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Version 1.0.4 (December 1995) of the Nonlinear Science Frequently
Asked Questions Document. This FAQ is posted monthly to the
sci.nonlinear newsgroup.
This FAQ is maintained by Jim Meiss {\tt jdm@boulder.colorado.edu}.
\begin{quote}
Copyright \copyright 1995 by James D. Meiss,
all rights reserved. This FAQ may be
posted to any USENET newsgroup, on-line service, or BBS as long as it is
posted in its entirety and includes this copyright statement. This FAQ may not
be distributed for financial gain. This FAQ may not be included in commercial
collections or compilations without express permission from the author.
\end{quote}
\noindent Table of Contents
\begin{enumerate}
\item What is nonlinear?
\item What is nonlinear science?
\item What is a dynamical system?
\item What is phase space?
\item What is a degree of freedom?
\item What is a map?
\item How are maps related to flows (differential equations)?
\item What is chaos?
\item What is sensitive dependence on initial conditions?
\item What are Lyapunov exponents?
\item What is Generic?
\item What is the minimum phase space dimension for chaos?
\item What are complex systems?
\item What are fractals?
\item What do fractals have to do with chaos?
\item What are topological and fractal dimension?
\item What is quantum chaos?
\item How do I know if my data is deterministic?
\item What is the control of chaos?
\item How can I build a chaotic circuit?
\item What are simple experiments that I can do to demonstrate chaos?
\item What is targeting?
\item What is time series analysis?
\item Is there chaos in the stock market?
\item What are solitons?
\item What should I read to learn more?
\item What technical journals have nonlinear science articles?
\item What are net sites for nonlinear science materials?
\item What nonlinear science software is available?
\item Acknowledgments
\end{enumerate}
\section{What is nonlinear?}
In geometry, linearity refers to Euclidean objects: lines, planes, (flat)
three dimensional space, etc.--these objects appear the same no matter how we
examine them. A nonlinear object, a sphere for example, looks different on
different scales--when looked at closely enough it looks like a plane, and
from a far enough distance it looks like a point.
In algebra, we define linearity in terms of functions which have the property
$f(x+y) = f(x)+f(y)$ and $f(ax) = af(x)$. Nonlinear is defined as the negation of
linear. This means that the result f may be out of proportion to the input $x$
or y. The result may be more than linear, as when a diode begins to pass
current; or less than linear, as when finite resources limit Malthusian
population growth. Thus the fundamental simplifying tools of linear analysis
are no longer available: for example, for a linear system, if we have two
zeros, $f(x) = 0$ and $f(y) = 0$, then we automatically have a third zero
$f(x+y) = 0$
(in fact there are infinitely many zeros as well, since linearity implies
that $f(ax+by) = 0$ for any $a$ and $b$). This is called the principle of
superposition--it gives many solutions from a few. For nonlinear systems, each
solution much be fought for (generally) with unvarying ardor!
\section{What is nonlinear science?}
Stanislaw Ulam reportedly said (something like) ``Calling a science
`{\it nonlinear}' is like calling zoology
`{\it the study of non-human animals}' ''. So why
do we have a name that appears to be merely a negative?
Firstly, linearity is rather special, and no model of a real system is truly
linear (you might protest that quantum mechanics is an exception, however this
is at the expense of infinite dimensionality which is just as bad or worse--
and ``any'' finite dimensional nonlinear model can be turned into an infinite
dimensional linear one). Some things are profitably studied as linear
approximations to the real models--for example the fact that Hooke's law, the
linear law of elasticity (strain is proportional to stress) is approximately
valid for a pendulum of small amplitude implies that its period is
approximately independent of amplitude (i.e.\ Period(Amplitude) =
Period($2\times$Amplitude)). However, as the amplitude gets large the period gets
longer, a fundamental effect of nonlinearity in the pendulum equations.
Secondly, nonlinear systems have been shown to exhibit surprising and complex
effects that would never be anticipated by a scientist trained only in linear
techniques. Prominent examples of these include bifurcation, chaos and
solitons. Nonlinearity has its most profound effects on dynamical systems
({\it see question 3}).
Further, while we can enumerate the linear objects, nonlinear ones are
nondenumerable, and as of yet mostly unclassified. We currently have no
general techniques (and very few special ones) for telling whether a
particular nonlinear system will exhibit the complexity of chaos, or the
simplicity of order. Thus since we cannot yet subdivide nonlinear science into
proper subfields, it exists as a whole.
Nonlinear science has applications to a wide variety of fields, from
mathematics, physics, biology, and chemistry, to engineering, economics, and
medicine. This is one of its most exciting aspects--that it brings researchers
from many disciplines together with a common language.
\section{What is a dynamical system?}
A dynamical system consists of an abstract phase space or state space, whose
coordinates describe the dynamical state at any instant; and a dynamical rule
which specifies the immediate future trend of all state variables, given only
the present values of those same state variables. Mathematically, a dynamical
system is described by an initial value problem.
Dynamical systems are ``deterministic'' if there is a unique consequent to every
state, and ``stochastic'' or ``random'' if there is more than one consequent
chosen from some probability distribution (the coin toss has two consequents
with equal probability for each initial state). Most of nonlinear science--and
everything in this FAQ--deals with deterministic systems.
A dynamical system can have discrete or continuous time. The discrete case is
defined by a map, $z_1 = f(z_0)$, that gives the state $z_1$
resulting from the
initial state $z_0$ at the next time value. The continuous case is defined by a
``flow'', $z(t) = \phi_t(z_0)$, which gives the state at time $t$,
given that the state was $z_0$ at time $0$.
A smooth flow can be differentiated w.r.t. time to
give a differential equation, $dz/dt = F(z)$. In this case we call $F(z)$ a
``vector field'', it gives a vector pointing in the direction of the velocity at
every point in phase space.
\section{What is phase space?}
Phase space is the collection of possible states of a dynamical system. A
phase space can be finite (e.g. for the coin toss, we have two states heads
and tails), countably infinite (e.g. state variables are integers), or
uncountably infinite (e.g. state variables are real numbers). Implicit in the
notion is that a particular state in phase space specifies the system
completely; it is all we need to know about the system to have complete
knowledge of the immediate future. Thus the phase space of the planar pendulum
is two dimensional, consisting of the position (angle) and velocity. According
to Newton, specification of these two variables uniquely determines the
subsequent motion of the pendulum.
Note that if we have a non-autonomous system, where the map or vector field
depends explicitly on time (e.g. a model for plant growth depending on solar
flux), then according to our definition of phase space, we must include time
as a phase space coordinate--since one must specify a specific time (e.g.
3 P.M.
on Tuesday) to know the subsequent motion. Thus $dz/dt = F(z,t)$ is a
dynamical system on the phase space consisting of $(z,t)$, with the addition
the new dynamical equation $dt/dt = 1$.
The path in phase space traced out by a solution of an initial value problem
is called an orbit or trajectory of the dynamical system. If the state
variables take real values in a continuum, the orbit of a continuous-time
system is a curve, while the orbit of a discrete-time system is a sequence of
points.
\section{What is a degree of freedom?}
The notion of ``degrees of freedom'' as it is used for Hamiltonian systems means
one canonical conjugate pair, a configuration, q, and its conjugate momentum
p. Hamiltonian systems (sometimes mistakenly identified with the notion of
conservative systems) always have such pairs of variables, and so the phase
space is even dimensional.
