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(need not have any "regions" where the property is true for every system). 
Generic is much weaker than "almost everywhere" (occurs with probability 1), 
in fact, it is possible to have generic properties which occur with 
probability zero. But it is as strong a property as one can define 
topologically, without having to have a property hold true in a region, or 
talking about measure (probability), which isn't a topological property (a 
property preserved by a continuous function).


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

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

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

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

For homeomorphisms, we must have at least two-dimensional phase space for 
chaos. This is equivalent to the flow result, since a three-dimensional flow 
gives rise to a two-dimensional homeomorphism by Poincaré section (see [2.7]).

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



[3]   Applications and Advanced Theory
[3.1] What are complex systems?
(Thanks to Troy Shinbrot for contributing to this answer)

Complex systems are spatially and/or temporally extended nonlinear systems 
characterized by collective properties associated with the system as a whole--
and that are different from the characteristic behaviors of the constituent 
parts.

While, chaos is the study of how simple systems can generate complicated 
behavior, complexity is the study of how complicated systems can generate 
simple behavior. An example of complexity is the synchronization of biological 
systems ranging from fireflies to neurons (e.g. Matthews, PC, Mirollo, RE & 
Strogatz, SH "Dynamics of a large system of coupled nonlinear oscillators," 
Physica 52D (1991) 293-331). In these problems, many individual systems 
conspire to produce a single collective rhythm.

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

   http://www.calresco.org/index.htm The Complexity & Artificial Life Web Site 
   http://www.calresco.org/sos/sosfaq.htm The self-organized systems FAQ


[3.2] What are fractals?

One way to define "fractal" is as a negation: a fractal is a set that does not 
look like a Euclidean object (point, line, plane, etc.) no matter how closely 
you look at it. Imagine focusing in on a smooth curve (imagine a piece of 
string in space)--if you look at any piece of it closely enough it eventually 
looks like a straight line (ignoring the fact that for a real piece of string 
it will soon look like a cylinder and eventually you will see the fibers, then 
the atoms, etc.). A fractal, like the Koch Snowflake, which is topologically 
one dimensional, never looks like a straight line, no matter how closely you 
look. There are indentations, like bays in a coastline; look closer and the 
bays have inlets, closer still the inlets have subinlets, and so on. Simple 
examples of fractals include Cantor sets (see [3.5], Sierpinski curves, the 
Mandelbrot set  and (almost surely) the Lorenz attractor (see [2.12]). 
Fractals also approximately describe many real-world objects, such as clouds 
(see http://makeashorterlink.com/?Z50D42C16)  mountains, turbulence, 
coastlines, roots and branches of trees and veins and lungs of animals.

"Fractal" is a term which has undergone refinement of definition by a lot of 
people, but was first coined by B. Mandelbrot, 
http://physics.hallym.ac.kr/reference/physicist/Mandelbrot.html,  and defined 
as a set with fractional (non-integer) dimension (Hausdorff dimension, see 
[3.4]). Mandelbrot defines a fractal in the following way:

    A geometric figure or natural object is said to be fractal if it
    combines the following characteristics: (a) its parts have the same
    form or structure as the whole, except that they are at a different
    scale and may be slightly deformed; (b) its form is extremely irregular,
    or extremely interrupted or fragmented, and remains so, whatever the scale
    of examination; (c) it contains "distinct elements" whose scales are very
    varied and cover a large range." (Les Objets Fractales 1989, p.154) 

See the extensive FAQ from sci.fractals at
   4. The standard example of a Cantor set is the "middle thirds" set 
constructed on the interval between 0 and 1. First, remove the middle third. 
Two intervals remain, each one of length one third. From each remaining 
interval remove the middle third. Repeat the last step infinitely many times. 
What remains is a Cantor set.

More generally (and abstrusely) a Cantor set is defined topologically as a 
nonempty, compact set which is perfect (every point is a limit point) and 
totally disconnected (every pair of points in the set are contained in 
disjoint covering neighborhoods).

See also
   http://www.shu.edu/html/teaching/math/reals/topo/defs/cantor.html
   http://personal.bgsu.edu/~carother/cantor/Cantor1.html
   http://mizar.uwb.edu.pl/JFM/Vol7/cantor_1.html

Georg Ferdinand Ludwig Philipp Cantor was born 3 March 1845 in St Petersburg, 
Russia, and died 6 Jan 1918 in Halle, Germany. To learn more about him see:
   http://turnbull.dcs.st-and.ac.uk/history/Mathematicians/Cantor.html
   http://www.shu.edu/html/teaching/math/reals/history/cantor.html

To read more about the Cantor function (a function that is continuous, 
differentiable, increasing, non-constant, with a derivative that is zero 
everywhere except on a set with length zero) see
http://www.shu.edu/projects/reals/cont/fp_cantr.html


