\i x}. Then we define a function\par
\pard\plain \s9 \f4\fs18 1 n\par
L(x,v) = lim --- ln |( Df (x)v )|\par
n -> oo n\par
\pard\plain \f20 Now the Multiplicative Ergodic Theorem of Oseledec states that this limit exists for almost all points x and all tangent vectors {\i v}. There are at most m distinct values of{\i L} as we let {\i v}
range over the tangent space. These are the Lyapunov exponents at {\i x}.\par
\par
For more information on computing the exponents see\par
\par
\pard\plain \s5\tx180 \f20 \tab Wolf, A., J. B. Swift, et al. (1985). "Determining Lyapunov Exponents from a Time Series." Physica D {\b 16}: 285-317.\par
\tab Eckmann, J.-P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from time series." Phys. Rev. A {\b 34}: 4971-4979.\par
\pard\plain \f20 \par
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\pard\plain \s254\sb120 \b\f21 [2.12] What is a Strange Attractor?\par
\pard\plain \f20 \tab Before Chaos (BC?), the only known attractors (see {\cf5 [2.8]}) were fixed points, periodic orbits (limit cycles), and invariant tori (quasiperiodic orbits). In fact the famous Poincar\'8e
-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). \par
\tab 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 {\cf5 [2.10]}
)) is the "icon" of chaos {\cf5 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," whi
ch means that it is a fractal (see {\cf5 [3.2]}).\par
\tab 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 {\cf5
[2.14]}) result will be a geometrically strange object.\par
\tab 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." {\ul Physica D} {\b 13}: 261-268).\par
\pard\plain \s254\sb120 \b\f21 \par
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\pard\plain \s254\sb120 \b\f21 [2.13] Can computers simulate chaos?\par
\pard\plain \f20 \par
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 eve
ntually 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 d
ouble 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 give
n 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.,\par
\par
\pard\plain \s5\tx180 \f20 \tab Earn, D. J. D. and S. Tremaine, "Exact Numerical Studies of Hamiltonian Maps: Iterating without Roundoff Error," Physica D {\b 56}, 1-22 (1992). \par
\tab Binder, P. M. and R. V. Jensen, "Simulating Chaotic Behavior with Finite State Machines," Phys. Rev. {\b 34A}, 4460-3 (1986).\par
\tab Rannou, F., "Numerical Study of Discrete Plane Area-Preserving Mappings," Astron. and Astrophys. {\b 31}, 289-301 (1974).\par
\pard\plain \f20 \par
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\pard\plain \s254\sb120 \b\f21 [2.14] What is generic?\par
\pard\plain \f20 (Thanks to Hawley Rising for contributing to this answer)\par
\par
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.\par
\par
The formal definition is done in the language of topology: Consider the class to be a space of systems, and suppose it has a {\i topology} (some notion of a neighborhood, or an open set). A subset of this space is {\i dense} if its {\i closure}
(the subset plus the limits of all sequences in the subset) is the whole space. It is {\i open} and {\i dense} if it is also an open set (union of neighborhoods). A set is {\i countable}
if it can be put into 1-1 correspondence with the counting numbers. A {\i 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 {\i 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 {\i generic}.\par
\par
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 {\i 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 h
ave 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).\par
\par
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\pard\plain \s254\sb120 \b\f21 [2.15] What is the minimum phase space dimension for chaos?\par
\pard\plain \f20 \par
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 Poincar\'8e
-Bendixson theorem tells us that there is no chaos in one or two-dimensional phase spaces. Chao
s 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').\par
\par
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.)\par
\par
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\par
\pard\plain \s9 \f4\fs18 x' = f(x) = rx(1-x).\par
\pard\plain \f20 This is provably chaotic for {\i r} = 4, and many other values of r as well (see e.g. {\v\uldb #Devaney}{\uldb Devaney}). Note that every point {\i x} < {\i f}(1/2) has two preimages, so this map is not invertible.\par
\par
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 Poincar\'8e section (see {\cf5 [2.7]}).\par
\par
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, {\i dx/dt }= {\i rx}(1-{\i x}
), is equivalent to the logistic map. See e.g. S. Ushiki, "Central difference scheme and chaos," Physica {\b 4D} (1982) 407-424.\par
\par
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\pard\plain \s255\sb240\tx540 \b\f21 \page [3]\tab Applications and Advanced Theory\par
\pard\plain \s254\sb120 \b\f21 [3.1] What are complex systems?\par
\pard\plain \f20 (Thanks to Troy Shinbrot for contributing to this answer)\par
\par
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.\par
\par
While, chaos is the study of how simple systems can generate complicated behavior, compl
exity 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 o
f coupled nonlinear oscillators," Physica {\b 52D} (1991) 293-331). In these problems, many individual systems conspire to produce a single collective rhythm.\par
\par
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\par
\par
\pard\plain \s5\tx180 \f20 \tab {\cf5 http://www.calresco.org/index.htm} The Complexity & Artificial Life Web Site \par
\tab {\cf5 http://www.calresco.org/sos/sosfaq.htm} The self-organized systems FAQ\par
\pard\plain \f20 \par
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\pard\plain \s254\sb120 \b\f21 [3.2] What are fractals?\par
\pard\plain \f20 \par
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 (ima
gine 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 a
toms, 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 {\cf5 [3.5]}, Sierpinski curves, the Mandelbrot set and (almost surely) the Lorenz attractor (see {\cf5 [2.12]}
). Fractals also approximately describe many real-world objects, such as clouds (see {\cf5 http://makeashorterlink.com/?Z50D42C16}){\v } mountains, turbulence, coastlines, roots and branches of trees and veins and lungs of animals.\par
\par
"Fractal" is a term which has undergone refinement of definition by a lot of people, but was first coined by B. Mandelbrot, {\cf5 http://physics.hallym.ac.kr/reference/physicist/Mandelbrot.html}
, and defined as a set with fractional (non-integer) dimension (Hausdorff dimension, see {\cf5 [3.4]}). Mandelbrot defines a fractal in the following way:\par
\par
\pard\plain \s9 \f4\fs18 A geometric figure or natural object is said to be fractal if it\par
combines the following characteristics: (a) its parts have the same\par
form or structure as the whole, except that they are at a different\par
scale and may be slightly deformed; (b) its form is extremely irregular,\par
or extremely interrupted or fragmented, and remains so, whatever the scale\par
of examination; (c) it contains "distinct elements" whose scales are very\par
varied and cover a large range." (Les Objets Fractales 1989, p.154) \par
\pard\plain \f20 \par
See the extensive FAQ from sci.fractals at\par
\pard\plain \s5\tx180 \f20 \tab <{\cf5 ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq}\par
\pard\plain \f20 \par
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\pard\plain \s254\sb120 \b\f21 [3.3] What do fractals have to do with chaos?\par
\pard\plain \f20 \par
Often chaotic dynamical systems exhibit fractal structures in phase space. H
owever, 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 Chaot
ic." Physica 1{\b 3D}: 261-268.)\par
\par
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\pard\plain \s254\sb120 \b\f21 [3.4] What are topological and fractal dimension?\par
\pard\plain \f20 \par
See the fractal FAQ:\par
\pard\plain \s5\tx180 \f20 \tab {\cf5 ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq}\par
\pard\plain \f20 or the site\par
\pard\plain \s5\tx180 \f20 \tab {\cf5 http://pro.wanadoo.fr/quatuor/mathematics.htm}\par
\pard\plain \f20 \par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s254\sb120 \b\f21 [3.5] What is a Cantor set?\par
\pard\plain \f20 (Thanks to Pavel Pokorny for contributing to this answer)\par
\par
A Cantor set is a surprising set of points that is both infinite (uncountably so, see {\cf5 [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 {\cf5 [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.\par
\par
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).