In the study of dissipative systems the term ``degree of freedom'' is often used
differently, to mean a single coordinate dimension of the phase space. This
can lead to confusion, and it is advisable the check which meaning of the term
is intended in a particular context.
Those with a physics background generally prefer to stick with the Hamiltonian
definition of the term ``degree of freedom''. For a more general system the
proper term is ``order'' which is equal to the dimension of the phase space.
Note that a Hamiltonian $H(q,p)$ with $N$ d.o.f. nominally moves in a $2N$
dimensional phase space. However, energy is conserved, and therefore the
motion is really on a $2N-1$ dimensional energy surface, $H(q,p) = E$.
Thus e.g.
the planar, circular restricted 3 body problem is 2 d.o.f., and motion is on
the 3D energy surface of constant ``Jacobi constant''. It can be reduced to a
2D area preserving map by Poincare section ({\it see question 6}).
If the Hamiltonian is time dependent, then we generally say it has an
additional $1/2$ degree of freedom, since this adds one dimension to the phase
space. (i.e. $1$ and $1/2$ d.o.f. means three variables, $q$,$p$ and $t$,
and energy is no longer conserved).
\section{What is a map?}
A map is simply a function, $f$, on the phase space that gives the next state,
$f(z)$, (the image) of the system given its current state, $z$.
(Often you will
find the notation $z' = f(z)$, where the prime means the next point, not the
derivative.)
Now a function must have a single value for each state, but there could be
several different states that give rise to the same image. Maps that allow
every state in the phase space to be accessed (onto) and which have precisely
one pre-image for each state (one-to-one) are invertible. If in addition the
map and its inverse are continuous (with respect to the phase space coordinate
$z$), then it is called a homeomorphism. A homeomorphism that has at least one
continuous derivative (w.r.t. $z$) and a continuously differentiable inverse is
a diffeomorphism.
Iteration of a map means repeatedly applying the map to the consequents of the
previous application. Thus we get a sequence
\[
z = f(z_n) = f(f(z_{n-1}) \ldots = f^n (z_0 )
\]
This sequence is the orbit or trajectory of the dynamical system with initial
condition $z_0$.
\section{How are maps related to flows (differential equations)?}
Every differential equation gives rise to a map, the time one map, defined by
advancing the flow one unit of time. This map may or may not be useful. If the
differential equation contains a term or terms periodic in time, then the time
$T$ map (where $T$ is the period) is very useful--it is an example of a Poincare
section. The time $T$ map in a system with periodic terms is also called a
stroboscopic map, since we are effectively looking at the location in phase
space with a stroboscope tuned to the period $T$. This map is useful because it
permits us to dispense with time as a phase space coordinate: the remaining
coordinates describe the state completely so long as we agree to consider the
same instant within every period.
In autonomous systems (no time-dependent terms in the equations), it may also
be possible to define a Poincare section and again reduce the phase space
dimension by one. Here the Poincare section is defined not by a fixed time
interval, but by successive times when an orbit crosses a fixed surface in
phase space. (Surface here means a manifold of dimension one less than the
phase space dimension).
However, not every flow has a global Poincare section (e.g. any flow with an
equilibrium point), which would need to be transverse to every possible orbit.
Maps arising from stroboscopic sampling or Poincare section of a flow are
necessarily invertible, because the flow has a unique solution through any
point in phase space--the solution is unique both forward and backward in
time. However, noninvertible maps can be relevant to differential equations:
Poincare maps are sometimes very well approximated by noninvertible maps. For
example, the Henon map $
(x,y) \rightarrow (-y-a+x^2,bx)$ with small $\left|b \right|$ is close to the
logistic map, $x \rightarrow -a+x^2$.
It is often (though not always) possible to go backwards, from an invertible
map to a differential equation having the map as its Poincare map. This is
called a suspension of the map. One can also do this procedure approximately
for maps that are close to the identity, giving a flow that approximates the
map to some order. This is extremely useful in bifurcation theory.
Note that any numerical solution procedure for a differential initial value
problem which uses discrete time steps in the approximation is effectively a
map. This is not a trivial observation; it helps explain for example why a
continuous-time system which should not exhibit chaos may have numerical
solutions which do--{\it see question 12}.
\section{What is chaos?}
Roughly speaking, chaos is effectively unpredictable long time behavior
arising in a deterministic dynamical system because of sensitivity to initial
conditions. It must be emphasized that a deterministic dynamical system is
perfectly predictable given perfect knowledge of the initial condition, and
further is in practice always predictable in the short term. The key to long-
term unpredictability is a property known as sensitivity to (or sensitive
dependence on) initial conditions.
For a dynamical system to be chaotic it must have a ``large'' set of initial
conditions which are highly unstable. No matter how precisely you measure the
initial condition in these systems, your prediction of its subsequent motion
goes radically wrong after a short time. Typically ({\it see
question 20} for one
definition of {\it typical}), the predictability horizon grows only
logarithmically with the precision of measurement (for positive Lyapunov
exponents, {\it see question 10}).
Thus for each increase in precision by a factor of
10, say, you may only be able to predict two more time units.
More precisely: A map $f$ is chaotic on a compact invariant set $S$ if:
\begin{itemize}
\item i) $f$ is transitive on $S$ (there is a point $x$ whose orbit
is dense in S), and
\item (ii) $f$ exhibits sensitive dependence on $S$ ({\it see question 9})
\end{itemize}
To these two requirements
Devaney adds the requirement that periodic points are dense in $S$, but this
doesn't seem to be really in the spirit of the notion, and is probably better
treated as a theorem (very difficult and very important), and not part of the
definition.
Usually we would like the set $S$ to be a large set. It is too much to hope for
except in special examples that $S$ be the entire phase space. If the dynamical
system is dissipative then we hope that S is an attractor with a large basin.
However, this need not be the case--we can have a chaotic saddle, an orbit
that has some unstable directions as well as stable directions.
As a consequence of long-term unpredictability, time series from chaotic
systems may appear irregular and disorderly. However, chaos is definitely not
(as the name might suggest) complete disorder; it is disorder in a
deterministic dynamical system, which is always predictable for short times.
The possibility of a predictability horizon in a deterministic system came as
something of a shock to mathematicians and physicists, long used to a notion
attributed to Laplace that, given precise knowledge of the initial conditions,
it should be possible to predict the future of the universe. This mistaken
faith in predictability was engendered by the success of Newton's mechanics
applied to planetary motions, which happen to be regular on human historic
time scales, but chaotic on the 5 million year time scale (see e.g.
``Newton's
Clock'', by Ivars Peterson (1993 W.H. Freeman) .
\section{What is sensitive dependence on initial conditions?}
Consider a boulder precariously perched on the top of an ideal hill. The
slightest push will cause the boulder to roll down one side of the hill or the
other: the subsequent behavior depends sensitively on the direction of the
push--and the push can be arbitrarily small. If you are standing at the bottom
of the hill on one side, then you would dearly like to know which direction
the boulder will fall.
Sensitive dependence is the equivalent behavior for every initial condition--
every point in the phase space is effectively perched on the top of a hill.
