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TUCoPS :: Cyber Culture :: nlscienc.txt

Nonlinear Frequently Asked Questions

Archive-name: sci/nonlinear-faq
Posting-Frequency: monthly

This is version 1.0.4  (December 1995) of the Frequently Asked Questions 
document for the newsgroup sci.nonlinear. This document can also be found in 
html format as:
and in Microsoft Word format as:
and in text form as:

What's New:

	Question [17] Added URL for quantum chaos
	Question [20] Added new book about nonlinear circuits
	Question [21] A new question: simple experimental demonstrations. 
	              Please send suggestions!
    Question [25] Soliton WWW site added.
    Question [28] Updates to bibliography sites, Electronic Texts, 
                  Conference Announcements
    Question [29] Addional Software sites added, 
	              and updated some outdated links

This FAQ is maintained by Jim Meiss <>.

Copyright (c) 1995 by James D. Meiss, all rights reserved. This FAQ may be 
posted to any USENET newsgroup, on-line service, or BBS as long as it is 
posted in its entirety and includes this copyright statement. This FAQ may not 
be distributed for financial gain. This FAQ may not be included in commercial 
collections or compilations without express permission from the author.

Table of Contents
[1]  What is nonlinear?
[2]  What is nonlinear science?
[3]  What is a dynamical system?
[4]  What is phase space?
[5]  What is a degree of freedom?
[6]  What is a map?
[7]  How are maps related to flows (differential equations)?
[8]  What is chaos?
[9]  What is sensitive dependence on initial conditions?
[10] What are Lyapunov exponents?
[11] What is Generic?
[12] What is the minimum phase space dimension for chaos?
[13] What are complex systems?
[14] What are fractals?
[15] What do fractals have to do with chaos?
[16] What are topological and fractal dimension?
[17] What is quantum chaos?
[18] How do I know if my data is deterministic?
[19] What is the control of chaos?
[20] How can I build a chaotic circuit?
[21] What are simple experiments that I can do to demonstrate chaos?
[22] What is targeting?
[23] What is time series analysis?
[24] Is there chaos in the stock market?
[25] What are solitons?
[26] What should I read to learn more?
[27] What technical journals have nonlinear science articles?
[28] What are net sites for nonlinear science materials?
[29] What nonlinear science software is available?
[30] Acknowledgments

[1] What is nonlinear?

In geometry, linearity refers to Euclidean objects: lines, planes, (flat) 
three dimensional space, etc.--these objects appear the same no matter how we 
examine them. A nonlinear object, a sphere for example, looks different on 
different scales--when looked at closely enough it looks like a plane, and 
from a far enough distance it looks like a point.

In algebra, we define linearity in terms of functions which have the property 
f(x+y) = f(x)+f(y) and f(ax) = af(x).  Nonlinear is defined as the negation of 
linear. This means that the result f may be out of proportion to the input x 
or y. The result may be more than linear, as when a diode begins to pass 
current; or less than linear, as when finite resources limit Malthusian 
population growth. Thus the fundamental simplifying tools of linear analysis 
are no longer available: for example, for a linear system, if we have two 
zeros,  f(x) = 0 and  f(y) = 0, then we automatically have a third zero f(x+y) 
= 0 (in fact there are infinitely many zeros as well, since linearity implies 
that  f(ax+by) = 0 for any a and b). This is called the principle of 
superposition--it gives many solutions from a few. For nonlinear systems, each 
solution much be fought for (generally) with unvarying ardor!

[2] What is nonlinear science?

Stanislaw Ulam reportedly said (something like)  "Calling a science 
'nonlinear' is like calling zoology 'the study of non-human animals'. So why 
do we have a name that appears to be merely a negative?

Firstly, linearity is rather special, and no model of a real system is truly 
linear (you might protest that quantum mechanics is an exception, however this 
is at the expense of infinite dimensionality which is just as bad or worse--
and 'any' finite dimensional nonlinear model can be turned into an infinite 
dimensional linear one). Some things are profitably studied as linear 
approximations to the real  models--for example the fact that Hooke's law, the 
linear law of elasticity (strain is proportional to stress)  is approximately 
valid for a pendulum of small amplitude implies that its period is 
approximately independent of amplitude (i.e. Period(Amplitude) = 
Period(2xAmplitude)). However, as the amplitude gets large the period gets 
longer, a fundamental effect of nonlinearity in the pendulum equations.

Secondly, nonlinear systems have been shown to exhibit surprising and complex 
effects that would never be anticipated by a scientist trained only in linear 
techniques. Prominent examples of these include bifurcation, chaos and 
solitons. Nonlinearity has its most profound effects on dynamical systems 
([Q3]see [Q3]).

Further, while we can enumerate the linear objects, nonlinear ones are 
nondenumerable, and as of yet mostly unclassified. We currently have no 
general techniques (and very few special ones) for telling whether a 
particular nonlinear system will exhibit the complexity of chaos, or the 
simplicity of order. Thus since we cannot yet subdivide nonlinear science into 
proper subfields, it exists has a whole.

Nonlinear science has applications to a wide variety of fields, from 
mathematics, physics, biology, and chemistry, to engineering, economics, and 
medicine. This is one of its most exciting aspects--that it brings researchers 
from many disciplines together with a common language.

[3]What is a dynamical system?

A dynamical system consists of an abstract phase space or state space, whose 
coordinates describe the dynamical state at any instant; and a dynamical rule 
which specifies the immediate future trend of all state variables, given only 
the present values of those same state variables. Mathematically, a dynamical 
system is described by an initial value problem.