\par
\par
See also\par
\pard\plain \s5\tx180 \f20 \tab {\cf5 http://www.shu.edu/html/teaching/math/reals/topo/defs/cantor.html}\par
\tab {\cf5 http://personal.bgsu.edu/~carother/cantor/Cantor1.html}\par
\tab {\cf5 http://mizar.uwb.edu.pl/JFM/Vol7/cantor_1.html}\par
\pard\plain \f20 \par
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:\par
\pard\plain \s5\tx180 \f20 \tab {\cf5 http://turnbull.dcs.st-and.ac.uk/history/Mathematicians/Cantor.html}\par
\tab {\cf5 http://www.shu.edu/html/teaching/math/reals/history/cantor.html}\par
\pard\plain \f20 \par
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\line {\cf5
http://www.shu.edu/projects/reals/cont/fp_cantr.html}\par
\par
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\pard\plain \s254\sb120 \b\f21 [3.6] What is quantum chaos?\par
\pard\plain \f20 (Thanks to Leon Poon for contributing to this answer)\par
\par
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 classi
cal 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 probl
em to the statistics of energy-level spacings in the semiclassical quantum regime.\par
\par
Classical chaos can be used to analyze such ostensibly quantum systems as the hydrogen atom, where classical predictions of microwave ionization thresholds agree with exper
iments. See Koch, P. M. and K. A. H. van Leeuwen (1995). "Importance of Resonances in Microwave Ionization of Excited Hydrogen Atoms." Physics Reports {\b 255}: 289-403.\par
\par
See also: \par
\pard\plain \s5\tx180 \f20 \tab {\cf5 http://sagar.physics.neu.edu/qchaos/qc.html} Quantum Chaos\par
\tab {\cf5 http://www.mpipks-dresden.mpg.de/~noeckel/microlasers.html} Microlaser Cavities\par
\pard\plain \f20 \par
\par
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\pard\plain \s254\sb120 \b\f21 [3.7] How do I know if my data are deterministic?\par
\pard\plain \f20 (Thanks to Justin Lipton for contributing to this answer)\par
\par
How can I tell if my data is deterministi
c? 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 o
r quantization. Thus any real time series, even if mostly deterministic, will be a stochastic processes\par
\par
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\par
\pard\plain \s5\tx180 \f20 \tab (1) pick a test state\par
\tab (2) search the time series for a similar or 'nearby' state and\par
\tab (3) compare their respective time evolution.\par
\pard\plain \f20 \par
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.\par
\par
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 embeddi
ng methods, see {\cf5 [3.14]}.\par
\par
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 y
ou 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 an
d 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 conv
enient anyway.\par
\par
See e.g.,\par
\pard\plain \s5\tx180 \f20 \tab Sugihara, G. and R. M. May (1990). "Nonlinear Forecasting as a Way of \line Distinguishing Chaos from Measurement Error in Time Series." Nature {\b 344}: 734-740.\par
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\pard\plain \s254\sb120 \b\f21 [3.8] What is the control of chaos?\par
\pard\plain \f20 \par
Control of chaos has come to mean the two things:\par
\pard\plain \s4 \f20 \tab stabilization of unstable periodic orbits,\par
\tab use of recurrence to allow stabilization to be applied locally.\par
\pard\plain \f20 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.\par
\par
\pard\plain \s5\tx180 \f20 \tab Hubler, A. W. (1989). "Adaptive Control of Chaotic Systems." Helv. Phys. Acta {\b 62}: 343-346.\par
\tab Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review Letters {\b 64}(11): 1196-1199. {\cf5 http://www-chaos.umd.edu/publications/abstracts.html#prl64.1196}\par
\pard\plain \f20 \par
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 expe
rimentalists 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.\par
\par
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:\par
\par
\pard\plain \s5\tx180 \f20 \tab Shinbrot, T. Ott, E., Grebogi, C. & Yorke, J.A., "Using Small Perturbations to Control Chaos," Nature, {\b 363} (1993) 411-7.\par
\tab Shinbrot, T., "Chaos: Unpredictable yet Controllable?" Nonlin. Sciences Today, {\b 3:2} (1993) 1-8.\par
\tab Shinbrot, T., "Progress in the Control of Chaos," Advance in Physics (in press).\par
\tab Ditto, WL & Pecora, LM "Mastering Chaos," Scientific American ({\b Aug. 1993}), 78-84.\par
\tab Chen, G. & Dong, X, "From Chaos to Order -- Perspectives and Methodologies in Controlling Chaotic Nonlinear Dynamical Systems," Int. J. Bif. & Chaos {\b 3} (1993) 1363-1409.\par
\pard\plain \f20 \par
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:\par
\par
\pard\plain \s5\tx180 \f20 \tab Auerbach, D., Ott, E., Grebogi, C., and Yorke, J.A. "Controlling Chaos in\line High Dimensional Systems," Phys. Rev. Lett. {\b 69} (1992) 3479-82\line {\cf5
http://www-chaos.umd.edu/publications/abstracts.html#prl69.3479}\par
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\pard\plain \s254\sb120 \b\f21 [3.9] How can I build a chaotic circuit?\par
\pard\plain \f20 (Thanks to Justin Lipton and Jose Korneluk for contributing to this answer)\par
\par
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.\par
\par
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 {\cf5 [2.8]}
) (the most famous known as the double scroll attractor) that can be displayed on a CRO.\par
\par
For more information on building such a circuit try: see\par
\par
\pard\plain \s5\tx180 \f20 \tab {\cf5 http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html} Chua's Circuit Applet\par
\pard\plain \f20 \par
References\par
\pard\plain \s5\tx180 \f20 \tab Matsumoto T. and Chua L.O. and Komuro M. "Birth and Death of the Double \line Scroll" Physica {\b D24} 97-124, 1987.\par
\tab Kennedy M. P., "Robust OP Amp Realization of Chua's Circuit", Frequenz \line {\b 46}, no. 3-4, 1992\par
\tab Madan, R. A., {\ul Chua's Circuit: A paradigm for chaos}, ed. R. A. Madan, \line Singapore: World Scientific, 1993.\par
\tab Pecora, L. and Carroll, T. {\ul Nonlinear Dynamics in Circuits}, Singapore: \line World Scientific, 1995.\par
\tab {\ul Nonlinear Dynamics of Electronic Systems, Proceedings of the Workshop \line } {\ul NDES 1993}, A.C.Davies and W.Schwartz, eds., World Scientific, 1994, \line ISBN 981-02-1769-2.\par
\tab Parker, T.S., and L.O.Chua, {\ul Practical Numerical Algorithms for Chaotic \line } {\ul Systems}, Springer-Verlag, 1989, ISBN's: 0-387-96689-7 \line and 3-540-96689-7.\par
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\pard\plain \s254\sb120 \b\f21 [3.10] What are simple experiments to demonstrate chaos?\par
\pard\plain \f20 \par
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 \par
\pard\plain \s5\tx180 \f20 \tab {\cf5 http://quasar.mathstat.uottawa.ca/~selinger/lagrange/doublependulum.html} Experimental Pendulum Designs\par
\tab {\cf5 http://www.maths.tcd.ie/~plynch/SwingingSpring/doublependulum.html} Java Applet\par
\tab {\cf5 http://monet.physik.unibas.ch/~elmer/pendulum/} Java Applets Pendulum Lab\par
\pard\plain \f20 \par
My favorite double pendulum consists of two identical planar pendula, so that you can demonstrate sensitive dependence {\cf5 [2.10]}, for a Java applet simulation see {\cf5 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:\par
\par
\pard\plain \s5\tx180 \f20 \tab A. Wolf & T. Bessoir, Diagnosing Chaos in the Space Circle, Physica {\b 50D}, 1991.\par
\pard\plain \f20 \par
One of the simplest chemical systems that shows chaos is the Belousov-Zhabotinsky reaction. The book by Strogatz {\cf5 [4.1]} has a good introduction to this subject,. For the recipe see {\cf5 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\par
\par
\pard\plain \s5\tx180 \f20 \tab Nicolis, Gregoire & Prigogine, (1989) {\ul Exploring Complexity: An \line } {\ul Introduction} W. H. Freeman\par
\pard\plain \f20 \par
The Chaotic waterwheel, while not so simple to build, is an exact realization of Lorenz famous equations. This is nicely discussed in Strogatz book {\cf5 [4.1]} as well.\par
\par