More precisely a set $S$ exhibits sensitive dependence if there is an r such
that for any $\epsilon > 0$ and for each $x \in S$, there is a $y$ such
that $|x - y| < \epsilon$, and $|x_n - y_n| > r$ for some $n > 0$.
That is there is a fixed
distance $r$ (say 1), such that no matter how precisely one specifies an initial
state there are nearby states that eventually get a distance r away.
Note: sensitive dependence does not require exponential growth of
perturbations (positive Lyapunov exponent), but this is typical ({\it
see question 20})
for chaotic systems. Note also that we most definitely do not require ALL
nearby initial points diverge--generically ({\it see question 20})
this does not happen--some
nearby points may converge. (We may modify our hilltop analogy slightly and
say the every point in phase space acts like a high mountain pass.) Finally,
the words ``initial conditions'' are a bit misleading: a typical small
disturbance introduced at any time will grow similarly. Think of ``initial''
as
meaning ``a time when a disturbance or error is introduced'', not necessarily
time zero.
\section{What are Lyapunov exponents?}
The hardest thing to get right about Lyapunov exponents is the spelling of
Lyapunov, which you will variously find as Liapunov, Lyapunof and even
Liapunoff. Of course Lyapunov is really spelled in the Cyrillic alphabet.
Now that there is an ANSI standard of
transliteration for Cyrillic, we expect all references to converge on the
version Lyapunov.
Lyapunov was born in Russia in 6 June 1857. He was greatly influenced by
Chebyshev and was a student with Markov. He was also a passionate man:
Lyapunov shot himself the day his wife died. He died 3 Nov. 1918, three
days later. According to the request on a note he left, Lyapunov was
buried with his wife. (biographical data from a biography by A.
T.Grigorian).
Lyapunov left us with more than just a simple note. He left a collection of
papers on the equilibrium shape of rotating liquids, on probability, and on
the stability of low-dimensional dynamical systems. It was from his
dissertation that the notion of Lyapunov exponent emerged. Lyapunov was
interested in showing how to discover if a solution to a dynamical system is
stable or not for all times. The usual method of studying stability ---
linearizing around the solution --- was not good enough. If you waited long
enough the small errors due to linearization would pile up and make the
approximation invalid. Lyapunov developed concepts to overcome these
difficulties.
Lyapunov exponents measure the rate of divergence of nearby orbits. Roughly
speaking the (maximal) Lyapunov exponent is the time average logarithmic
growth rate of the distance between two nearby orbits. Positive Lyapunov
exponents indicate sensitive dependence on initial conditions, since the
distance then grows (on average in time and locally in phase space)
exponentially in time.
There are basically two ways to compute Lyapunov exponents. In one way one
chooses two nearby points, evolves them in time, measuring the growth rate of
the distance between them. This is useful when one has a time series, but has
the disadvantage that the growth rate is really not a local effect as the
points separate. A better way is to measure the growth rate of tangent vectors
to a given orbit.
More precisely, consider a map $f$ in an $m$ dimensional phase space, and its
derivative matrix $Df(x)$. Let $v$ be a tangent vector at the point $x$.
Then we define a function
\[
L(x,v) = \lim_{n \rightarrow \infty} \frac{1}{n} \ln
\left| \left( D f^n ( x) v \right) \right|
\]
Now the Multiplicative Ergodic Theorem of Oseledec states that this limit
exists for almost all points $x$ and all tangent vectors $v$. There are at most m
distinct values of $L$ as we let v range over the tangent space. These are the
Lyapunov exponents at $x$.
For more information on computing the exponents see
\begin{itemize}
\item Wolf, A., J. B. Swift, et al. (1985). ``Determining Lyapunov Exponents from
a Time Series.'' Physica D 16: 285-317.
\item Eckmann, J.-P., S. O. Kamphorst, et al. (1986). ``Liapunov exponents from
time series.'' Phys. Rev. A 34: 4971-4979.
\end{itemize}
\section{What is Generic?}
Generic in dynamical systems is intended to convey ``usual'' or, more properly,
``observable''. Roughly speaking, a property is generic over a class if any
system in the class can be modified ever so slightly (perturbed), into one
with that property.
The formal definition is done in the language of topology: Consider the class
to be a space of systems, and suppose it has a topology (some notion of a
neighborhood, or an open set). A subset of this space is {\it dense} if its
closure (the subset plus the limits of all sequences in the subset) is the
whole space. It is {\it open and dense} if it is also an open set (union of
neighborhoods). A set is {\it countable}
if it can be put into 1-1 correspondence
with the counting numbers. A {\it 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
{\it Baire }
space, which basically means its big enough. If we have such a Baire space of
dynamical systems, and there is a property which is true on a countable
intersection of open dense sets, them that property is {\it generic}.
If all this sounds too complicated, think of it as a precise way of defining a
set which is near every system in the collection (dense), which isn't too big
(needn't have any ``regions'' where the property is true for {\it every} system).
Generic is much weaker than ``almost everywhere'' (occurs with probability 1),
in fact, it is possible to have generic properties which occur with
probability zero. But it is as strong a property as one can define
topologically, without having to have a property hold true in a region, or
talking about measure (probability), which isn't a topological property (a
property preserved by a continuous function).
\section{What is the minimum phase space dimension for chaos?}
This is a slightly confusing topic, since the answer depends on the type of
system considered. First consider a flow (or system of differential
equations). In this case the Poincare-Bendixson theorem tells us that there is
no chaos in one or two dimensional phase spaces. Chaos is possible in three
dimensional flows--standard examples such as the Lorenz equations are indeed
three dimensional, and there are mathematical 3D flows that are provably
chaotic (e.g. the {\it solenoid}).
Note: if the flow is non-autonomous then time is a phase space coordinate, so
a system with two physical variables + time becomes three dimensional, and
chaos is possible (i.e. Forced second-order oscillators do exhibit chaos.)
For maps, it is possible to have chaos in one dimension, but only if the map
is not invertible. A prominent example is the Logistic map $x' = f(x) = rx(1-
x)$. This is provably chaotic for $r = 4$, and many other values of $r$ as well
(see e.g. Devaney). Note that every point has two preimages, except for the
image of the critical point $x=1/2$, so this map is not invertible.
For homeomorphisms, we must have at least two dimensional phase space for
chaos. This is equivalent to the flow result, since a three dimensional flow
gives rise to a two dimensional homeomorphism by Poincare section ({\it
see question 6}).
Note that a numerical algorithm for a differential equation is a map, because
time on the computer is necessarily discrete. Thus numerical solutions of two
and even one dimensional systems of ordinary differential equations may
exhibit chaos. Usually this results from choosing the size of the time step
too large. For example Euler discretization of the Logistic differential
equation, $dx/dt = rx(1-x)$, is equivalent to the logistic map. See e.g.
S.~Ushiki, {\it Central difference scheme and chaos},
Physica D, vol. 4 (1982) 407-424.
\section{What are complex systems?}
A complex system, as I understand it, is a system with many inequivalent
degrees of freedom. While, chaos is the study of how simple systems can
generate complicated behavior, complexity is the study of how complicated
systems can generate simple behavior. An example of complexity is the
synchronization of biological systems ranging from fireflies to neurons (e.g.
Matthews, PC, Mirollo, RE \& Strogatz, SH ``Dynamics of a large system of
coupled nonlinear oscillators,'' {\it Physica D} {\it 52} (1991) 293-331).