[3.6] What is quantum chaos?
(Thanks to Leon Poon for contributing to this answer)

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

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

See also: 
   http://sagar.physics.neu.edu/qchaos/qc.html Quantum Chaos
   http://www.mpipks-dresden.mpg.de/~noeckel/microlasers.html  Microlaser 
Cavities



[3.7] How do I know if my data are deterministic?
(Thanks to Justin Lipton for contributing to this answer)

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

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

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

Essentially all measures of determinism taken from time series rely upon 
finding the closest states to a given 'test' state (i.e., correlation 
dimension, Lyapunov exponents, etc.). To define the state of a system one 
typically relies on phase space embedding methods, see [3.14].

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

See e.g.,
   Sugihara, G. and R. M. May (1990). "Nonlinear Forecasting as a Way of 
      Distinguishing Chaos from Measurement Error in Time Series." Nature 
344: 734-740.


[3.8] What is the control of chaos?

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

   Hubler, A. W. (1989). "Adaptive Control of Chaotic Systems." Helv. Phys. 
Acta 62: 343-346.
   Ott, E., C. Grebogi, et al. (1990). "Controlling Chaos." Physical Review 
Letters 64(11): 1196-1199. http://www-
chaos.umd.edu/publications/abstracts.html#prl64.1196

The idea that chaotic systems can in fact be controlled may be 
counterintuitive--after all they are unpredictable in the long term. 
Nevertheless, numerous theorists have independently developed methods which 
can be applied to chaotic systems, and many experimentalists have demonstrated 
that physical chaotic systems respond well to both simple and sophisticated 
control strategies. Applications have been proposed in such diverse areas of 
research as communications, electronics, physiology, epidemiology, fluid 
mechanics and chemistry.

The great bulk of this work has been restricted to low-dimensional systems; 
more recently, a few researchers have proposed control techniques for 
application to high- or infinite-dimensional systems. The literature on the 
subject of the control of chaos is quite voluminous; nevertheless several 
reviews of the literature are available, including:

   Shinbrot, T. Ott, E., Grebogi, C. & Yorke, J.A., "Using Small Perturbations 
to Control Chaos," Nature, 363 (1993) 411-7.
   Shinbrot, T., "Chaos: Unpredictable yet Controllable?" Nonlin. Sciences 
Today, 3:2 (1993) 1-8.
   Shinbrot, T., "Progress in the Control of Chaos," Advance in Physics (in 
press).
   Ditto, WL & Pecora, LM "Mastering Chaos," Scientific American (Aug. 1993), 
78-84.
   Chen, G. & Dong, X, "From Chaos to Order -- Perspectives and Methodologies 
in Controlling Chaotic Nonlinear Dynamical Systems," Int. J. Bif. & Chaos 3 
(1993) 1363-1409.

It is generically quite difficult to control high dimensional systems; an 
alternative approach is to use control to reduce the dimension before applying 
one of the above techniques. This approach is in its infancy; see:

   Auerbach, D., Ott, E., Grebogi, C., and Yorke, J.A. "Controlling Chaos in
   High Dimensional Systems," Phys. Rev. Lett. 69  (1992) 3479-82
   http://www-chaos.umd.edu/publications/abstracts.html#prl69.3479


[3.9] How can I build a chaotic circuit?
(Thanks to Justin Lipton and Jose Korneluk for contributing to this answer)

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

The circuit he designed, now known as Chua's circuit, consists of a piecewise 
linear resistor as its nonlinearity (making analysis very easy) plus two 
capacitors, one resistor and one inductor--the circuit is unforced 
(autonomous). In fact the chaotic aspects (bifurcation values, Lyapunov 
exponents, various dimensions etc.) of this circuit have been extensively 
studied in the literature both experimentally and theoretically. It is 
extremely easy to build and presents beautiful attractors (see [2.8]) (the 
most famous known as the double scroll attractor) that can be displayed on a 
CRO.

For more information on building such a circuit try: see

   http://www.cmp.caltech.edu/~mcc/chaos_new/Chua.html  Chua's Circuit  Applet

References
   Matsumoto T. and Chua L.O. and Komuro M. "Birth and Death of the Double 
      Scroll" Physica D24 97-124, 1987.
   Kennedy M. P., "Robust OP Amp Realization of Chua's Circuit", Frequenz 
      46, no. 3-4, 1992
   Madan, R. A., Chua's Circuit: A paradigm for chaos, ed. R. A. Madan, 
      Singapore: World Scientific, 1993.
   Pecora, L. and Carroll, T. Nonlinear Dynamics in Circuits, Singapore: 
      World Scientific, 1995.
   Nonlinear Dynamics of Electronic Systems, Proceedings of the Workshop 
      NDES 1993, A.C.Davies and W.Schwartz, eds., World Scientific, 1994, 
      ISBN 981-02-1769-2.
   Parker, T.S., and L.O.Chua, Practical Numerical Algorithms for Chaotic 
      Systems, Springer-Verlag, 1989, ISBN's: 0-387-96689-7 
      and 3-540-96689-7.