Billiard tables can exhibit chaotic motion, see {\cf5 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!\par
\par
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\pard\plain \s254\sb120 \b\f21 [3.11] What is targeting?\par
\pard\plain \f20 (Thanks to Serdar Iplik\'8di for contributing to this answer)\par
\par
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, {\cf5 [3.8]} a targeting strategy is important. See:\par
\par
\pard\plain \s5\tx180 \f20 \tab Kostelich, E., C. Grebogi, E. Ott, and J. A. Yorke, "Higher\line Dimensional Targeting," Phys Rev. E,. {\b 47}, , 305-310 (1993).\par
\tab Barreto, E., E. Kostelich, C. Grebogi, E. Ott, and J. A. Yorke, "Efficient\line Switching Between Controlled Unstable Periodic Orbits in Higher\line Dimensional Chaotic Systems," Phys Rev E, {\b 51}, 4169-4172 (1995).\par
\pard\plain \f20 \par
One application of targeting is to control a space
craft'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,\par
\par
\pard\plain \s5\tx180 \f20 \tab E. Bollt and J. D. Meiss, "Targeting Chaotic Orbits to the Moon \line Through Recurrence," Phys. Lett. A {\b 204}, 373-378 (1995).\par
\tab {\cf5 http://www.cds.caltech.edu/~marsden/software/spacecraft_orbits.html}\par
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\pard\plain \s254\sb120 \b\f21 [3.12] What is time series analysis?\par
\pard\plain \f20 (Thanks to Jim Crutchfield for contributing to this answer)\par
\par
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 {\cf5 [2.12]})in three-dimensional sp
ace by measuring only one of its coordinates, say x(t), and plotting the delay coordinates ({\i x}({\i t}), {\i x}({\i t}+{\i h}), {\i x}({\i t}+2{\i h})) for a fixed {\i h}.\par
\par
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 temp
oral 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 geo
metry and the geometric invariants of dynamical systems. (Packard, Crutchfield, Farmer & Shaw)\par
\par
\pard\plain \s5\tx180 \f20 \tab G.U. Yule, Phil. Trans. R. Soc. London A {\b 226} (1927) p. 267.\par
\tab N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, "Geometry\line from a time series", Phys. Rev. Lett. {\b 45}, no. 9 (1980) 712.\par
\tab F. Takens, "Detecting strange attractors in fluid turbulence", in: {\ul Dynamical \line } {\ul Systems and Turbulence}, eds. D. Rand and L.-S. Young \line (Springer, Berlin, 1981)\par
\tab Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.Sh.T. \line "The analysis of observed chaotic data in physical systems", \line Rev. Modern Physics {\b 65 }(1993) 1331-1392.\par
\tab D. Kaplan and L. Glass (1995). {\ul Understanding Nonlinear Dynamics}, \line Springer-Verlag {\cf5 http://www.cnd.mcgill.ca/books_understanding.html}\par
\tab E. Peters (1994) {\ul Fractal Market Analysis : Applying Chaos Theory to \line } {\ul Investment and Economics}, Wiley\line {\cf5 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471585246.html}\par
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\pard\plain \s254\sb120 \b\f21 [3.13] Is there chaos in the stock market?\par
\pard\plain \f20 (Thanks to Bruce Stewart for Contributions to this answer)\par
\par
In order to address this question, we must first agree what we mean by chaos, see {\cf5 [2.9]}.\par
\par
In dynamical systems theory, chaos means irregular fluctuations in a deterministic system (see {\cf5 [2.3]} and {\cf5 [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 {\cf5 [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?\par
\par
To be useful, economic chaos would have to involve some kind of collective behavior which can be fully described by a small numb
er 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 st
ate 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 i
dentification very difficult.\par
\par
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 pa
rticipants 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 actio
ns; or the patterns last only a very short time following some upset to the markets; or both.\par
\par
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 wh
ether they are looking for low-dimensional chaos. In fact it seems unlikely that markets remain stationary long enough to identify a chaotic attractor (see {\cf5 [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 :-)
!\par
\par
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.\par
\par
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 {\cf5 [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 an
d 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, "Cha
otic Economic Dynamics," Oxford UP, 1990, begins to explore the implications for business cycles.\par
\par
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\pard\plain \s254\sb120 \b\f21 [3.14] What are solitons?\par
\pard\plain \f20 \par
The process of obtaining a solution of a linear (constant coefficient) differential equ
ations 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.\par
\par
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 continuo
us 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 soliton
s will emerge virtually unaffected from interactions with anything, giving them great stability.\par
\par
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.\par
\par
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 impo
rtant 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.\par
\par
The soliton literature is by now vast. Two books which contain clear discussions of solitons as well as references to original papers are\par
\pard\plain \s5\tx180 \f20 \tab A. C. Newell, {\ul Solitons in Mathematics and Physics}, SIAM, Philadelphia,\line Penn. (1985)\par
\tab M.J. Ablowitz and P.A.Clarkson, {\ul Solitons, nonlinear evolution equations and inverse\line } {\ul scattering}, Cambridge (1991). {\cf5 http://www.cup.org/titles/catalogue.asp?isbn=0521387302}\par
\tab See {\cf5 http://www.ma.hw.ac.uk/solitons/}\par
\pard\plain \f20 \par
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\pard\plain \s254\sb120 \b\f21 [3.15] What is spatio-temporal chaos?\par
\pard\plain \f20 \par
\tab
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 ext
remely active, and rather unsettled area of research. For an introduction see:\par
\pard\plain \s5\tx180 \f20 \tab Cross, M. C. and P. C. Hohenberg (1993). "Pattern Formation outside of\line Equilibrium." Rev. Mod. Phys. {\b 65}: 851-1112.\par
\tab {\cf5 http://www.cmp.caltech.edu/~mcc/st_chaos.html} Spatio-Temporal Chaos\par
\pard\plain \f20 \par
An interesting application which exhibits pattern formation and spatio-temporal chaos is to excitable media in biological or chemical systems. See\par
\par
\pard\plain \s5\tx180 \f20 \tab Chaos, Solitions and Fractals {\b 5} #3&4 (1995) Nonlinear Phenomena in Excitable \line Physiological System, {\cf5 http://www.elsevier.nl/locate/chaos}\par
\tab {\cf5 http://ojps.aip.org/journal_cgi/dbt?KEY=CHAOEH&Volume=8&Issue=1} \line Chaos focus issue on Fibrillation\par
\pard\plain \f20 \par
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\pard\plain \s254\sb120 \b\f21 [3.16] What are cellular automata?\par
\pard\plain \f20 (Thanks to Pavel Pokorny for Contributions to this answer)\par
\par
\tab A Cellular automaton (CA) is a dynamical system with discrete time (like a map, see {\cf5 [2.6]}), discrete state space and discrete geometrical space (like an ODE), see {\cf5 [2.7]}). Thus they can be represented by a state {\i s}({\i i,j}
) for spatial state {\i i}, at time {\i j}, where s is taken from some finite set. The update rule is that the new state is some function of the old states, {\i s}({\i i},{\i j}+1) = {\i f}({\i 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):\par
\pard\plain \s9 \f4\fs18 time state space geometrical space\par
PDE C C C\par
ODE C C D\par
CML D C D\par
CA D D D\par
\pard\plain \f20 \par
\tab Perhaps the most famous CA is Conway's game "life." This CA evolves according to a deterministic rule which gives the st
ate 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.\par
\par
For further reading see\par
\par