In these
problems, many individual systems conspire to produce a single collective
rhythm.
The notion of complex systems has received lots of popular press, but it is
not really clear as of yet if there is a ``theory''
about a ``concept''. We are
withholding judgement.
\section{What are fractals?}
One way to define ``fractal'' is as a negation: a fractal
is a set that does not
look like a Euclidean object (point, line, plane, etc.) no matter how closely
you look at it. Imagine focusing in on a smooth curve (imagine a piece of
string in space)--if you look at any piece of it closely enough it eventually
looks like a straight line (ignoring the fact that for a real piece of string
it will soon look like a cylinder and eventually you will see the fibers, then
the atoms, etc.). A fractal, like the Koch Snowflake, which is topologically
one dimensional, never looks like a straight line, no matter how closely you
look. There are indentations, like bays in a coastline; look closer and the
bays have inlets, closer still the inlets have subinlets, and so on.
``Fractal'' is a term which has undergone refinement of definition by a lot of
people, but was first coined by B. Mandelbrot and defined as a set with
fractional (non-integer) dimension (Hausdorff dimension,
{\it see question 16}). While
this definition has a lot of drawbacks, note that it says nothing about self-
similarity--even though the most commonly known fractals are indeed self-
similar.
See the extensive FAQ from sci.fractals at
\begin{verbatim}
ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq
http://www.cis.ohio-state.edu/hypertext/faq/usenet/fractal-faq/faq.html
\end{verbatim}
\section{What do fractals have to do with chaos?}
Often chaotic dynamical systems exhibit fractal structures in phase space.
However, there is no direct relation. There are chaotic systems that have
nonfractal limit sets (e.g. Arnold's cat map) and fractal structures that can
arise in nonchaotic dynamics (see e.g. Grebogi, C., et al. (1984). ``Strange
Attractors that are not Chaotic.'' Physica 13D: 261-268.)
\section{What are topological and fractal dimension?}
See the fractal FAQ:
\begin{verbatim}
ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq
http://www.cis.ohio-state.edu/hypertext/faq/usenet/fractal-faq/faq.html
\end{verbatim}
\section{What is quantum chaos?}
According to the correspondence principle, there is a limit where classical
behavior as described by Hamilton's equations becomes similar, in some
suitable sense, to quantum behavior as described by the appropriate wave
equation. Formally, one can take this limit to be $h \rightarrow 0$,
where $h$ is Planck's
constant; alternatively, one can look at successively higher energy levels,
etc. Such limits are referred to as ``semiclassical''. It has been found that
the semiclassical limit can be highly nontrivial when the classical problem is
chaotic. The study of how quantum systems, whose classical counterparts are
chaotic, behave in the semiclassical limit has been called quantum chaos. More
generally, these considerations also apply to elliptic partial differential
equations that are physically unrelated to quantum considerations. For
example, the same questions arise in relating classical acoustic waves to
their corresponding ray equations. Among recent results in quantum chaos is a
prediction relating the chaos in the classical problem to the statistics of
energy-level spacings in the semiclassical quantum regime.
Classical chaos can be used to analyze such ostensibly quantum systems as the
hydrogen atom, where classical predictions of microwave ionization thresholds
agree with experiments. See Koch, P. M. and K. A. H. van Leeuwen (1995).
``Importance of Resonances in Microwave Ionization of Excited Hydrogen Atoms.''
Physics Reports 255: 289-403.
See the Quantum Chaos Home Page:
{\tt http://sagar.cas.neu.edu/qchaos/qc.html}
\section{How do I know if my data is deterministic?}
How can I tell if my data is deterministic? This is a very tricky problem. It
is difficult because in practice no time series
consists of pure ``{\it signal}''.
There will always be some form of corrupting noise, even if it is present as
roundoff or truncation error or as a result of finite arithmetic or
quantization. Thus any real time series, even if mostly deterministic, will be
a stochastic processes
All methods for distinguishing deterministic and stochastic processes rely on
the fact that a deterministic system will always evolve in the same way from a
given starting point. Thus given a time series that we are testing for
determinism we (1) pick a test state (2) search the time series for a similar
or ``nearby'' state and (3) compare their respective time evolution.
Define the error as the difference between the time evolution of the ``test''
state and the time evolution of the nearby state. A deterministic system will
have an error that either remains small (stable, regular solution) or increase
exponentially with time (chaotic solution). A stochastic system will have a
randomly distributed error.
Essentially all measures of determinism taken from time series rely upon
finding the closest states to a given ``test'' state (i.e., correlation
dimension, Lyapunov exponents, etc.). To define the state of s system one
typically relies on phase space embedding methods, {\it see question
23}.
Typically one chooses an embedding dimension, and investigates the propagation
of the error between two nearby states. If the error looks random, one
increases the dimension. If you can increase the dimension to obtain a
deterministic looking error, then you are done. Though it may sound simple it
is not really! One complication is that as the dimension increases the search
for a nearby state requires a lot more computation time and a lot of data (the
amount of data required increases exponentially with embedding dimension) to
find a suitably close candidate. If the embedding dimension (number of
measures per state) is chosen too small (less than the ``true'' value)
deterministic data can appear to be random but in theory there is no problem
choosing the dimension too large--the method will work. Practically, anything
approaching about 10 dimensions is considered so large that a stochastic
description is probably more suitable and convenient anyway.
See e.g.,
Sugihara, G. and R. M. May (1990). ``Nonlinear Forcasting as a Way of
Distinguishing Chaos from Measurement Error in Time Series.'' Nature 344:
734-740.
\section{What is the control of chaos?}
Control of chaos has come to mean the two things: (1) stabilization of
unstable periodic orbits, (2) use of recurrence to allow stabilization to be
applied locally. Thus term ``control of chaos'' is somewhat of a misnomer--but
the name has stuck. The ideas for controlling chaos originated in the work of
Hubler followed by the Maryland Group.
Hubler, A. W. (1989). ``Adaptive Control of Chaotic Systems.'' Helv.
Phys. Acta 62: 343-346).
Ott, E., C. Grebogi, et al. (1990). ``Controlling Chaos.'' Physical Review
Letters 64(11): 1196-1199.
\noindent {\tt http://www-chaos.umd.edu/
publications/abstracts.html\#prl64.1196}
The idea that chaotic systems can in fact be controlled may be
counterintuitive -- after all they are unpredictable in the long term.
Nevertheless, numerous theorists have independently developed methods which
can be applied to chaotic systems, and many experimentalists have demonstrated
that physical chaotic systems respond well to both simple and sophisticated
control strategies. Applications have been proposed in such diverse areas of
research as communications, electronics, physiology, epidemiology, fluid
mechanics and chemistry.
The great bulk of this work has been restricted to low-dimensional systems;
more recently, a few researchers have proposed control techniques for
application to high- or infinite-dimensional systems. The literature on the
subject of the control of chaos is quite voluminous; nevertheless several
reviews of the literature are available, including:
Shinbrot, T. Ott, E., Grebogi, C. \& Yorke, J.A., ``Using Small Perturbations
to Control Chaos,'' Nature, 363 (1993) 411-7.
Shinbrot, T., ``Chaos: Unpredictable yet Controllable?'' Nonlin. Sciences
Today, 3:2 (1993) 1-8.