Dynamical systems are "deterministic" if there is a unique consequent to every 
state, and "stochastic" or "random" if there is more than one consequent 
chosen from some probability distribution (the coin toss has two consequents 
with equal probability for each initial state). Most of nonlinear science--and 
everything in this FAQ--deals with deterministic systems.

A dynamical system can have discrete or continuous time. The discrete case is 
defined by a map, z_1 = f(z_0), that gives the state z_1 resulting from the 
initial state z_0 at the next time value. The continuous case is defined by a 
"flow",  z(t) = \phi_t(z_0), which gives the state at time t, given that the 
state was z_0 at time 0. A smooth flow can be differentiated w.r.t. time to 
give a differential equation, dz/dt = F(z). In this case we call F(z) a 
"vector field," it gives a vector pointing in the direction of the velocity at 
every point in phase space.

[4] What is phase space?

Phase space is the collection of possible states of a dynamical system. A 
phase space can be finite (e.g. for the coin toss, we have two states heads 
and tails), countably infinite (e.g. state variables are integers), or 
uncountably infinite (e.g. state variables are real numbers). Implicit in the 
notion is that a particular state in phase space specifies the system 
completely; it is all we need to know about the system to have complete 
knowledge of the immediate future. Thus the phase space of the planar pendulum 
is two dimensional, consisting of the position (angle) and velocity. According 
to Newton, specification of these two variables uniquely determines the 
subsequent motion of the pendulum.

Note that if we have a non-autonomous system, where the map or vector field 
depends explicitly on time (e.g. a model for plant growth depending on solar 
flux), then according to our definition of phase space, we must include time 
as a phase space coordinate--since one must specify a specific time (e.g. 3PM 
on Tuesday) to know the subsequent motion.  Thus  dz/dt = F(z,t) is a 
dynamical system on the phase space consisting of (z,t), with the  addition 
the new dynamical equation dt/dt = 1.

The path in phase space traced out by a solution of an initial value problem 
is called an orbit or trajectory of the dynamical system. If the state 
variables take real values in a continuum, the orbit of a continuous-time 
system is a curve, while the orbit of a discrete-time system is a sequence of 

[5] What is a degree of freedom?

The notion of "degrees of freedom" as it is used for Hamiltonian systems means 
one canonical conjugate pair, a configuration, q, and its conjugate momentum 
p. Hamiltonian systems (sometimes mistakenly identified with the notion of 
conservative systems) always have such pairs of variables, and so the phase 
space is even dimensional.

In the study of dissipative systems the term "degree of freedom" is often used 
differently, to mean a single coordinate dimension of the phase space. This 
can lead to confusion, and it is advisable the check which meaning of the term 
is intended in a particular context.

Those with a physics background generally prefer to stick with the Hamiltonian 
definition of the term "degree of freedom." For a more general system the 
proper term is "order" which is equal to the dimension of the phase space.

Note that a Hamiltonian H(q,p) with N d.o.f. nominally moves in a 2N 
dimensional phase space. However, energy is conserved, and therefore the 
motion is really on a 2N-1 dimensional energy surface, H(q,p) = E. Thus e.g. 
the planar, circular restricted 3 body problem is 2 d.o.f., and motion is on 
the 3D energy surface of constant  "Jacobi constant." It can be reduced to a 
2D area preserving map by Poincare section (see Q6]).

If the Hamiltonian is time dependent, then we generally say it has an 
additional 1/2 degree of freedom, since this adds one dimension to the phase 
space. (i.e. 1 1/2 d.o.f. means three variables, q,p and t, and energy is no 
longer conserved). 

[6] What is a map?

A map is simply a function, f,  on the phase space that gives the next state, 
f(z), (the image) of the system given its current state, z. (Often you will 
find the notation z' = f(z), where the prime  means the next point, not the 

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   )  = f(f(z   ).... = f (z )
             n      n-1          n-2            0

This sequence is the orbit or trajectory of the dynamical system with initial 
condition z_0.

[7] How are maps related to flows (differential equations)? 

Every differential equation gives rise to a map, the time one map, defined by 
advancing the flow one unit of time. This map may or may not be useful. If the 
differential equation contains a term or terms periodic in time, then the time 
T map (where T is the period) is very useful--it is an example of a Poincare 
section. The time T map in a system with periodic terms is also called a 
stroboscopic map, since we are effectively looking at the location in phase 
space with a stroboscope tuned to the period T. This map is useful because it 
permits us to dispense with time as a phase space coordinate: the remaining 
coordinates describe the state completely so long as we agree to consider the 
same instant within every period. 

In autonomous systems (no time-dependent terms in the equations), it may also 
be possible to define a Poincare section and again reduce the phase space 
dimension by one. Here the Poincare section is defined not by a fixed time 
interval, but by successive times when an orbit crosses a fixed surface in 
phase space. (Surface here means a manifold of dimension one less than the 
phase space dimension).

However, not every flow has a global Poincare section (e.g. any flow with an 
equilibrium point), which would need to be transverse to every possible orbit.

Maps arising from stroboscopic sampling or Poincare section of a flow are 
necessarily invertible, because the flow has a unique solution through any 
point in phase space--the solution is unique both forward and backward in 
time. However, noninvertible maps can be relevant to differential equations: 
Poincare maps are sometimes very well approximated by noninvertible maps. For 
example, the Henon map (x,y) -> (-y-a+x^2,bx) with small |b| is close to the 
logistic map, x -> -a+x^2.