\pard\plain \s5\tx180 \f20 \tab S. Wolfram (1986) Theory and Application of Cellular Automata, World Scientific Singapore.\par
\tab Physica {\b 10D} (1984)--the entire volume\par
\par
\pard\plain \f20 Some programs that do CA, as well as more generally "artificial life" are available at \par
\pard\plain \s5\tx180 \f20 \tab {\cf5 http://www.alife.org/links.html}\par
\tab {\cf5 http://www.kasprzyk.demon.co.uk/www/ALHome.html}\par
\pard\plain \f20 \par
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\pard\plain \s254\sb120 \b\f21 [3.17] What is a Bifurcation?\par
\pard\plain \f20 (Thanks to Zhen Mei for Contributions to this answer)\par
\par
A bifurcation is a qualitative change in dynamics upon a small variation in the parameters of a system.\par
\par
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 solution
s 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 an
d 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 o
r a delay differential equation.\par
\par
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 dimension
al maps, chaos occurs by infinitely many period doubling bifurcations\par
\tab (See {\cf5 http://www.stud.ntnu.no/~berland/math/feigenbaum/})\par
\par
There are a number of well constructed computer tools for studying bifurcations. In particular see {\cf5 [5.2]} for AUTO and DStool.\par
\par
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\pard\plain \s254\sb120 \b\f21 [3.18] What is a Hamiltonian Chaos?\par
\pard\plain \f20 \par
The transition to chaos for a Hamiltonian (conservative) system is somewhat different than that for a dissipative system (recall {\cf5 [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 {\cf5 [2.12]}
). Geometrically the orbits move on tori, i.e. the mathematical generalization o
f 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 p
lanet around the sun. \par
\par
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 {\cf5 [2.11]} are not necessarily po
sitive. 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 thi
s 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 perturbatio
ns. These orbits fill out what are called KAM tori.\par
\par
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 {\cf5 [2.4]} becomes self-similar (has fractal structure {\cf5
[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 th
is point, but instead of tori they cover cantor sets {\cf5 [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.\par
\par
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 importa
nt 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 Poincar\'8e called the "homoclinic trellis."\par
\par
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 {\cf5 [4.1]}
. There are a number of excellent advanced texts on Hamiltonian dynamics, some of which are listed in {\cf5 [4.1]}, but we also mention\par
\par
\pard\plain \s5\tx180 \f20 \tab Meyer, K. R. and G. R. Hall (1992), Introduction to Hamiltonian dynamical systems and the N-body problem (New York, Springer-Verlag).\par
\pard\plain \f20 \par
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\pard\plain \s255\sb240\tx540 \b\f21 \page [4]\tab To Learn More\par
\pard\plain \s254\sb120 \b\f21 [4.1] What should I read to learn more?\par
\pard\plain \s5\tx180 \f20 \tab Popularizations\par
\pard\plain \s4 \f20 1\tab Gleick, J. (1987). {\ul Chaos, the Making of a New Science}. \line London, Heinemann. {\cf5 http://www.around.com/chaos.html}\par
2\tab Stewart, I. (1989). {\ul Does God Play Dice?} Cambridge, Blackwell.\line {\cf5 http://www.amazon.com/exec/obidos/ASIN/1557861064}\par
3\tab Devaney, R. L. (1990). C{\ul haos, Fractals, and Dynamics:} {\ul Computer} \line {\ul Experiments in Mathematics.} Menlo Park, Addison-Wesley\line {\cf5 http://www.amazon.com/exec/obidos/ASIN/1878310097}\par
4\tab Lorenz, E., (1994) {\ul The Essence of Chaos}, Univ. of Washington Press.\line {\cf5 http://www.amazon.com/exec/obidos/ASIN/0295975148}\par
5\tab Schroeder, M. (1991) {\ul Fractals, Chaos, Power: Minutes from an infinite paradise}\line W. H. Freeman New York: \par
\pard\plain \s5\tx180 \f20 \tab Introductory Texts\par
\pard\plain \s4 \f20 1\tab Abraham, R. H. and C. D. Shaw (1992) {\ul Dynamics: The Geometry of }\line {\ul Behavior}, 2nd ed. Redwood City, Addison-Wesley.\par
2\tab Baker, G. L. and J. P. Gollub (1990). {\ul Chaotic Dynamics}. \line Cambridge, Cambridge Univ. Press. \line {\cf5 http://www.cup.org/titles/catalogue.asp?isbn=0521471060}\par
3\tab {\v\fs20\up6 Devaney}Devaney, R. L. (1986). {\ul An Introduction to Chaotic Dynamical }\line {\ul Systems}. Menlo Park, Benjamin/Cummings.\line {\cf5 http://math.bu.edu/people/bob/books.html}\par
4\tab Kaplan, D. and L. Glass (1995). {\ul Understanding Nonlinear Dynamics}, \line Springer-Verlag New York. {\cf5 http://www.cnd.mcgill.ca/books_understanding.html}\par
5\tab Glendinning, P. (1994). {\ul Stability, Instability and Chaos.} \line Cambridge, Cambridge Univ Press. \line {\cf5 http://www.cup.org/Titles/415/0521415535.html} \par
6\tab Jurgens, H., H.-O. Peitgen, et al. (1993). {\ul Chaos and Fractals: New} \line {\ul Frontiers of Science}. New York, Springer Verlag.\line {\cf5 http://www.springer-ny.com/detail.tpl?isbn=0387979034}\par
7\tab Moon, F. C. (1992). {\ul Chaotic and Fractal Dynamics}. New York, John Wiley. \line {\cf5 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471545716.html} \par
8\tab Percival, I. C. and D. Richard (1982). {\ul Introduction to Dynamics}. Cambridge, \line Cambridge Univ. Press. {\cf5 http://www.cup.org/titles/catalogue.asp?isbn=0521281490} \par
9\tab Scott, A. (1999). {\ul NONLINEAR SCIENCE: Emergence and Dynamics of} \line {\ul Coherent Structures}, Oxford {\cf5 http://www4.oup.co.uk/isbn/0-19-850107-2}\line {\cf5 http://www.imm.dtu.dk/documents/users/acs/BOOK1.html} \par
10\tab Smith, P (1998) {\ul Explaining Chaos}, Cambridge \line {\cf5 http://us.cambridge.org/titles/catalogue.asp?isbn=0521477476}\par
11\tab Strogatz, S. (1994). {\ul Nonlinear Dynamics and Chaos}. Reading, \line Addison-Wesley\line {\cf5 http://www.perseusbooksgroup.com/perseus-cgi-bin/display/0-7382-0453-6}\par
12\tab Thompson, J. M. T. and H. B. Stewart (1986) {\ul Nonlinear Dynamics and }\line {\ul Chaos}. Chichester, John Wiley and Sons.\line {\cf5 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471876844.html}\par
13\tab Tufillaro, N., T. Abbott, et al. (1992). {\ul An Experimental Approach} \line t{\ul o Nonlinear Dynamics and Chaos}. Redwood City, Addison-Wesley. \line {\cf5 http://www.amazon.com/exec/obidos/ASIN/0201554410/}\par
14\tab Turcotte, Donald L. (1992). {\ul Fractals and Chaos in Geology and} \line {\ul Geophysics}, Cambridge Univ. Press. \line {\cf5 http://www.cup.org/titles/catalogue.asp?isbn=0521567335} \par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Introductory Articles\par
\pard\plain \s4 \f20 1\tab May, R. M. (1986). "When Two and Two Do Not Make Four." \line Proc. Royal Soc. B228: 241.\par
2\tab Berry, M. V. (1981). "Regularity and Chaos in Classical Mechanics, \line Illustrated by Three Deformations of a Circular Billiard." \line Eur. J. Phys. 2: 91-102.\par
3\tab Crawford, J. D. (1991). "Introduction to Bifurcation Theory." \line Reviews of Modern Physics 63(4): 991-1038.\par
3\tab Shinbrot, T., C. Grebogi, et al. (1992). "Chaos in a Double Pendulum." \line Am. J. Phys 60: 491-499.\par
5\tab David Ruelle. (1980). "Strange Attractors," \line The Mathematical Intelligencer 2: 126-37.\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Advanced Texts\par
\pard\plain \s4\fi-360\li720\tx720 \f20 1\tab Arnold, V. I. (1978). {\ul Mathematical Methods of Classical Mechanics}.\line New York, Springer.\line {\cf5 http://www.springer-ny.com/detail.tpl?isbn=038796890}\par
2\tab Arrowsmith, D. K. and C. M. Place (1990), {\ul An Introduction to Dynamical Systems}.\line Cambridge, Cambridge University Press.\line {\cf5 http://us.cambridge.org/titles/catalogue.asp?isbn=0521316502}\par
3\tab Guckenheimer, J. and P. Holmes (1983), {\ul Nonlinear Oscillations, Dynamical\line } {\ul Systems, and Bifurcation of Vector Fields}, Springer-Verlag New York.\par