Shinbrot, T., ``Progress in the Control of Chaos,'' Advance in Physics (in
press).
Ditto, WL \& Pecora, LM ``Mastering Chaos,'' Scientific American (Aug. 1993),
78-84.
Chen, G. \& Dong, X, ``From Chaos to Order -- Perspectives and Methodologies
in Controlling Chaotic Nonlinear Dynamical Systems,'' Int. J. Bif. \& Chaos 3
(1993) 1363-1409.
It is generically quite difficult to control high dimensional systems; an
alternative approach is to use control to reduce the dimension before applying
one of the above techniques. This approach is in its infancy; see:
Auerbach, D., Ott, E., Grebogi, C., and Yorke, J.A. ``Controlling Chaos in
High Dimensional Systems,'' Phys. Rev. Lett. 69 (1992) 3479-82
\noindent {\tt http://www-chaos.umd.edu/
publications/abstracts.html\#prl69.3479}
\section{How can I build a chaotic circuit?}
There are many different physical systems which display chaos, dripping
faucets, water wheels, oscillating magnetic ribbons etc. but the most simple
systems which can be easily implemented are chaotic circuits. In fact an
electronic circuit was one of the first demonstrations of chaos which showed
that chaos is not just a mathematical abstraction. Leon Chua designed the
circuit 1983.
The circuit he designed, now known as Chua's circuit, consists of a piecewise
linear resistor as its nonlinearity (making analysis very easy) plus two
capacitors, one resistor and one inductor--the circuit is unforced
(autonomous). In fact the chaotic aspects (bifurcation values, Lyapunov
exponents, various dimensions etc.) of this circuit have been extensively
studied in the literature both experimentally and theoretically. It is
extremely easy to build and presents beautiful attractors (the most famous
known as the double scroll attractor) that can be displayed on a CRO.
For more information on building such a circuit try:
Kennedy M. P., ``Robust OP Amp Realization of Chua's Circuit'', Frequenz,
vol. 46, no. 3-4, 1992.
Madan, R. A., ``Chua's Circuit: A paradigm for chaos'', ed. R. A. Madan,
Singapore: World Scientific, 1993.
Pecora, L. and Carroll, T. ``Nonlinear Dynamics in Circuits'',
Singapore: World Scientific, 1995.
\section{What are simple experiments that I can do to demonstrate
chaos?}
There are many ``chaos toys'' on the market. Most consist of some sort of
pendulum that is forced by an electromagnet. One can of course build a simple
double pendulum to observe beautiful chaotic behavior see
\noindent {\tt http://www.ibm.com/
Stretch/EOS/chaos.html }.
My favorite double pendulum
consists of two identical planar pendula, so that you can demonstrate
sensitive dependence ({\it see question 9}).
One of the simplest chemical systems that shows chaos is the Belousov-
Zhabotinsky reacation.The book by Strogatz ({\it see
question 26}) has a good introduction to
this subject, see also
\begin{verbatim}
http://taylor.mc.duke.edu/~rubin/BZ/BZexplain.html
\end{verbatim}
for some more information.
The Chaotic waterwheel, while not so simple to build, is an exact realization
of Lorenz famous equaions. This is nicely discussed in Strogatz book Q26 as
well.
Chua's nonlinear curcuit is also a good example. ({\it see
question 20} above.)
\section{What is targeting?}
To direct trajectories in chaotic systems, one can generically apply small
perturbations; see:
Shinbrot, T. Ott, E., Grebogi, C. \& Yorke, J.A., ``Using Small
Perturbations to Control Chaos,'' Nature, 363 (1993) 411-7).
We are still awaiting a good answer to this question.
\section{What is time series analysis?}
This is the application of dynamical systems techniques to a data series,
usually obtained by ``measuring'' the value of a single observable as a function
of time. The major tool in a dynamicists toolkit is ``delay coordinate
embedding'' which creates a phase space portrait from a single data series. It
seems remarkable at first, but one can reconstruct a picture equivalent
(topologically) to the full Lorenz attractor in three dimensional space by
measuring only one of its coordinates, say $x(t)$, and plotting the delay
coordinates $(x(t), x(t+h), x(t+2h))$ for a fixed $h$.
It is important to emphasize that the idea of using derivatives or delay
coordinates in time series modeling is nothing new. It goes back at least to
the work of Yule, who in 1927 used an autoregressive (AR) model to make a
predictive model for the sunspot cycle. AR models are nothing more than delay
coordinates used with a linear model. Delays, derivatives, principal
components, and a variety of other methods of reconstruction have been widely
used in time series analysis since the early 50's, and are described in
several hundred books. The new aspects raised by dynamical systems theory are
(i) the implied geometric view of temporal behavior and (ii) the existence of
``geometric invariants'', such as dimension and Lyapunov exponents. The central
question was not whether delay coordinates are useful for time series
analysis, but rather whether reconstruction methods preserve the geometry and
the geometric invariants of dynamical systems. (Packard, Crutchfield, Farmer \&
Shaw)
G.U. Yule, Phil. Trans. R. Soc. London A 226 (1927) p. 267.
N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, ``Geometry
from a time series'', Phys. Rev. Lett. vol. 45, no. 9 (1980) 712.
F. Takens, ``Detecting strange attractors in fluid turbulence'', in:
Dynamical Systems and Turbulence, eds. D. Rand and L.-S. Young (Springer,
Berlin, 1981)
Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.Sh.T.
``The analysis of observed chaotic data in physical systems'', Rev. of
Modern Physics 65 (1993) 1331-1392.
D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics,
Springer-Verlag
\section{Is there chaos in the stock market?}
In order to address this question, we must first agree what we mean by chaos,
{\it see question 8}.
In dynamical systems theory, chaos means irregular fluctuations in a
deterministic system ({\it see question 3} and {\it question 18}).
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--{\it see
question 4}) is vast, and
there is no hope of exploiting the determinism. This is high-dimensional
chaos, which might just as well be truly random behavior. In this sense, the
stock market is chaotic, but who cares?
To be useful, economic chaos would have to involve some kind of collective
behavior which can be fully described by a small number of variables. In the
lingo, the system would have to be self-organizing, resulting in low-
dimensional chaos. If this turns out to be true, then you can exploit the low-
dimensional chaos to make short-term predictions. The problem is to identify
the state variable which characterize the collective modes. Furthermore,
having limited the number of state variables, many events now become external
to the system, that is, the system is operating in a changing environment,
which makes the problem of system identification very difficult.
If there were such collective modes of fluctuation, market players would
probably know about them; economic theory says that if many people recognized
these patterns, the actions they would take to exploit them would quickly
nullify the patterns. Therefore if these patterns exist, they must be hard to
recognize because they do not emerge clearly from the sea of noise caused by
individual actions; or the patterns last only a very short time following some
upset to the markets; or both.
There are a number of people and groups trying to find these patterns. Some of
these groups are known to outsiders, because they include prominent
researchers in the field of chaos; we have no idea whether they are succeeding
or not. If you know chaos theory and would like to make yourself a slave to
the rhythms of market trading, you can probably find a major trading firm
which will give you a chance to try your ideas. But don't expect them to give
you a share of any profits you may make for them {\tt :-)} !
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.