It is often (though not always) possible to go backwards, from an invertible 
map to a differential equation having the map as its Poincare map. This is 
called a suspension of the map. One can also do this procedure approximately 
for maps that are close to the identity, giving a flow that approximates the 
map to some order. This is extremely useful in bifurcation theory.

Note that any numerical solution procedure for a differential initial value 
problem which uses discrete time steps in the approximation is effectively a 
map. This is not a trivial observation; it helps explain for example why a 
continuous-time system which should not exhibit chaos may have numerical 
solutions which do--[Q12]see [Q12].

[8] What is chaos?

Roughly speaking, chaos is effectively unpredictable long time behavior 
arising in a deterministic dynamical system because of sensitivity to initial 
conditions. It must be emphasized that a deterministic dynamical system is 
perfectly predictable given perfect knowledge of the initial condition, and 
further is in practice always predictable in the short term. The key to long-
term unpredictability is a property known as sensitivity to (or sensitive 
dependence on) initial conditions. 

For a dynamical system to be chaotic it must have a 'large' set of initial 
conditions which are highly unstable. No matter how precisely you measure the 
initial condition in these systems, your prediction of its subsequent motion 
goes radically wrong after a short time. Typically (see [Q20] for one 
definition of 'typical'), the predictability horizon grows only 
logarithmically with the precision of measurement (for positive Lyapunov 
exponents, see [Q10]). Thus for each increase in  precision by a  factor of 
10, say, you may only be able to predict two more time units.

More precisely: A map f is chaotic on a compact invariant set S if (i) f is 
transitive on S (there is a point x whose orbit is dense in S), and (ii) f 
exhibits sensitive dependence on S (see [Q9]). To these two requirements 
Devaney adds the requirement that periodic points are dense in S, but this 
doesn't seem to be really in the spirit of the notion, and is probably better 
treated as a theorem (very difficult and very important), and not part of the 

Usually we would like the set S to be a large set. It is too much to hope for 
except in special examples that S be the entire phase space. If the dynamical 
system is dissipative then we hope that S is an attractor with a large basin. 
However, this need not be the case--we can have a chaotic saddle, an orbit 
that has some unstable directions as well as stable directions.

As a consequence of long-term unpredictability, time series from chaotic 
systems may appear irregular and disorderly. However, chaos is definitely not 
(as the name might suggest) complete disorder; it is disorder in a 
deterministic dynamical system, which is always predictable for short times.

The possibility of a predictability horizon in a deterministic system came as 
something of a shock to mathematicians and physicists, long used to a notion 
attributed to Laplace that, given precise knowledge of the initial conditions, 
it should be possible to predict the future of the universe. This mistaken 
faith in predictability was engendered by the success of Newton's mechanics 
applied to planetary motions, which happen to be regular on human historic 
time scales, but chaotic on the 5 million year time scale (see e.g. "Newton's 
Clock", by Ivars Peterson (1993 W.H. Freeman) .
[9] What is sensitive dependence on initial conditions?

Consider a boulder precariously perched on the top of an ideal hill. The 
slightest push will cause the boulder to roll down one side of the hill or the 
other: the subsequent behavior depends sensitively on the direction of the 
push--and the push can be arbitrarily small. If you are standing at the bottom 
of the hill on one side, then you would dearly like to know which direction 
the boulder will fall.

Sensitive dependence is the equivalent behavior for every initial condition--
every point in the phase space is effectively perched on the top of a hill.

More precisely a set S exhibits sensitive dependence if there is an r such 
that for any epsilon > 0 and for each x in S, there is a y such that |x - y|  
<  epsilon,  and |x_n - y_n| > r for some n > 0. That is there is a fixed 
distance r (say 1), such that no matter how precisely one specifies an initial 
state there are nearby states that eventually get a distance r away. 

Note: sensitive dependence does not require exponential growth of 
perturbations (positive Lyapunov exponent), but this is typical (see Q[20]) 
for chaotic systems. Note also that we most definitely do not require ALL 
nearby initial points diverge--generically [Q20] this does not happen--some 
nearby points may converge. (We may modify our hilltop analogy slightly and 
say the every point in phase space acts like a high mountain pass.) Finally, 
the words "initial conditions" are a bit misleading: a typical small 
disturbance introduced at any time will grow similarly. Think of "initial" as 
meaning "a time when a disturbance or error is introduced," not necessarily 
time zero.

[10] What are Lyapunov exponents?

The hardest thing to get right about Lyapunov exponents is the spelling of 
Lyapunov, which you will variously find as Liapunov, Lyapunof and even 
Liapunoff.  Of course Lyapunov is really spelled in the Cyrillic  alphabet: 
(Lambda)(backwards r)(pi)(Y)(H)(0)(B). Now that there is an ANSI standard of 
transliteration for Cyrillic, we expect all references to converge on the 
version Lyapunov. 

Lyapunov was born in Russia in 6 June 1857. He was greatly influenced by 
Chebyshev and was a student with Markov. He was also a passionate man: 
Lyapunov shot himself the day his wife died. He died 3 Nov. 1918, three 
days later.  According to the request on a note he left, Lyapunov was 
buried with his wife. [biographical data from a biography by A.  

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 

Lyapunov exponents measure the rate of divergence of nearby orbits. Roughly 
speaking the (maximal) Lyapunov exponent is the time average logarithmic 
growth rate of the distance between two nearby orbits. Positive Lyapunov 
exponents indicate sensitive dependence on initial conditions, since the 
distance then grows (on average in time and locally in phase space) 
exponentially in time. 