4\tab Kantz, H., and T. Schreiber (1997). {\ul Nonlinear time series analysis}.\line Cambridge, Cambridge University Press\line {\cf5 http://www.mpipks-dresden.mpg.de/~schreibe/myrefs/book.html}\par
5\tab Katok, A. and B. Hasselblatt (1995), I{\ul ntroduction to the Modern\line } {\ul Theory of Dynamical Systems}, Cambridge, Cambridge Univ. Press.\line {\cf5 http://titles.cambridge.org/catalogue.asp?isbn=0521575575} \par
6\tab Hilborn, R. (1994), {\ul Chaos and Nonlinear Dyanamics: an Introduction for}\line {\ul Scientists and Engineers}, Oxford Univesity Press.\line {\cf5 http://www4.oup.co.uk/isbn/0-19-850723-2}\par
7\tab Lichtenberg, A.J. and M. A. Lieberman (1983), {\ul Regular and Chaotic Motion,} \line Springer-Verlag, New York .\par
8\tab Lind, D. and Marcus, B. (1995) {\ul An Introduction to Symbolic Dynamics and Coding},\line Cambridge University Press, Cambridge {\cf5 http://www.math.washington.edu/SymbolicDynamics/}\par
9\tab MacKay, R.S and J.D. Meiss (eds) (1987), {\ul Hamiltonian Dynamical Systems A reprint \line } {\ul selection,} , Adam Hilger, Bristol\par
10\tab Nayfeh, A.H. and B. Balachandran (1995), {\ul Applied Nonlinear Dynamics:}\line {\ul Analytical, Computational and Experimental Methods}\line John Wiley& Sons Inc., New York\line {\cf5
http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471593486.html}\par
11\tab Ott, E. (1993). {\ul Chaos in Dynamical Systems}. Cambridge University Press, \line Cambridge. {\cf5 http://us.cambridge.org/titles/catalogue.asp?isbn=0521010845}\par
12\tab L.E. Reichl, (1992), {\ul The Transition to Chaos, in Conservative and Classical Systems:} \line {\ul Quantum Manifestations} Springer-Verlag, New York\par
13\tab Robinson, C. (1999), {\ul Dynamical Systems}: {\ul Stability, Symbolic}\line {\ul Dynamics, and Chaos}, 2nd Edition, Boca Raton, CRC Press. \line {\cf5 http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=8495}\par
14\tab Ruelle, D. (1989), {\ul Elements of Differentiable Dynamics and Bifurcation Theory},\line Academic Press Inc.\par
15\tab Tabor, M. (1989), {\ul Chaos and Integrability in Nonlinear Dynamics}:\line {\ul an Introduction}, Wiley, New York.\line {\cf5 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471827282.html}\par
16\tab Wiggins, S. (1990), {\ul Introduction to Applied Nonlinear Dynamical} {\ul Systems and Chaos,}\line Springer-Verlag New York.\par
17\tab Wiggins, S. (1988), {\ul Global Bifurcations and Chaos}, Springer-Verlag New York.\par
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\pard\plain \s254\sb120 \b\f21 [4.2] What technical journals have nonlinear science articles?\par
\pard\plain \f20 \par
\pard\plain \s9 \f4\fs18 Physica D The premier journal in Applied Nonlinear Dynamics\par
Nonlinearity Good mix, with a mathematical bias\par
Chaos AIP Journal, with a good physical bent\par
SIAM J. of Dynamical Systems Online Journal with multimedia\par
{\cf5 http://www.siam.org/journals/siads/siads.htm}\par
Chaos Solitons and Fractals Low quality, some good applications\par
Communications in Math Phys an occasional paper on dynamics\par
Comm. in Nonlinear Sci. New Elsevier journal\par
and Num. Sim. {\cf5 http://www.elsevier.com/locate/cnsns}\par
Ergodic Theory and Rigorous mathematics, and careful work\par
Dynamical Systems\par
International J of lots of color pictures, variable quality.\par
Bifurcation and Chaos\par
J Differential Equations A premier journal, but very mathematical\par
J Dynamics and Diff. Eq. Good, more focused version of the above\par
J Dynamics and Stability Focused on Eng. applications. New editorial\par
of Systems board--stay tuned.\par
J Fluid Mechanics Some expt. papers, e.g. transition to turbulence\par
J Nonlinear Science a newer journal--haven't read enough yet.\par
J Statistical Physics Used to contain seminal dynamical systems papers\par
Nonlinear Dynamics Haven't read enough to form an opinion\par
Nonlinear Science Today Weekly News: {\cf5 http://www.springer-ny.com/nst/}\par
Nonlinear Processes in New, variable quality...may be improving\par
Geophysics\par
Physics Letters A Has a good nonlinear science section\par
Physical Review E Lots of Physics articles with nonlinear emphasis\par
Regular and Chaotic Dynamics Russian Journal {\cf5 http://web.uni.udm.ru/~rcd/\par
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\pard\plain \s254\sb120 \b\f21 [4.3] What are net sites for nonlinear science materials?\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Bibliography\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://www.uni-mainz.de/FB/Physik/Chaos/chaosbib.html} Mainz http site\par
\tab {\cf5 ftp://ftp.uni-mainz.de/pub/chaos/chaosbib/} Mainz ftp site\par
\tab {\cf5 http://www-chaos.umd.edu/publications/searchbib.html} Seach the Mainz Site{\cf5 \par
}\tab {\cf5 http://www-chaos.umd.edu/publications/references.html} Maryland{\cf5 \par
}\tab {\cf5 http://www.cpm.mmu.ac.uk/~bruce/combib/} Complexity Bibliography\par
\tab {\cf5 http://www.mth.uea.ac.uk/~h720/research/} Ergodic Theory and Dynamical Systems\par
\tab {\cf5 http://www.drchaos.net/drchaos/intro.html} Nonlinear Dynamics Resources (pdf file)\par
\tab {\cf5 http://www.nonlin.tu-muenchen.de/chaos/Projects/miguelbib} Sanjuan's Bibliography\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Preprint Archives\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://www.math.sunysb.edu/dynamics/preprints/} StonyBrook{\cf5 \par
}\tab {\cf5 http://cnls.lanl.gov/People/nbt/intro.html} Los Alamos Preprint Server\par
\tab {\cf5 http://xxx.lanl.gov/} Nonlinear Science Eprint Server{\cf5 \par
}\tab {\cf5 http://www.ma.utexas.edu/mp_arc/mp_arc-home.html} Math-Physics Archive{\cf5 \par
}\tab {\cf5 http://www.ams.org/global-preprints/} AMS Preprint Servers List\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Conference Announcements\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://at.yorku.ca/amca/conferen.htm} Mathematics Conference List\par
\tab {\cf5 http://www.math.sunysb.edu/dynamics/conferences/conferences.html} StonyBrook List\par
\tab {\cf5 http://www.nonlin.tu-muenchen.de/chaos/termine.html} Munich List\par
\tab {\cf5 http://xxx.lanl.gov/Announce/Conference/} Los Alamos List\par
\tab {\cf5 http://www.tam.uiuc.edu/Events/conferences.html} Theoretical & Applied Mechanics\par
\tab {\cf5 http://www.siam.org/meetings/ds99/index.htm} SIAM Dynamical Systems 1999 \par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Newsletters\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 gopher://gopher.siam.org:70/11/siag/ds} SIAM Dynamical Systems Group\par
\tab {\cf5 http://www.amsta.leeds.ac.uk/Applied/news.dir/} UK Nonlinear News\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Education Sites\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://math.bu.edu/DYSYS/} Devaney's Dynamical Systems Project\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Electronic Journals\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://www.springer-ny.com/nst/} Nonlinear Science Today\par
\tab {\cf5 http://www3.interscience.wiley.com/cgi-bin/jtoc?ID=38804} Complexity\par
\tab {\cf5 http://journal-ci.csse.monash.edu.au/} Complexity International Journal\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Electronic Texts\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://cnls.lanl.gov/People/nbt//Book/node1.html} An experimental approach \line to nonlinear dynamics and chaos\par
\tab {\cf5 http://www.nbi.dk/~predrag/QCcourse/} Lecture Notes on Periodic Orbits\par
\tab {\cf5 http://hypertextbook.com/chaos/} The Chaos HyperTextBook\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Institutes and Academic Programs\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://physicsweb.org/resources/dsearch.phtml} Physics Institutes\par
\tab {\cf5 http://ip-service.com/WiW/institutes.html} Nonlinear Groups\par
\tab {\cf5 http://www-chaos.engr.utk.edu/related.html} Research Groups in Chaos\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Java Applets Sites\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://physics.hallym.ac.kr/education/TIPTOP/VLAB/about.html} Virtual Laboratory\par