\section{What are solitons?}
Consider this frequently asked question: The Fourier transform can simplify
the evolution of linear differential equations; is there a counterpart which
similarly simplifies nonlinear equations? The answer is No. Nonlinear
equations are qualitatively more complex than linear equations, and a
procedure which gives the dynamics as simply as for linear equations must
contain a mistake. There are, however, exceptions to any rule.
Certain nonlinear differential equations can be fully solved by, e.g., the
``inverse scattering method''. Examples are the Korteweg-de Vries, nonlinear
Schrodinger, and sine-Gordon equations. In these cases the real space maps, in
a rather abstract way, to an inverse space, which is comprised of continuous
and discrete parts and evolves linearly in time. The continuous part typically
corresponds to radiation and the discrete parts to stable solitary waves, i.e.
pulses, which are called solitons. The linear evolution of the inverse space
means that solitons will emerge virtually unaffected from interactions with
anything, giving them great stability.
More broadly, there is a wide variety of systems which support stable solitary
waves through a balance of dispersion and nonlinearity. Though these systems
may not be integrable as above, in many cases they are close to systems which
are, and the solitary waves may share many of the stability properties of true
solitons, especially that of surviving interactions with other solitary waves
(mostly) unscathed. It is widely accepted to call these solitary waves
solitons, albeit with qualifications.
Why solitons? Solitons are simply a fundamental nonlinear wave phenomenon.
Many very basic linear systems with the addition of the simplest possible or
first order nonlinearity support solitons; this universality means that
solitons will arise in many important physical situations. Optical fibers can
support solitons, which because of their great stability are an ideal medium
for transmitting information. In a few years long distance telephone
communications will likely be carried via solitons.
The soliton literature is by now vast. Two books which contain clear
discussions of solitons as well as references to original papers are
Alan C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia,
Penn. (1985)
Mark J. Ablowitz, Solitons, nonlinear evolution equations and inverse
scattering, Cambridge (1991).
See the Soliton Home page:
{\tt http://www.ma.hw.ac.uk/
solitons/ }
\section{What should I read to learn more?}
Popularizations
\begin{enumerate}
\item Gleick, J. (1987). Chaos, the Making of a New Science. London,
Heinemann.
\item Stewart, I. (1989). Does God Play Dice? Cambridge, Blackwell.
\item Devaney, R. L. (1990). Chaos, Fractals, and Dynamics : Computer
Experiments in Mathematics. Menlo Park, Addison-Wesley Pub. Co.
\item Lorenz, E., (1994) The Essence of Chaos, University of Washington Press.
\end{enumerate}
\noindent Introductory Texts
\begin{enumerate}
\item Percival, I. C. and D. Richard (1982). Introduction to Dynamics.
Cambridge, Cambridge Univ. Press.
\noindent {\tt http://www.cup.org/
Titles/28/0521281490.html }
\item Devaney, R. L. (1986). An Introduction to Chaotic Dynamical Systems.
Menlo Park, Benjamin/Cummings.
\noindent {\tt http://www.aw.com/
he/Math/MathCategories/ABP/devaney13046.html }
\item Baker, G. L. and J. P. Gollub (1990). Chaotic Dynamics. Cambridge,
Cambridge Univ. Press.
\noindent {\tt http://www.cup.org/ Titles/38/052138897X.html }
\item Tufillaro, N., T. Abbott, et al. (1992). An Experimental Approach to
Nonlinear Dynamics and Chaos. Redwood City, Addison-Wesley.
\noindent {\tt http://www.aw.com/
he/Math/MathCategories/ABP/tufillaro55441.html }
\item Jurgens, H., H.-O. Peitgen, et al. (1993). Chaos and Fractals: New
Frontiers of Science. New York, Springer Verlag.
\noindent {\tt http://www.springer-ny.com }
\item Glendinning, P. (1994). Stability, Instability and Chaos. Cambridge,
Cambridge Univ Press.
\noindent {\tt http://www.cup.org/
Titles/415/0521415535.html }
\item Strogatz, S. (1994). Nonlinear Dynamics and Chaos. Reading, Addison-
Wesley.
\noindent {\tt http://www.aw.com/
he/Math/MathCategories/Chaos/strogatz54344.html }
\item Moon, F. C. (1992). Chaotic and Fractal Dynamics. New York, John Wiley.
\noindent {\tt gopher://gopher.infor.com:6000/
0exec\%3A-v\%20a\%20R9469895-9471436-/
.text/Main\%3A/.bin/aview }
\item Turcotte, Donald L. (1992). Fractals and Chaos in Geology and
Geophysics, Cambridge Univ. Press.
\noindent {\tt http://www.wiley.com }
\item Ott, Edward (1993). Chaos in Dynamical Systems. Cambridge,
Cambridge University Press.
\noindent {\tt http://www.cup.org/
Titles/43/0521432154.html }
\item D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics,
Springer-Verlag New York.
\noindent {\tt http://www.cnd.mcgill.ca/
Understanding/ }
\end{enumerate}
\noindent Introductory Articles
\begin{enumerate}
\item May, R. M. (1986). ``When Two and Two Do Not Make Four.'' Proc. Royal Soc.
B228: 241.
\item Berry, M. V. (1981). ``Regularity and Chaos in Classical Mechanics,
Illustrated by Three Deformations of a Circular Billiard.'' Eur. J.
Phys. 2: 91-102.
\item Crawford, J. D. (1991). ``Introduction to Bifurcation Theory.'' Reviews of
Modern Physics 63(4): 991-1038.
\item Shinbrot, T., C. Grebogi, et al. (1992). ``Chaos in a Double Pendulum.''
Am. J. Phys 60: 491-499.
\item David Ruelle. (1980). ``Strange Attractors,'' The Mathematical
Intelligencer 2: 126-37.
\end{enumerate}
\section{What technical journals have nonlinear science articles?}
\begin{itemize}
\item {\sf Physica D }
The premier journal in Nonlinear Dynamics
\item {\sf Nonlinearity}
Good mix, with a mathematical bias
\item {\sf Chaos}
AIP Journal, with a good physical bent
\item {\sf Physics Letters A}
Has a good nonlinear science section
\item {\sf Physical Review E}
Lots of Physics articles with nonlinear emphasis
\item {\sf Ergodic Theory and Dynamical Systems }
Rigorous mathematics, and careful work
\item {\sf J Differential Equations}
A premier journal, but very mathematical
\item {\sf J Dynamics and Diff. Eq.}
Good, more focused version of the above
\item {\sf J Dynamics and Stability of Systems}
Focused on Eng. applications. New editorial
board--stay tuned.
\item {\sf J Statistical Physics}
Used to contain seminal dynamical systems papers
\item {\sf SIAM Journals}
Only the odd dynamical systems paper
\item {\sf J Fluid Mechanics}
Some expt. papers, e.g. transition to turbulence
\item {\sf Nonlinear Dynamics}
Haven't read enough to form an opinion
\item {\sf J Nonlinear Science}
a newer journal--haven't read enough yet.
\item {\sf Nonlinear Science Today }
News of the week
see:
{\tt http://www.springer-ny.com/nst }
\item {\sf International J of Bifurcation and Chaos}
lots of color pictures, variable quality.