There are basically two ways to compute Lyapunov exponents. In one way one 
chooses two nearby points, evolves them in time, measuring the growth rate of 
the distance between them. This is useful when one has a time series, but has 
the disadvantage that the growth rate is really not a local effect as the 
points separate. A better way is to measure the growth rate of tangent vectors 
to a given orbit.

More precisely, consider a map f in an m dimensional phase space, and its 
derivative matrix Df(x). Let v be a tangent vector at the point x. Then we 
define a function 
                              1          n 
        L(x,v)  =    lim     --- ln |( Df (x)v )|
                   n -> oo    n                     

Now the Multiplicative Ergodic Theorem of Oseledec states that this limit 
exists for almost all points x and all tangent vectors v. There are at most m 
distinct values of L as we let v range over the tangent space. These are the 
Lyapunov exponents at x. 

For more information on computing the exponents see

   Wolf, A., J. B. Swift, et al. (1985). "Determining Lyapunov Exponents from 
     a Time Series." Physica D 16: 285-317.
   Eckmann, J.-P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from 
     time series." Phys. Rev. A 34: 4971-4979.

[11] What is Generic?

Generic in dynamical systems is intended to convey "usual" or, more properly, 
"observable". Roughly speaking, a property is generic over a class if any 
system in the class can be modified ever so slightly (perturbed), into one 
with that property. 

The formal definition is done in the language of topology: Consider the class 
to be a space of systems, and suppose it has a topology (some notion of a 
neighborhood, or an open set). A subset of this space is *dense* if its 
closure (the subset plus the limits of all sequences in the subset) is the 
whole space. It is *open and dense* if it is also an open set (union of 
neighborhoods). A set is *countable* if it can be put into 1-1 correspondence 
with the counting numbers. A *countable intersection of open dense sets* is 
the intersection of a countable number of open dense sets. If all such 
intersections in a space are also dense, then the space is called a *Baire* 
space, which basically means its big enough. If we have such a Baire space of 
dynamical systems, and there is a property which is true on a countable 
intersection of open dense sets, them that property is *generic*. 

If all this sounds too complicated, think of it as a precise way of defining a 
set which is near every system in the collection (dense), which isn't too big 
(needn't have any "regions" where the property is true for *every* system). 
Generic is much weaker than "almost everywhere" (occurs with probability 1), 
in fact, it is possible to have generic properties which occur with 
probability zero. But it is as strong a property as one can define 
topologically, without having to have a property hold true in a region, or 
talking about measure (probability), which isn't a topological property (a 
property preserved by a continuous function).

[12] What is the minimum phase space dimension for chaos?

This is a slightly confusing topic, since the answer depends on the type of 
system considered. First consider a flow (or system of differential 
equations). In this case the Poincare-Bendixson theorem tells us that there is 
no chaos in one or two dimensional phase spaces. Chaos is possible in three 
dimensional flows--standard examples such as the Lorenz equations are indeed 
three dimensional, and there are mathematical 3D flows that are provably 
chaotic (e.g. the 'solenoid').

Note: if the flow is non-autonomous then time is a phase space coordinate, so 
a system with two physical variables + time becomes three dimensional, and 
chaos is possible (i.e. Forced second-order oscillators do exhibit chaos.)

For maps, it is possible to have chaos in one dimension, but only if the map 
is not  invertible. A prominent example is the Logistic map x' = f(x) = rx(1-
x). This is provably chaotic for r = 4, and many other values of r as well 
(see e.g. Devaney). Note that every point has two preimages, except for the 
image of the critical point x=1/2, so this map is not invertible.

For homeomorphisms, we must have at least two dimensional phase space for 
chaos. This is equivalent to the flow result, since a three dimensional flow 
gives rise to a two dimensional homeomorphism by Poincare section (see [Q6]).

Note that a numerical algorithm for a differential equation is a map, because 
time on the computer is necessarily discrete. Thus  numerical solutions of two 
and even one dimensional systems of ordinary differential equations may 
exhibit chaos. Usually this results from choosing the size of the time step 
too large.  For example Euler discretization of the Logistic differential 
equation, dx/dt = rx(1-x), is equivalent to the logistic map. See e.g. S. 
Ushiki, Central difference scheme and chaos, Physica D, vol. 4 (1982) 407-424.

[13] What are complex systems?

A complex system, as I understand it, is a system with many inequivalent 
degrees of freedom. While, chaos is the study of how simple systems can 
generate complicated behavior, complexity is the study of how complicated 
systems can generate simple behavior.  An example of complexity is the 
synchronization of biological systems ranging from fireflies to neurons (e.g. 
Matthews, PC, Mirollo, RE & Strogatz, SH "Dynamics of a large system of 
coupled nonlinear oscillators," Physica D _52_ (1991) 293-331).  In these 
problems, many individual systems conspire to produce a single collective 

The notion of complex systems has received lots of popular press, but it is 
not really clear as of yet if there is a "theory" about a "concept". We are 
withholding judgement.

[14] What are fractals?

One way to define "fractal" is as a negation: a fractal is a set that does not 
look like a Euclidean object (point, line, plane, etc.) no matter how closely 
you look at it.  Imagine focusing in on a smooth curve (imagine a piece of 
string in space)--if you look at any piece of it closely enough it eventually 
looks like a straight line (ignoring the fact that for a real piece of string 
it will soon look like a cylinder and eventually you will see the fibers, then 
the atoms, etc.). A fractal, like the Koch Snowflake, which is topologically 
one dimensional, never looks like a straight line, no matter how closely you 
look.  There are indentations, like bays in a coastline; look closer and the 
bays have inlets, closer still the inlets have subinlets, and so on.