\tab {\cf5 http://monet.physik.unibas.ch/~elmer/pendulum/} Java Pendulum\par
\tab {\cf5 http://kogs-www.informatik.uni-hamburg.de/~wiemker/applets/fastfrac/fastfrac.html} \line Java Fractal Explorer\par
\tab {\cf5 http://www.apmaths.uwo.ca/~bfraser/index.html} B. Fraser\rquote s Nonlinear Lab\par
\tab {\cf5 http://www.cmp.caltech.edu/~mcc/Chaos_Course/} Mike Cross' Demos\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Who is Who in Nonlinear Dynamics\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://www.chaos-gruppe.de/wiw/wiw.html} Munich List{\cf5 \par
}\tab {\cf5 http://www.math.sunysb.edu/dynamics/people/list.html} Stonybrook List\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Lists of Nonlinear sites\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://makeashorterlink.com/?C58C23C16} Netscape\rquote s List\par
\tab {\cf5 http://cnls.lanl.gov/People/nbt/sites.html} Tufillaro's List\par
\tab {\cf5 http://cires.colorado.edu/people/peckham.scott/chaos.html} Peckham's List\par
\tab {\cf5 http://members.tripod.com/~IgorIvanov/physics/nonlinear.html} Physics Encyclopedia\par
\tab {\cf5 http://www.maths.ex.ac.uk/~hinke/dss/index.html} Osinga's Software List{\cf5 \par
}\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Dynamical Systems\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://www.math.sunysb.edu/dynamics/} Dynamical Systems Home Page\par
\tab {\cf5 http://www.math.psu.edu/gunesch/entropy.html} Entropy and Dynamics\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Chaos sites\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://www.industrialstreet.net/chaosmetalink/} Chaos Metalink\par
\tab {\cf5 http://bofh.priv.at/ifs/} Iterated Function Systems Playground\par
\tab {\cf5 http://www.xahlee.org/PageTwo_dir/more.html} Xah Lee's dynamics and Fractals pages\par
\tab {\cf5 http://acl2.physics.gatech.edu/tutorial/outline.htm} Tutorial on Control of Chaos\par
\tab {\cf5 http://www.mathsoft.com/mathresources/constants/wellknown/article/0,,2090,00.html}\line All about Feigenbaum Constants\par
\tab {\cf5 http://www.stud.ntnu.no/~berland/math/feigenbaum/} The Feigenbaum Fractal\par
\tab {\cf5 http://members.aol.com/MTRw3/index.html} Mike Rosenstein's Chaos Page.\par
\tab {\cf5 http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/cspls.html} Chaos in Psychology\par
\tab {\cf5 http://www.eie.polyu.edu.hk/~cktse/NSR/} Movies and Demonstrations\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Time Series\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://www.drchaos.net/drchaos/refs.html} Dynamics and Time Series\par
\pard \s11\qj\tx540\tqc\tx3680 \tab {\cf5 http://astro.uni-tuebingen.de/groups/time/} Time series Analysis\par
\pard \s11\tx540\tqc\tx3680 \tab {\cf5 http://www-personal.buseco.monash.edu.au/~hyndman/TSDL/index.htm} \line Time Series Data Library\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Complex Systems Sites\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://www.math.upatras.gr/~mboudour/nonlin.html} Complexity Home Page\par
\pard \s11\tx540\tqc\tx3680\tx9000 \tab {\cf5 http://www.calresco.org/} The Complexity & Artificial Life Web Site\par
\pard \s11\tx540\tqc\tx3680 \tab {\cf5 http://www.physionet.org/} Complexity and Physiology Site\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Fractals Sites\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\cf5 http://forum.swarthmore.edu/advanced/robertd/index.html#frac} A Fractal Gallery\par
\tab {\cf5 http://spanky.triumf.ca/www/welcome1.html} The Spanky Fractal DataBase\par
\tab {\cf5 http://sprott.physics.wisc.edu/fractals.htm} Sprott's Fractal Gallery\par
\tab {\cf5 http://fractales.inria.fr/} Projet Fractales\par
\tab {\cf5 http://force.stwing.upenn.edu/~lau/fractal.html} Lau's Fractal Stuff\par
\tab {\cf5 http://skal.planet-d.net/quat/f_gal.html} 3D Fractals\par
\tab {\cf5 http://www.cnam.fr/fractals.html} Fractal Gallery\par
\tab {\cf5 http://www.fractaldomains.com/} Fractal Domains Gallery\par
\tab {\cf5 http://home1.swipnet.se/~w-17723/fracpro.html} Fractal Programs\par
\tab {\cf5 http://xahlee.org/PageTwo_dir/MathPrograms_dir/mathPrograms.html#Fractals}\line Fractal Programs\par
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\pard\plain \s255\sb240\tx540 \b\f21 \page [5]\tab Computational Resources\par
\pard\plain \f20 \par
\pard\plain \s254\sb120 \b\f21 [5.1] What are general computational resources?\par
\pard\plain \s5\tx180 \f20 \tab CAIN Europe Archives\line {\cf5 http://www.can.nl/education/material/software.html} Software Area\par
\tab FAQ guide to packages from sci.math.num-analysis\line {\cf5 ftp://rtfm.mit.edu/pub/usenet/news.answers/num-analysis/faq/part1}\par
\tab NIST Guide to Available Mathematical Software\line {\cf5 http://gams.cam.nist.gov/}\par
\tab Mathematics Archives Software \line {\cf5 http://archives.math.utk.edu/software.html}\par
\tab Matpack, C++ numerical methods and data analysis library\line {\cf5 http://www.matpack.de/}\par
\tab Numerical Recipes Home Page\line {\cf5 http://www.nr.com/}\par
\pard\plain \f20 \par
\pard\plain \s254\sb120 \b\f21 [5.2] Where can I find specialized programs for nonlinear science?\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab {\b The Academic Software Library:}\par
\pard\plain \s11\tx540\tqc\tx3680 \f20 \tab {\b Chaos Simulations}\line Bessoir, T., and A. Wolf, 1990. Demonstrates logistic map, Lyapunov exponents, billiards in a stadium, sensitive dependence, n-body gravitational motion.\par
\tab {\b Chaos Data Analyser}\line A PC program for analyzing time series. By Sprott, J.C. and G. Rowlands. \line {\i For more info}:{\cf5 http://sprott.physics.wisc.edu/cda.htm}\par
{\b \tab Chaos Demonstrations}\line A PC program for demonstrating chaos, fractals, cellular automata, and related nonlinear phenomena. By J. C. Sprott and G. Rowlands.\line {\i System}: IBM PC or compatible with at least 512K of memory.\line {\i
Available}: The Academic Software Library, (800) 955-TASL. $70.\par
{\b \tab Chaotic Dynamics Workbench}\line Performs interactive numerical experiments on systems modeled by ordinary differential equations, including: four versions of driven Duffi
ng oscillators, pendulum, Lorenz, driven Van der Pol osc., driven Brusselator, and the Henon-Heils system. By R. Rollins.\line {\i System}: IBM PC or compatible, 512 KB memory.\line {\i Available}: The Academic Software Library, (800) 955-TASL, $70\line
\par
\pard\plain \s5\tx180 \f20 {\b \tab Applied Chaos Tools}\line Software package for time series analysis based on the UCSD group's, work. This package is a companion for Abarbanel's book {\ul Analysis of Observed Chaotic Data}, Springer-Verlag.\line {\i
System}: Unix-Motif, Windows 95/NT\line {\i For more info see}: {\cf5 http://www.zweb.com/apnonlin/csp.html}\line \par
{\b \tab AUTO}\line
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.\line {\i System}
: versions to run under X windows--SUN or sgi or LINUX\line {\i Available}: anonymous ftp from {\cf5 ftp://ftp.cs.concordia.ca/pub/doedel/auto} \line \par
\tab {\b BZphase\line }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.\line {\i System}: {\f2010
DOS 6 and higher + PMODE/W DOS Extender}. Also openGL version\line {\i Available}: {\cf5 http://members.tripod.com/~RedAndr/BZPhase.htm}\line \par
{\b \tab Chaos\line }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\line {\i System}
: Silicon Graphics workstations, IBM RISC workstations with GL\line {\i Available}: {\cf5 http://msg.das.bnl.gov/~bstewart/software.html}\line \par
{\b \tab Chaos\line }
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 i
terative maps, an "electronic chaos-generator", the Mandelbrot set, and ODEs.\line {\i System}: IBM PC or compatible.\line {\i Available}: $${\cf5 http://www.springer-ny.com/catalog/np/updates/0-387-57457-3.html}\line \par