\item {\sf Chaos Solitons and Fractals}
Variable quality, some good applications
\item {\sf Communications in Math Phys}
an occasional paper on dynamics
\item {\sf Nonlinear Processes in Geophysics}
New, variable quality...may be improving
\end{itemize}
\section{What are net sites for nonlinear science materials?}
\begin{verbatim}
Bibliography
http://www.uni-mainz.de/FB/Physik/Chaos/chaosbib.html
ftp://ftp.uni-mainz.de/pub/chaos/chaosbib/
http://t13.lanl.gov/ronnie/cabinet.html
http://www-chaos.umd.edu/publications/references.html
http://www-chaos.umd.edu/~msanjuan/biblio.html
\end{verbatim}
\begin{verbatim}
Preprint Archives
http://cnls-www.lanl.gov/nbt/intro.html
Los Alamos Preprint Server
http://xyz.lanl.gov/
Nonlinear Science Eprint Server
http://www.ma.utexas.edu/mp_arc/mp_arc-home.html
Math-Physics Archive
http://e-math.ams.org/web/preprints/preprints-home.html
AMS Preprint
\end{verbatim}
\begin{verbatim}
Conference Announcements
http://t13.lanl.gov/~nxt/meet.html
http://www.nonlin.tu-muenchen.de/chaos/termine.html
http://xxx.lanl.gov/Announce/Conference/
http://www.math.psu.edu/weiss/conf.html
\end{verbatim}
\begin{verbatim}
Newsletters
gopher://gopher.siam.org:70/11/siag/ds
SIAM Dynamical Systems Group
http://www.amsta.leeds.ac.uk/Applied/news.dir/
UK Nonlinear News
\end{verbatim}
\begin{verbatim}
Electronic Journals
http://www.springer-ny.com/nst/
Nonlinear Science Today
http://www.santafe.edu/sfi/Complexity
The Complexity Journal
http://www.csu.edu.au/ci/ci.html
Complexity International Journal
\end{verbatim}
\begin{verbatim}
Electronic Texts
http://www.lib.rmit.edu.au/fractals/exploring.html
Exploring Chaos \& Fractals
http://www.nbi.dk/~predrag/QCcourse/
Cvitanovic's Lecture Notes
http://www.students.uiuc.edu/~ag-ho/chaos/chaos.html
Chaos Intro
\end{verbatim}
\begin{verbatim}
Institutes and Academic Programs
http://www.physics.mcgill.ca/physics-services/physics_complex.html
http://www.physics.mcgill.ca/physics-services/physics_complex2.html
Extensive List of Physics Groups in Nonlinear Phenonmena
http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/institutes.html
Extensive List of Nonlinear Groups
\end{verbatim}
\begin{verbatim}
Who is Who in Nonlinear Dynamics
http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/wiw.html
\end{verbatim}
\begin{verbatim}
Nonlinear Lists
http://cnls-www.lanl.gov/nbt/sites.html
Extensive List of Nonlinear
http://www.ar.com/ger/sci.nonlinear.html
URLs from Sci.nonlinear
http://www.industrialstreet.com/chaos/metalink.htm\#SCIENCE
Chaos URLs
\end{verbatim}
\begin{verbatim}
Time Series sites
http://cnls-www.lanl.gov/nbt/intro.html
Dynamics and Time Series
http://chuchi.df.uba.ar/series.html
time series
http://chuchi.df.uba.ar/tools/tools.html
ftp://ftp.cs.colorado.edu/pub/Time-Series/TSWelcome.html
Santa Fe Time Series Competition
\end{verbatim}
\begin{verbatim}
Chaos Sites
http://ucmp1.berkeley.edu/henon.html
Expt. henon attractor
http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html
All about Feigenbaum Constants
http://members.aol.com/MTRw3/w3/sw/sw00.html
Mike Rosenstein's Chaos Page.
http://www.prairienet.org/business/ptech/full/chaostry.html
Chaos Network
gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/lorenz.gif
Lorenz Attractor
\end{verbatim}
\begin{verbatim}
Complexity Sites
http://life.anu.edu.au/complex_systems/complex.html
Complex Sytems
http://www.cc.duth.gr/~mboudour/nonlin.html
Complexity Home Page
\end{verbatim}
\begin{verbatim}
Fractals Sites
ftp://spanky.triumf.ca/fractals/
The Spanky Fractal DataBase
http://sprott.physics.wisc.edu/fractals.htm
Sprott's Fractal Gallery
http://www-syntim.inria.fr/fractales/
Groupe Fractales
http://acacia.ens.fr:8080/home/massimin/quat/f_gal.ang.html
3D Fractals
http://www.cnam.fr/fractals.html
Fractal Gallery
http://homepage.seas.upenn.edu/~lau/fractal.html
http://homepage.seas.upenn.edu/~rajiyer/math480.html
Course on Fractal Geometry
\end{verbatim}
\section{What nonlinear science software is available?}
\noindent General Resources
``Guide to Available Mathematical Software'' maintained by NIST:
\noindent {\tt http://gams.cam.nist.gov/ }
``Mathematics Archives Software''
\noindent {\tt http://archives.math.utk.edu/
software.html }
\begin{itemize}
\item {\sf dstool}
Free software from Guckenheimer's group at Cornell; DSTool has lots of
examples of chaotic systems, Poincare' sections, bifurcation diagrams.
System: Unix, X windows.
Available by anonymous ftp:
\noindent {\tt ftp://macomb.tn.cornell.edu/
pub/dstool/ }
\item {\sf AUTO}
Bifurcation/Continuation Software (THE standard). AUTO94 with a GUI
requires X and Motif to be present. There is also a command line version
AUTO86 The softare is transported as a compressed, encoded file
called auto.tar.Z.uu. You should describe your UNIX server in the email.
System: versions to run under X windows--SUN or sgi
Available: send email to doedel@cs.concordia.ca
\item {\sf Chaos}
Visual simulation in two- and three-dimensional phase space; based on
visual algorithms rather than canned numerical algorithms; well-suited for
educational use; comes with tutorial exercises.
System: Silicon Graphics workstations,
IBM RISC workstations with GL
Available by anonymous ftp:
\noindent {\tt http://msg.das.bnl.gov/
~bstewart/software.html }
\item {\sf Xphased}
Phase Plane plotter for x-windows systems
System: X-windows, Unix, SunOS 4 binary
Available by anonymous ftp:
\noindent {\tt http://www.ama.caltech.edu/
~tpw/xphased.html }
\item {\sf StdMap}
Iterates Area Preserving Maps, by J. D. Meiss.
Iterates 8 different maps. It will find periodic orbits, cantori, stable
and unstable manifolds, and allows you to iterate curves.
System: Macintosh
Available by anonymous ftp:
\noindent {\tt ftp://amath.colorado.edu/
pub/dynamics/programs/ }
\item {\sf Lyapunov Exponents and Time Series}
Based on Alan Wolf's algorithm, {\it see question 10},
but a more efficient version.
System: Comes as C source, Fortran source, PC executable, etc
Available by anonymous ftp:
\noindent {\tt http://www.users.interport.net/
~wolf/ }
\item {\sf Lyapunov Exponents}
Keith Briggs Fortran codes for Lyapunov exponents
System: any with a Fortran compiler
Available by anonymous ftp:
\noindent {\tt http:www.pd.uwa.edu.au/
Keith/homepage.html }
\item {\sf MTRChaos}
{\sf MTRCHAOS} and {\sf MTRLYAP} compute correlation dimension
and largest Lyapunov exponents, delay portraits. By Mike Rosenstein.