"Fractal" is a term which has undergone refinement of definition by a lot of 
people, but was first coined by B. Mandelbrot and defined as a set with 
fractional (non-integer) dimension (Hausdorff dimension, see [Q16]).  While 
this definition has a lot of drawbacks, note that it says nothing about self-
similarity--even though the most commonly known fractals are indeed self-

See the extensive FAQ from sci.fractals at

[15] What do fractals have to do with chaos?

Often chaotic dynamical systems exhibit fractal structures in phase space. 
However, there is no direct relation. There are chaotic systems that have 
nonfractal limit sets (e.g. Arnold's cat map) and fractal structures that can 
arise in nonchaotic dynamics (see e.g. Grebogi, C., et al. (1984). "Strange 
Attractors that are not Chaotic." Physica 13D: 261-268.)

[16] What are topological and fractal dimension?

See the fractal FAQ:

[17] What is quantum chaos?

 According to the correspondence principle, there is a limit where classical 
behavior as described by Hamilton's equations becomes similar, in some 
suitable sense, to quantum behavior as described by the appropriate wave 
equation. Formally, one can take this limit to be h -> 0, where h is Planck's 
constant; alternatively, one can look at successively higher energy levels, 
etc. Such limits are referred to as "semiclassical". It has been found that 
the semiclassical limit can be highly nontrivial when the classical problem is 
chaotic. The study of how quantum systems, whose classical counterparts are 
chaotic, behave in the semiclassical limit has been called quantum chaos. More 
generally, these considerations also apply to elliptic partial differential 
equations that are physically unrelated to quantum considerations. For 
example, the same questions arise in relating classical acoustic waves to 
their corresponding ray equations. Among recent results in quantum chaos is a 
prediction relating the chaos in the classical problem to the statistics of 
energy-level spacings in the semiclassical quantum regime. 

Classical chaos can be used to analyze such ostensibly quantum systems as the 
hydrogen atom, where classical predictions of microwave ionization thresholds 
agree with experiments. See Koch, P. M. and K. A. H. van Leeuwen (1995). 
"Importance of Resonances in Microwave Ionization of Excited Hydrogen Atoms." 
Physics Reports 255: 289-403.

     <> Quantum Chaos Home Page

[18] How do I know if my data is deterministic?

How can I tell if my data is deterministic? This is a very tricky problem. It 
is difficult because in practice no time series consists of pure 'signal.' 
There will always be some form of corrupting noise, even if it is present as 
roundoff or truncation error or as a result of finite arithmetic or 
quantization. Thus any real time series, even if mostly deterministic, will be 
a stochastic processes

All methods for distinguishing deterministic and stochastic processes rely on 
the fact that a deterministic system will always evolve in the same way from a 
given starting point. Thus given a time series that we are testing for 
determinism we (1) pick a test state (2) search the time series for a similar 
or 'nearby' state and (3) compare their respective time evolution. 

Define the error as the difference between the time evolution of the 'test' 
state and the time evolution of the nearby state. A deterministic system will 
have an error that either remains small (stable, regular solution) or increase 
exponentially with time (chaotic solution). A stochastic system will have a 
randomly distributed error.

Essentially all measures of determinism taken from time series rely upon 
finding the closest states to a given 'test' state (i.e., correlation 
dimension, Lyapunov exponents, etc.). To define the state of s system one 
typically relies on phase space embedding methods, see [23]. 

Typically one chooses an embedding dimension, and investigates the propagation 
of the error between two nearby states. If the error looks random, one 
increases the dimension. If you can increase the dimension to obtain a 
deterministic looking error, then you are done. Though it may sound simple  it 
is not really! One complication is that as the dimension increases the search 
for a nearby state requires a lot more computation time and a lot of data (the 
amount of data required increases exponentially with embedding dimension) to 
find a suitably close candidate. If the embedding dimension (number of 
measures per state) is chosen too small (less than the 'true' value) 
deterministic data can appear to be random but in theory there is no problem 
choosing the dimension too large--the method will work. Practically, anything 
approaching about 10 dimensions is considered so large that a stochastic 
description is probably more suitable and convenient anyway.

See e.g.,

    Sugihara, G. and R. M. May (1990). "Nonlinear Forcasting as a Way of 
    Distinguishing Chaos from Measurement Error in Time Series." Nature 344: 

[19] What is the control of chaos?

Control of chaos has come to mean the two things: (1) stabilization of 
unstable periodic orbits, (2) use of recurrence to allow stabilization to be 
applied locally. Thus term "control of chaos" is somewhat of a misnomer--but 
the name has stuck. The ideas for controlling chaos originated in the work of 
Hubler followed by the Maryland Group.

    Hubler, A. W. (1989). "Adaptive Control of Chaotic Systems." Helv. 
   Phys. Acta 62: 343-346).

   Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review 
   Letters 64(11): 1196-1199.

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 

   Ditto, WL & Pecora, LM "Mastering Chaos," Scientific American (Aug. 1993), 

   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 

[20] How can I build a chaotic circuit?

There are many different physical systems which display chaos, dripping 
faucets, water wheels, oscillating magnetic ribbons etc. but the most simple 
systems which can be easily implemented are chaotic circuits. In fact an 
electronic circuit was one of the first demonstrations of chaos which showed 
that chaos is not just a mathematical abstraction. Leon Chua designed the 
circuit 1983. 