{\b \tab CHAOS II\line }Chaos Programs to go with Baker, G. L. and J. P. Gollub (1990) Chaotic Dynamics. Cambridge, Cambridge Univ. {\cf5 http://www.cup.org/titles/catalogue.asp?isbn=0521471060}\line {\i System}
: IBM, 512K memory, CGA or EGA graphics, True Basic\line {\i For more info}: contact Gregory Baker, P.O. Box 278 ,Bryn Athyn, PA, 19009\line \par
{\b \tab Chaos Analyser\line }Programs to Time delay embedding, Attractor (3d) viewing and animation, Poincar\'8e
sections, Mutual information, Singular Value Decomposition embedding, Full Lyapunov spectra (with noise cancellation), Local SVD analysis (for determining the systems dimension). By Mike Banbrook.\line {\i System}: Unix, X windows\line {\i For more info}
: {\cf5 http://www.ee.ed.ac.uk/~mb/analysis_progs.html}\line \par
{\b \tab Chaos Cookbook}\line These programs go with J. Pritchard's book, {\ul The Chaos Cookbook} System: Programs written in Visual Basic & Turbo Pascal\line {\i Available}: $${\cf5 http://www.amazon.com/exec/obidos/ASIN/0750617772}\line \par
{\b \tab Chaos Plot\line }ChaosPlot is a simple program which plots the chaotic behavior of a damped, driven anharmonic oscillator.\line {\i System}: Macintosh\line {\i For more info}: {\cf5 http://archives.math.utk.edu/
software/mac/diffEquations/.directory.html}\line \par
{\b \tab Cubic Oscillator Explorer\line }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.\line {\i System}: Macintosh + Digidesign DSP
\~card, Digisystem init 2.6 and (optional) MIDI Manager\line {\i Available}: (Missing??) Fractal Music\line \par
{\b \tab DataPlore\line }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.\line {\i Systems}
: MS Windows, Linux, SUN Solaris 2.6\line {\i Available}: $${\cf5 http://www.datan.de/dataplore/}\line \par
{\b \tab dstool\line }Free software from Guckenheimer's group at Cornell; DSTool has lots of examples of chaotic systems, Poincar\'8e sections, bifurcation diagrams.\line {\i System}: Unix, X windows.\line {\i Available}: {\cf5
ftp://cam.cornell.edu/pub/dstool/}\line \par
{\b \tab Dynamical Software Pro\line }Analyze non-linear dynamics and chaos. Includes ODEs, delay differential equations, discrete maps, numerical integration, time series embedding, etc. \line {\i System}
: DOS. Microsoft Fortran compiler for user defined equations.\line {\i Available}: SciTech {\cf5 http://www.scitechint.com/}\line \par
{\b \tab Dynamics: Numerical Explorations.\line }A book + disk by H. Nusse, and J.Yorke. A hands on approach to learning the concepts and the many aspects in computing relevant quantities in chaos\line {\i System}
: PC-compatible computer or X-windows system on Unix computers\line {\i Available}: $$ {\fs22\cf5 http://www.springer-ny.com/detail.tpl?isbn=0387982647 }\line \par
\tab {\b Dynamics Solver\line }Dynamics Solver solve numerically both initial-value problems and boundary-value problems for continuous and discrete dynamical systems.\line {\i System}: Windows 3.1 or Windows 95/98/NT{\b \line }{\i Available}: {\cf5
http://tp.lc.ehu.es/jma/ds/ds.html}{\b \line }\par
\tab {\b DynaSys\line }Phase plane portraits of 2D ODEs by Etienne Dupuis\line {\i System}: Windows 95/98\line {\i Available}: (Missing??)\par
\par
{\b \tab FD3\line }A program to estimate fractal dimensions of a set. By DiFalco/Sarraille \line {\i System}: C source code, suitable for compiling for use on a Unix or DOS platform.\line {\i Available}: {\cf5 ftp://ftp.cs.csustan.edu/pub/fd3/}\line
\par
{\b \tab FracGen\line }FracGen is a freeware program to create fractal images using Iterated Function Systems. A tutorial is provided with the program. By Patrick Bangert \line {\i System}: PC-compatible computer, Windows 3.1\line {\i Available}: {\cf5
http://212.201.48.1/pbangert/site/fracgen.html}\line \par
{\b \tab Fractal Domains\line }Generates of Mandelbrot and Julia sets. By Dennis C. De Mars\line {\i System}: Power Macintosh\line {\i Available}: {\cf5 http://www.fractaldomains.com/}\line \par
\tab {\b Fractal Explorer\line }Generates Mandelbrot and Newton's method fractals. By Peter Stone\line {\i System: }Power Macintosh\line {\i Available}: {\cf5 http://usrwww.mpx.com.au/~peterstone/index.html}\line \par
{\b \tab GNU Plotutils\line }
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 gr
aph, which is based on libplot, and ODE integration software.\line {\i System}: GNU/Linux, FreeBSD, and Unix systems.{\b \line }{\i Available}: {\cf5 http://www.gnu.org/software/plotutils/plotutils.html}\line \par
{\b \tab Ilya}\line A program to visually study a reaction-diffusion model based on the Brusselator from Future Skills Software, Herber Sauro.\line {\i System}: Requires Windows 95, at least 256 colours\line {\i Available} : {\cf5
http://www.fssc.demon.co.uk/rdiffusion/ilya.htm}\line \par
{\b \tab INSITE\line }(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 sof
tware 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, {\b 75}, 1081-1089 (1987).
\line {\i System}: C codes in Unix Tar or DOS format (later requires QuickWindowC\line or MetaWINDOW/Plus 3.7C. and MS C compiler 5.1)\line {\i Available}: INSITE SOFTWARE, p.o. Box 9662, Berkeley, CA , U.S.A.\line \par
{\b \tab Institut fur ComputerGraphik}\line A collection of programs for developing advanced visualization techniques in the field of three-dimensional dynamical systems. By L\'9affelmann H., Gr\'9aller E.\line {\i System}: various, requires AVS\line {\i
Available}: {\cf5 http://www.cg.tuwien.ac.at/research/vis/dynsys/}\line \par
{\b \tab KAOS1D\line }A tool for studying one-dimensional (1D) discrete dynamical systems. Does bifurcation diagrams, etc. for a number of maps\line {\i System}: PC compatible computer, DOS, VGA graphics\line {\i Available}: {\cf5
http://www.if.ufrgs.br/~arenzon/jsoftw.html}\line \par
{\b \tab LOCBIF\line }An interactive tool for bifurcation analysis of non-linear ordinary differential equations ODE's and maps{\b . }By Khibnik, Nikolaev, Kuznetsov and V. Levitin \line {\i System}: Now part of XPP (See below)\line {\i Available}: {\cf5
http://www.math.pitt.edu/~bard/classes/wppdoc/locbif.html}\line \par
{\b \tab Lyapunov Exponents\line }Keith Briggs Fortran codes for Lyapunov exponents\line {\i System}: any with a Fortran compiler\line {\i Available}: {\cf5 http://more.btexact.com/people/briggsk2/}\line \par
{\b \tab Lyapunov Exponents and Time Series\line }Based on Alan Wolf's algorithm, see {\cf5 [2.11]}, but a more efficient version.\line {\i System}: Comes as C source, Fortran source, PC executable, etc\line {\i Available}: {\cf5
http://www.cooper.edu/engineering/physics/wolf/} (Seems to be missing?)\line \par
{\b \tab Lyapunov Exponents and Time Series\line }Michael Banbrook's C codes for Lyapunov exponents & time series analysis\line {\i System}: Sun with X windows.\line {\i Available}: {\cf5 http://www.see.ed.ac.uk/~mb/analysis_progs.html}\line \par
\tab {\b Lyapunov Exponents Toolbox (LET)\line 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.
\line {\i System}: MATLAB 5\line {\i Available}: {\cf5 ftp://ftp.mathworks.com/pub/contrib/v5/misc/let}\line \par
\tab {\b Lyapunov.m\line }A Matlab program based on the {\f2010 QR Method , by von Bremen, Udwadia, and Proskurowski, Physica D, vol. }{\b\f2010 101}{\f2010 , 1-16, (1997)\line }{\i\f2010 System}{\f2010 : Matlab\line }{\i\f2010 Available}{\f2010 : }{
\f2010\cf5 http://www.usc.edu/dept/engineering/mecheng/DynCon/}{\b \line }\par
{\b \tab Macintosh Dynamics Programs\line }{\i Lists available at}: {\cf5 http://hypertextbook.com/chaos/92.shtml\line }and {\cf5 http://www.xahlee.org/PageTwo_dir/MathPrograms_dir/mathPrograms.html}\line \par
{\b \tab MacMath\line }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.\line {\i
System}: Macintosh\line {\i For more info}: {\cf5 http://www.math.hmc.edu/codee/solvers/mac-math.html}\line {\i Available}: $$ Springer-Verlag {\fs22\cf5 http://www.springer-ny.com/detail.tpl?isbn=0387941355 }\line \par
{\b \tab Madonna\line }Solves Differential and Difference Equations. Runs STELLA. Has a parser with a control language. By Robert Macey and George Oster at Berkeley\line {\i System}: Macintosh or Windows 95 or later\line {\i Available} : $$ {\cf5