System: PC-compatible computer running DOS 3.1 or higher,
640K RAM, and EGA display. VGA \& coprocessor recommended
Available by anonymous ftp:
\noindent {\tt ftp://spanky.triumf.ca/
pub/fractals/programs/ibmpc/ }
\item {\sf Chaos Plot}
ChaosPlot is a simple program which plots the chaotic behavior of a damped,
driven anharmonic oscillator.
System: Macintosh
Available from:
\noindent {\tt ftp://archives.math.utk.edu/
software/mac/diffEquations/ChaosPlot/ChaosPlot.sea.hqx }
\item {\sf MatLab Chaos}
A collection of routines from the Mathworks folks for generating diagrams
which illustrate chaotic behavior associated with the logistic equation.
System: Requires MatLab.
Available by anonymous ftp:
\noindent {\tt ftp://ftp.mathworks.com/
pub/contrib/misc/chaos/ }
\item {\sf SciLab}
A simulation program similar in intent to MatLab. It's primarily designed
for systems/signals work, and is large. From INRIA in France.
System: Unix, X Windows, 20 Meg Disk space.
Available by anonymous ftp:
\noindent {\tt ftp://ftp.inria.fr/
INRIA/Projects/Meta2/Scilab }
\item {\sf Cubic Oscillator Explorer}
The CUBIC OSCILLATOR EXPLORER is a Macintosh application which allows
interactive exploration of the chaotic processes of the Cubic Oscillator,
commonly known as Duffing's System.
System: Macintosh
Available from WWW FRACTAL MUSIC PROJECT at:
\noindent {\tt http://www-ks.rus.uni-stuttgart.de/
people/schulz/fmusic/ }
\item {\sf Dynamics: Numerical Explorations.}
Nusse, Helena E. and J.E. Yorke, 1994. book + diskette. A hands on approach
to learning the concepts and the many aspects in computing relevant
quantities in chaos
System: PC-compatible computer or X-windows system on Unix computers
Available: Springer-Verlag
\item {\sf PHASER}
Kocak, H., 1989. Differential and Difference Equations through Computer
Experiments: with a supplementary diskette comtaining PHASER: An
Animator/Simulator for Dynamical Systems emonstrates a large number of 1D-
4D differential equations--many not chaotic--and 1D-3D difference
equations.
System: PC-compatible computer + ???
Available: Springer-Verlag
\end{itemize}
The Academic Software Library:
\begin{itemize}
\item {\sf Chaos Simulations}
Bessoir, T., and A. Wolf, 1990. Demonstrates logistic map, Lyapunov
exponents, billiards in a stadium, sensitive dependence,
n-body gravitational motion.
Available: The Academic Software Library, (800) 955-TASL. \$ 70.
\item {\sf Chaos Data Analyser}
A PC program for analyzing time series. By Sprott, J.C. and G. Rowlands.
Available: The Academic Software Library, (800) 955-TASL. \$ 70.
For more information see:
\noindent {\tt http://sprott.physics.wisc.edu/cda.htm }
\item {\sf Chaos Demonstrations}
A PC program for demonstrating chaos, fractals, cellular automata,
and related nonlinear phenomena. By J. C. Sprott and G. Rowlands.
System: IBM PC or compatible with at least 512K of memory.
Available: The Academic Software Library, (800) 955-TASL. \$ 70.
\item {\sf Chaotic Dynamics Workbench}
Performs interactive numerical experiments on systems
modeled by ordinary differential equations, including: four versions of
driven Duffing oscillators, pendulum, Lorenz, driven Van der Pol osc.,
driven Brusselator, and the Henon-Heils system. By R. Rollins.
System: IBM PC or compatible, 512 KB memory.
Available: The Academic Software Library, (800) 955-TASL, \$ 70.
\item {\sf Chaos}
A Program Collection for the PC by Korsch, H.J. and H-J. Jodl, 1994,
A book/disk combo that gives a hands-on, computer experiment approach to
learning nonlinear dynamics. Some of the modules cover billiard systems,
double pendulum, Duffing oscillator, 1D iterative maps, an ``electronic
chaos-generator'', the Mandelbrot set, and ODEs.
System: IBM PC or compatible.
Available: Springer-Verlag
\item {\sf MacMath}
Comes on a disk with the book MacMath, by Hubbard and West. A
collection of programs for dynamical systems (1 \& 2 D maps, 1 to 3D flows).
Quality is uneven, and expected Macintosh features (color, resizeable
windows) are not always supported (in version 9.0).
System: Macintosh
See: {\tt http://archives.math.utk.edu/
cgibin/fife.test/mkTxtPage.pl?/
ftp/software/mac/calculus/MacMath/MacMath.abstract }
Available: Springer-Verlag
\item {\sf Tufillaro's Programs}
From the book Nonlinear Dynamics and Chaos by Tufillaro, Abbot and Reilly
(1992). A collection of programs for the Macintosh.
System: Macintosh
Available: Addison-Wesley
For more info see:
\noindent {\tt http://cnls-www.lanl.gov/nbt/qm.html }
\noindent {\tt http://cnls-www.lanl.gov/nbt/bb.html }
\item {\sf Applied Chaos Tools}
Software package for time series analysis based on the UCSD group's,
work. This package is a companion for Abarbanel's book ``Analysis of
Observed Chaotic Data'', Springer-Verlag.
System: Unix, and soon Windows 95
For more info see:
\noindent {\tt http://pm.znet.com/apchaos/csp.html }
\end{itemize}
\section{Acknowledgments}
Thanks to
\begin{itemize}
\item Hawley Rising {\tt mailto://rising@crl.com },
\item Bruce Stewart {\tt mailto://bstewart@bnlux1.bnl.gov }
\item Alan Champneys {\tt mailto://a.r.champneys@bristol.ac.uk }
\item Michael Rosenstein {\tt mailto://MTR1a@aol.com }
\item Troy Shinbrot {\tt mailto://shinbrot@bart.chem-eng.nwu.edu }
\item Matt Kennel {\tt mailto://kennel@msr.epm.ornl.gov }
\item Lou Pecora {\tt mailto://pecora@zoltar.nrl.navy.mil }
\item Richard Tasgal {\tt mailto://tasgal@math.tau.ac.il }
\item Wayne Hayes {\tt mailto://wayne@cs.toronto.edu }
\item S. H. Doole {\tt mailto://Stuart.Doole@Bristol.ac.uk }
\item Pavel Pokorny {\tt mailto://pokornp@tiger.vscht.cz },
\item Gerard Middleton {\tt mailto://middleto@mcmail.CIS.McMaster.CA }
\item Ronnie Mainieri {\tt mailto://ronnie@cnls.lanl.gov }
\item Leon Poon {\tt mailto://lpoon@Glue.umd.edu }
\item Justin Lipton {\tt mailto://JML@basil.eng.monash.edu.au }
\end{itemize}
Anyone else who would like to contribute, please do! Send me your comments:
Jim Meiss
{\tt mailto://jdm@boulder.colorado.edu }
\end{document}