The circuit he designed, now known as Chua's circuit, consists of a piecewise 
linear resistor as its nonlinearity (making analysis very easy) plus two 
capacitors, one resistor and one inductor--the circuit is unforced 
(autonomous). In fact the chaotic aspects (bifurcation values, Lyapunov 
exponents, various dimensions etc.) of this circuit have been extensively 
studied in the literature both experimentally and theoretically. It is 
extremely easy to build and presents beautiful attractors (the most famous 
known as the double scroll attractor) that can be displayed on a CRO. 

For more information on building such a circuit try: 

   Kennedy M. P., "Robust OP Amp Realization of Chua's Circuit", Frequenz, 
   vol. 46, no. 3-4, 1992.

   Madan, R. A., "Chua's Circuit: A paradigm for chaos", ed. R. A. Madan, 
   Singapore: World Scientific, 1993.

   Pecora, L. and Carroll, T. "Nonlinear Dynamics in Circuits", 
   Singapore: World Scientific, 1995.

[21] What are simple experiments that I can do to demonstrate 

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 
<>.  My favorite double pendulum 
consists of  two identical planar pendula, so that you can demonstrate 
sensitive dependence [Q9].

One of the simplest chemical systems that shows chaos is the Belousov-
Zhabotinsky reacation.The book by Strogatz [Q26] has a good introduction to 
this subject, see also <>
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 

Chua's nonlinear curcuit is also a good example. See [Q20] above.

[22] What is targeting?

To direct trajectories in chaotic systems, one can generically apply small
perturbations; see:

   Shinbrot, T. Ott, E., Grebogi, C. & Yorke, J.A., "Using Small 
   Perturbations to Control Chaos," Nature, 363 (1993) 411-7). 

We are still awaiting a good answer to this question.

[23] What is time series analysis?

This is the application of dynamical systems techniques to a data series, 
usually obtained by "measuring" the value of a single observable as a function 
of time. The major tool in a dynamicists toolkit is "delay coordinate 
embedding" which creates a phase space portrait from a single data series. It 
seems remarkable at first, but one can reconstruct a picture equivalent 
(topologically) to the full Lorenz attractor in three dimensional space by 
measuring only one of its coordinates, say x(t), and plotting the delay 
coordinates (x(t), x(t+h), x(t+2h)) for a fixed h.

It is important to emphasize that the idea of using derivatives or delay 
coordinates in time series modeling is nothing new.  It goes back at least to 
the work of Yule, who in 1927 used an autoregressive (AR) model to make a 
predictive model for the sunspot cycle. AR models are nothing more than delay 
coordinates used with a linear model.  Delays, derivatives, principal 
components, and a variety of other methods of reconstruction have been widely 
used in time series analysis since the early 50's, and are described in 
several hundred books.  The new aspects raised by dynamical systems theory are 
(i) the implied geometric view of temporal behavior and (ii) the existence of 
"geometric invariants", such as dimension and Lyapunov exponents.  The central 
question was not whether delay coordinates are useful for time series 
analysis, but rather whether reconstruction methods preserve the geometry and 
the geometric invariants of dynamical systems. (Packard, Crutchfield, Farmer & 

   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,

[24] Is there chaos in the stock market?

In order to address this question, we must first agree what we mean by chaos, 
see [Q8].

In dynamical systems theory, chaos means irregular fluctuations in a 
deterministic system (see [Q3] and [Q18]). This means the system behaves 
irregularly because of its own internal logic, not because of random forces 
acting from outside. Of course if you define your dynamical system to be the 
socio-economic behavior of the entire planet, nothing acts randomly from 
outside (except perhaps the occasional meteor), so you have a dynamical 
system. But its dimension (number of state variables--see [Q4]) is vast, and 
there is no hope of exploiting the determinism. This is high-dimensional 
chaos, which might just as well be truly random behavior. In this sense, the 
stock market is chaotic, but who cares? 

To be useful, economic chaos would have to involve some kind of collective 
behavior which can be fully described by a small number of variables. In the 
lingo, the system would have to be self-organizing, resulting in low-
dimensional chaos. If this turns out to be true, then you can exploit the low-
dimensional chaos to make short-term predictions. The problem is to identify 
the state variable which characterize the collective modes. Furthermore, 
having limited the number of state variables, many events now become external 
to the system, that is, the system is operating in a changing environment, 
which makes the problem of system identification very difficult. 

If there were such collective modes of fluctuation, market players would 
probably know about them; economic theory says that if many people recognized 
these patterns, the actions they would take to exploit them would quickly 
nullify the patterns. Therefore if these patterns exist, they must be hard to 
recognize because they do not emerge clearly from the sea of noise caused by 
individual actions; or the patterns last only a very short time following some 
upset to the markets; or both.

There are a number of people and groups trying to find these patterns. Some of 
these groups are known to outsiders, because they include prominent 
researchers in the field of chaos; we have no idea whether they are succeeding 
or not. If you know chaos theory and would like to make yourself a slave to 
the rhythms of market trading, you can probably find a major trading firm 
which will give you a chance to try your ideas. But don't expect them to give 
you a share of any profits you may make for them :-) !

In short, anyone who tells you about the secrets of chaos in the stock market 
doesn't know anything useful, and anyone who knows will not tell. It's an 
interesting question, but you're unlikely to find the answer.

[25] What are solitons?