http://www.berkeleymadonna.com/}\line \par
{\b \tab MatLab Chaos\line }A collection of routines for generate diagrams which illustrate chaotic behavior associated with the logistic equation.\line {\i System}: Requires MatLab.\line {\i Available} : {\cf5
ftp://ftp.mathworks.com/pub/contrib/misc/chaos/}\line \par
{\b \tab MTRChaos\line }MTRCHAOS and MTRLYAP compute correlation dimension and largest Lyapunov exponents, delay portraits. By Mike Rosenstein. \line {\i System}
: PC-compatible computer running DOS 3.1 or higher, 640K RAM, and EGA display. VGA & coprocessor recommended\line {\i Available}: {\cf5 ftp://spanky.triumf.ca/pub/fractals/programs/ibmpc/}\line \par
{\b \tab Nonlinear Dynamics Toolbox\line }{\f2010
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.}\line {\i System}: {\f2010
Windows 95, 98, or NT}\line {\i Available} : Missing??\line {\b \par
\tab NLD Toolbox}\line This toolbox has many of the standard dynamical systems, By Jeff Brush{\b \line }{\i System}: PC, MS-DOS.\line {\i Available:} {\cf5 http://www.physik.tu-darmstadt.de/nlp/nldtools/nldtools.html}\line \par
{\b \tab ODECalc\line }A program for integrating boundary value and initial value Problems for up to 9th order ODEs. By Optimal Designs.\line {\i System}: PC 386+, DOS 3.3+, 16 bit arch.\line {\i Available} : {\cf5
ftp://ftp.mecheng.asme.org/pub/EDU_TOOL/Ode200.exe}\line \par
{\b \tab PHASER\line }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.\line {\i System}: PC-compatible\line {\i Available}: Springer-Verlag {\fs22\cf5 http://www.springer-ny.com/detail.tpl?isbn=0387142029}\line \par
\tab {\b PhysioToolkit\line }{\f2010 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\line }{
\i\f2010 System}{\f2010 : Unix\line }{\i\f2010 Available}{\f2010 : }{\f2010\cf5 http://www.physionet.org/physiotools/}{\f2010 \line }{\b \par
}\tab {\b Recurrence Quantification Analysis\line }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.\line {\i
System}: DOS executable\line {\i Available}:{\cf5 http://homepages.luc.edu/~cwebber/}{\b \line }\par
{\b \tab SciLab\line }A simulation program similar in intent to MatLab. It's primarily designed for systems/signals work, and is large. From INRIA in France.\line {\i System}: Unix, X Windows, 20 Meg Disk space.\line {\i Available} : {\cf5
ftp://ftp.inria.fr/INRIA/Projects/Meta2/Scilab}\line \par
{\b \tab StdMap\line }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.\line {\i System}: Macintosh\line {\i Available}: {\cf5
http://amath.colorado.edu/faculty/jdm/stdmap.html}\line \par
{\b \tab STELLA\line }Simulates dynamics for Biological and Social systems modelling. Uses a building block metaphor constructing models.\line {\i System}: Macintosh and Windows PC\line {\i Available}: $$ {\cf5 http://www.hps-inc.com/edu/stella/stella.htm
}\line \par
{\b \tab Time Series Tools\line }An extensive list of Unix tools for Time Series analysis\line {\i System}: Unix\line {\i For more info}: {\cf5 http://chuchi.df.uba.ar/guille/TS/tools/tools.html} (Link down??)\line \par
{\b \tab Time Series Analysis from Darmstadt\line }Four prgrams Time Series analysis and Dimension calculation from the Institute of Applied Physics at Darmstadt.\line {\i System}: OS2 or Solaris/Linux/Win9X/NT + Fortran source\line {\i For more info}: {
\cf5 http://www.physik.tu-darmstadt.de/nlp/distribution.html}\line \par
{\b \tab Time Series Analysis from Kennel\line }The program mkball finds the minimum embedding dimension using the false strands enhancement of the false neighbors algorithm of Kennel & Abarbanel.\line {\i System}: any C compiler\line {\i Available}: {
\cf5 ftp://lyapunov.ucsd.edu/pub/nonlinear/mbkall.tar.gz}\line \par
{\b \tab TISEAN} {\b Time Series Analysis\line A}gorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing. By Rainer Hegger, Holger Kantz and Thomas Schreiber\line {\i System}
: C, C++ and Fortran Codes for Unix,\line {\i Available}: {\cf5 http://www.mpipks-dresden.mpg.de/~tisean/}\line {\b \par
\tab Tufillaro's Programs\line }From the book Nonlinear Dynamics and Chaos by Tufillaro, Abbot and Reilly (1992) (for a sample section see {\cf5 http://www.drchaos.net/drchaos/Book/node1.html}). A collection of programs for the Macintosh.\line {\i System}
: Macintosh\line {\i Available}: {\cf5 http://www.drchaos.net/drchaos/bb.html}\line \par
{\b \tab Unified Life Models (ULM)\line }ULM, by Stephane Legendre, is a program to study population dynamics and more generally, discrete dynamical systems. It models any s
pecies life cycle graph (matrix models) inter- and intra-specific competition (non linear systems), environmental stochasticity, demographic stochasticity (branching processes), and metapopulations, migrations (coupled systems).\line {\i System}
: PC/Windows 3.X\line {\i Available}: from {\cf5 http://www.snv.jussieu.fr}\line \par
\tab {\b Virtual Laboratory\line }Simulations of 2D active media by the Complex Systems Group at the Max Planck Inst. in Berlin.{\b \line }{\i System}: Requires PV-Wave by Visual Numerics $${\cf5 http://www.vni.com/products/wave/}\line {\i Available}: $$
{\cf5 http://w3.rz-berlin.mpg.de/~mik/oertzen/vlm/m_contents.htm}\line \par
{\b \tab VRA (Visual Recurrence Analysis)\line }
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 ti
me series under study. By Eugene Kononov\line {\i Stystem}: Windows 95\line {\i Available}: {\cf5 http://pweb.netcom.com/~eugenek/download.html}{\i \line }\par
{\b \tab Xphased\line }Phase 3D plane program for X-windows systems (for systems like Lorenz, Rossler). Plot, rotate in 3-d, Poincar\'8e sections, etc. By Thomas P. Witelski\line {\i System}: X-windows, Unix, SunOS 4 binary\line {\i Available}: {\cf5
http://www.alumni.caltech.edu/~witelski/xphased.html}\line \par
{\b \tab XPP-Aut\line }Differential equations and maps for x-windows systems. Links to Auto for bifurcation analysis. By Bard Ermentrout\line {\i System}: X-windows, Binaries for many unix systems\line {\i Available} : {\cf5
ftp://ftp.math.pitt.edu/pub/bardware/tut/start.html}\line \par
{\b \tab XSpiral\line }Simulate pattern formation in 2-D excitable media (in particular 2 models, one of them the FitzHugh-Nagumo). By Flavio Fenton.\line {\i System}: X-windows\line {\i Available} : (Missing??)\line \par
\pard\plain \f20 \par
\pard\plain \s10\brdrb\brdrs \f20 back to {\v\uldb faq-Contents.html}{\uldb table of contents}\par
\pard\plain \s255\sb240\tx540 \b\f21 \page [6] Acknowledgments\par
\pard\plain \f20 \par
\pard\plain \s5\tx180 \f20 \tab Alan Champneys {\cf5 a.r.champneys@bristol.ac.uk}\par
\tab Jim Crutchfield {\cf5 chaos@gojira.Berkeley.EDU}\par
\tab S. H. Doole {\cf5 Stuart.Doole@Bristol.ac.uk}\par
\tab David Elliot {\cf5 delliott@isr.umd.edu}\par
\tab Fred Klingener {\cf5 klingener@BrockEng.com}\par
\tab Matt Kennel {\cf5 kennel@msr.epm.ornl.gov}\par
\tab Jose Korneluk {\cf5 jose.korneluk@sfwmd.gov}\par
\tab Wayne Hayes {\cf5 wayne@cs.toronto.edu}\par
\tab Justin Lipton {\cf5 JML@basil.eng.monash.edu.au}\par
\tab Ronnie Mainieri {\cf5 ronnie@cnls.lanl.gov}\par
\tab Zhen Mei {\cf5 meizhen@mathematik.uni-marburg.de}\par
\tab Gerard Middleton {\cf5 middleto@mcmail.CIS.McMaster.CA}\par
\tab Andy de Paoli {\cf5 andrea.depaoli@mail.esrin.esa.it}\par
\tab Lou Pecora {\cf5 pecora@zoltar.nrl.navy.mil}\par
\tab Pavel Pokorny {\cf5 pokornp@tiger.vscht.cz},\par
\tab Leon Poon {\cf5 lpoon@Glue.umd.edu}\par
\tab Hawley Rising {\cf5 rising@crl.com},\par
\tab Michael Rosenstein {\cf5 MTR1a@aol.com}\par
\tab Harold Ruhl {\cf5 hjr@connix.com}\par
\tab Troy Shinbrot {\cf5 shinbrot@bart.chem-eng.nwu.edu}\par
\tab Viorel Stancu {\cf5 vstancu@sb.tuiasi.ro}\par
\tab Jaroslav Stark {\cf5 j.stark@ucl.ac.uk}\par
\tab Bruce Stewart {\cf5 bstewart@bnlux1.bnl.gov}\par
\tab Richard Tasgal {\cf5 tasgal@math.tau.ac.il}\par
\pard\plain \f20 \par
Anyone else who would like to contribute, please do! Send me your comments:{\v\uldb http://amath.colorado.edu/appm/faculty/jdm/ }{\uldb Jim Meiss} at {\cf5 jdm@boulder.colorado.edu}\par
}