Consider this frequently asked question: The Fourier transform can simplify 
the evolution of linear differential equations; is there a counterpart which 
similarly simplifies nonlinear equations? The answer is No. Nonlinear 
equations are qualitatively more complex than linear equations, and a 
procedure which gives the dynamics as simply as for linear equations must 
contain a mistake. There are, however, exceptions to any rule. 

Certain nonlinear differential equations can be fully solved by, e.g., the 
"inverse scattering method." Examples are the Korteweg-de Vries, nonlinear 
Schrodinger, and sine-Gordon equations. In these cases the real space maps, in 
a rather abstract way, to an inverse space, which is comprised of continuous 
and discrete parts and evolves linearly in time. The continuous part typically 
corresponds to radiation and the discrete parts to stable solitary waves, i.e. 
pulses, which are called solitons. The linear evolution of the inverse space 
means that solitons will emerge virtually unaffected from interactions with 
anything, giving them great stability. 

More broadly, there is a wide variety of systems which support stable solitary 
waves through a balance of dispersion and nonlinearity. Though these systems 
may not be integrable as above, in many cases they are close to systems which 
are, and the solitary waves may share many of the stability properties of true 
solitons, especially that of surviving interactions with other solitary waves 
(mostly) unscathed. It is widely accepted to call these solitary waves 
solitons, albeit with qualifications. 

Why solitons? Solitons are simply a fundamental nonlinear wave phenomenon. 
Many very basic linear systems with the addition of the simplest possible or 
first order nonlinearity support solitons; this universality means that 
solitons will arise in many important physical situations. Optical fibers can 
support solitons, which because of their great stability are an ideal medium 
for transmitting information. In a few years long distance telephone 
communications will likely be carried via solitons. 

The soliton literature is by now vast. Two books which contain clear 
discussions of solitons as well as references to original papers are 
   Alan C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, 
   Penn. (1985)

   Mark J. Ablowitz, Solitons, nonlinear evolution equations and inverse 
   scattering, Cambridge (1991). 
See the Soliton Home page:

[26] What should I read to learn more?

1.	Gleick, J. (1987). Chaos, the Making of a New Science. London, 
2.	Stewart, I. (1989). Does God Play Dice? Cambridge, Blackwell.
3.	Devaney, R. L. (1990). Chaos, Fractals, and Dynamics :  Computer 
         Experiments in Mathematics. Menlo Park, Addison-Wesley Pub. Co.
4.	Lorenz, E., (1994) The Essence of Chaos, University of Washington Press.

Introductory Texts
1.	Percival, I. C. and D. Richard (1982). Introduction to Dynamics. 
         Cambridge, Cambridge Univ. Press. 
2.	Devaney, R. L. (1986). An Introduction to Chaotic Dynamical Systems.
         Menlo Park, Benjamin/Cummings.
3.	Baker, G. L. and J. P. Gollub (1990). Chaotic Dynamics. Cambridge, 
         Cambridge Univ. Press. <>
4.	Tufillaro, N., T. Abbott, et al. (1992). An Experimental Approach to 
         Nonlinear Dynamics and Chaos. Redwood City, Addison-Wesley.
5.	Jurgens, H., H.-O. Peitgen, et al. (1993).  Chaos and Fractals: New 
         Frontiers of Science. New York, Springer Verlag. 
6.	Glendinning, P. (1994). Stability, Instability and Chaos. Cambridge, 
         Cambridge Univ Press.
7.	Strogatz, S. (1994). Nonlinear Dynamics and Chaos. Reading, Addison-
8.	Moon, F. C. (1992). Chaotic and Fractal Dynamics. New York, John Wiley.
9.	Turcotte, Donald L. (1992). Fractals and Chaos in Geology and 
         Geophysics, Cambridge Univ. Press.
10.    Ott, Edward (1993). Chaos in Dynamical Systems. Cambridge,
         Cambridge University Press.
11.    D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics, 
         Springer-Verlag New York.

Introductory Articles
1.	May, R. M. (1986). "When Two and Two Do Not Make Four." Proc. Royal Soc. 
         B228: 241.
2.	Berry, M. V. (1981). "Regularity and Chaos in Classical Mechanics, 
         Illustrated by Three Deformations of a Circular Billiard." Eur. J. 
         Phys. 2: 91-102.
3.	Crawford, J. D. (1991). "Introduction to Bifurcation Theory." Reviews of 
         Modern Physics 63(4): 991-1038.
4.	Shinbrot, T., C. Grebogi, et al. (1992). "Chaos in a Double Pendulum." 
         Am. J. Phys 60: 491-499.
5.	David Ruelle. (1980). "Strange Attractors," The Mathematical 
                  Intelligencer 2:  126-37.

[27] What technical journals have nonlinear science articles?

Physica D                    The premier journal in Nonlinear Dynamics
Nonlinearity                 Good mix, with a mathematical bias 
Chaos                        AIP Journal, with a good physical bent
Physics Letters A            Has a good nonlinear science section
Physical Review E            Lots of Physics articles with nonlinear emphasis
Ergodic Theory and           Rigorous mathematics, and careful work
      Dynamical Systems  
J Differential Equations     A premier journal, but very mathematical
J Dynamics and Diff. Eq.     Good, more focused version of the above
J Dynamics and Stability     Focused on Eng. applications. New editorial
      of Systems              board--stay tuned. 
J Statistical Physics        Used to contain seminal dynamical systems papers 
SIAM Journals                Only the odd dynamical systems paper
J Fluid Mechanics            Some expt. papers, e.g. transition to turbulence
Nonlinear Dynamics           Haven't read enough to form an opinion
J Nonlinear Science          a newer journal--haven't read enough yet. 
Nonlinear Science Today      News of the week see:
International J of           lots of color pictures, variable quality.
    Bifurcation and Chaos 
Chaos Solitons and Fractals  Variable quality, some good applications
Communications in Math Phys  an occasional paper on dynamics
Nonlinear Processes in       New, variable quality...may be improving

[28] What are net sites for nonlinear science materials? 

Preprint Archives
   <> Los Alamos Preprint Server
   <> Nonlinear Science Eprint Server
   <>  Math-Physics Archive
   <> AMS Preprint

Conference Announcements

   <gopher://>  SIAM Dynamical Systems Group
   <> UK Nonlinear News

Electronic Journals
   <>  Nonlinear Science Today
   <> The Complexity Journal
   <> Complexity International Journal

Electronic Texts
        Exploring Chaos & Fractals
   <> Cvitanovic's Lecture Notes
   <> Chaos Intro

Institutes and Academic Programs
       Extensive List of Physics Groups in Nonlinear Phenonmena
       Extensive List of Nonlinear Groups

Who is Who in Nonlinear Dynamics

Nonlinear Lists
   <> Extensive List of Nonlinear
   <> URLs from Sci.nonlinear
   <> Chaos URLs

Time Series sites
   <> Dynamics and Time Series
   <>  time series
   <> Santa Fe 
         Time Series Competition

Chaos Sites
   <>  Expt. henon attractor
   <> All about
         Feigenbaum Constants
   <> Mike Rosenstein's Chaos Page.
   <> Chaos Network
         Lorenz Attractor

Complexity Sites
   <> Complex Sytems
   <> Complexity Home Page

Fractals Sites
   <> The Spanky Fractal DataBase
   <> Sprott's Fractal Gallery
   <> Groupe Fractales
   <> 3D Fractals
   <>  Fractal Gallery>
   <> Course on Fractal 

[29] What nonlinear science software is available?

General Resources
      "Guide to Available Mathematical Software" maintained by NIST: 
      "Mathematics Archives Software" 
   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:

   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

   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:

   Phase Plane plotter for x-windows systems
      System: X-windows, Unix, SunOS 4 binary
      Available by anonymous ftp:

   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:

Lyapunov Exponents and Time Series
   Based on Alan Wolf's algorithm, see[Q10], but a more efficient version.
      System: Comes as C source, Fortran source, PC executable, etc
      Available by anonymous ftp:

Lyapunov Exponents
   Keith Briggs Fortran codes for Lyapunov exponents
      System: any with a Fortran compiler
      Available by anonymous ftp:

  MTRCHAOS and MTRLYAP compute correlation dimension
  and largest Lyapunov exponents, delay portraits. By Mike Rosenstein.
      System: PC-compatible computer running DOS 3.1 or higher, 
             640K RAM, and EGA display. VGA & coprocessor recommended
      Available by anonymous ftp:

Chaos Plot
  ChaosPlot is a simple program which plots the chaotic behavior of a damped,
  driven anharmonic oscillator.
      System: Macintosh
      Available from: <
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:

   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:

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:

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

   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 
      System: PC-compatible computer + ???
      Available: Springer-Verlag

The Academic Software Library:
 Chaos Simulations
   Bessoir, T., and A. Wolf, 1990. Demonstrates logistic map, Lyapunov
   exponents, billiards in a stadium, sensitive dependence, 
   n-body gravitational motion.
     Available: The Academic Software Library, (800) 955-TASL. $70.
 Chaos Data Analyser 
   A PC program for analyzing time series. By Sprott, J.C. and G. Rowlands.
     Available: The Academic Software Library, (800) 955-TASL. $70.
     For more information see:
 Chaos Demonstrations
   A PC program for demonstrating chaos, fractals, cellular automata, 
   and related nonlinear phenomena.  By J. C. Sprott and G. Rowlands.
     System: IBM PC or compatible with at least 512K of memory.
     Available: The Academic Software Library, (800) 955-TASL. $70.
 Chaotic Dynamics Workbench
   Performs interactive numerical experiments on systems
   modeled by ordinary differential equations, including: four versions of    
   driven Duffing oscillators, pendulum, Lorenz, driven Van der Pol osc., 
   driven Brusselator, and the Henon-Heils system.  By R. Rollins. 
     System: IBM PC or compatible, 512 KB memory.
     Available: The Academic Software Library, (800) 955-TASL, $70.

   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

   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: <
	Available: Springer-Verlag

Tufillaro's Programs
   From the book Nonlinear Dynamics and Chaos by Tufillaro, Abbot and Reilly
   (1992). A collection of programs for the Macintosh.
      System: Macintosh
      Available: Addison-Wesley
      For more info see:

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:

[30] Acknowledgments

Thanks to 
    Hawley Rising <mailto://>,
    Bruce Stewart <mailto://>
    Alan Champneys <mailto://>
    Michael Rosenstein <mailto://>
    Troy Shinbrot <mailto://>
    Matt Kennel <mailto://>
    Lou Pecora <mailto://>
    Richard Tasgal <mailto://>
    Wayne Hayes <mailto://>
    S. H. Doole <mailto://>
    Pavel Pokorny <mailto://>, 
    Gerard Middleton <mailto://middleto@mcmail.CIS.McMaster.CA>
    Ronnie Mainieri <mailto://>
    Leon Poon <mailto://>
    Justin Lipton <mailto://>

Anyone else who would like to contribute, please do! Send me your comments:

		Jim Meiss

	James Meiss
	Program in Applied Math

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