[3.10] What are simple experiments to demonstrate chaos?

 There are many "chaos toys" on the market. Most consist of some sort of 
pendulum that is forced by an electromagnet. One can of course build a simple 
double pendulum to observe beautiful chaotic behavior see 
   http://quasar.mathstat.uottawa.ca/~selinger/lagrange/doublependulum.html 
Experimental Pendulum Designs
   http://www.maths.tcd.ie/~plynch/SwingingSpring/doublependulum.html  Java 
Applet
   http://monet.physik.unibas.ch/~elmer/pendulum/ Java Applets Pendulum Lab

My favorite double pendulum consists of two identical planar pendula, so that 
you can demonstrate sensitive dependence [2.10], for a Java applet simulation 
see http://www.cs.mu.oz.au/~mkwan/pendulum/pendulum.html. Another cute toy is 
the "Space Circle" that you can find in many airport gift shops. This is 
discussed in the article:

   A. Wolf & T. Bessoir, Diagnosing Chaos in the Space Circle, Physica 50D, 
1991.

One of the simplest chemical systems that shows chaos is the Belousov-
Zhabotinsky reaction. The book by Strogatz [4.1] has a good introduction to 
this subject,. For the recipe see 
http://www.ux.his.no/~ruoff/BZ_Phenomenology.html. Chemical chaos is modeled 
(in a generic sense) by the "Brusselator" system of differential equations. 
See

   Nicolis, Gregoire & Prigogine, (1989) Exploring Complexity: An 
      Introduction W. H. Freeman

The Chaotic waterwheel, while not so simple to build, is an exact realization 
of Lorenz famous equations. This is nicely discussed in Strogatz book [4.1] as 
well.

Billiard tables can exhibit chaotic motion, see 
http://www.maa.org/mathland/mathland_3_3.html, though it might be hard to see 
this next time you are in a bar, since a rectangular table is not chaotic!


[3.11] What is targeting?
(Thanks to Serdar Iplikçi for contributing to this answer)

Targeting is the task of steering a chaotic system from any initial point to 
the target, which can be either an unstable equilibrium point or an unstable 
periodic orbit, in the shortest possible time, by applying relatively small 
perturbations. In order to effectively control chaos, [3.8] a targeting 
strategy is important. See:

   Kostelich, E., C. Grebogi, E. Ott, and J. A. Yorke, "Higher
      Dimensional Targeting," Phys Rev. E,. 47, , 305-310 (1993).
   Barreto, E., E. Kostelich, C. Grebogi, E. Ott, and J. A. Yorke, "Efficient
      Switching Between Controlled Unstable Periodic Orbits in Higher
      Dimensional Chaotic Systems," Phys Rev E, 51, 4169-4172 (1995).

One application of targeting is to control a spacecraft's trajectory so that 
one can find low energy orbits from one planet to another. Recently targeting 
techniques have been used in the design of trajectories to asteroids and even 
of a grand tour of the planets. For example,

   E. Bollt and J. D. Meiss, "Targeting Chaotic Orbits to the Moon 
      Through Recurrence," Phys. Lett. A  204, 373-378 (1995).
   http://www.cds.caltech.edu/~marsden/software/spacecraft_orbits.html


 [3.12] What is time series analysis?
(Thanks to Jim Crutchfield for contributing to this answer)

This is the application of dynamical systems techniques to a data series, 
usually obtained by "measuring" the value of a single observable as a function 
of time. The major tool in a dynamicist's toolkit is "delay coordinate 
embedding" which creates a phase space portrait from a single data series. It 
seems remarkable at first, but one can reconstruct a picture equivalent 
(topologically) to the full Lorenz attractor (see [2.12])in three-dimensional 
space by measuring only one of its coordinates, say x(t), and plotting the 
delay coordinates (x(t), x(t+h), x(t+2h)) for a fixed h.

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

   G.U. Yule, Phil. Trans. R. Soc. London A 226 (1927) p. 267.
   N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, "Geometry
      from a time series", Phys. Rev. Lett. 45, no. 9 (1980) 712.
   F. Takens, "Detecting strange attractors in fluid turbulence", in: Dynamical 
      Systems and Turbulence, eds. D. Rand and L.-S. Young 
      (Springer, Berlin, 1981)
   Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.Sh.T. 
      "The analysis of observed chaotic data in physical systems",  
      Rev. Modern Physics 65 (1993) 1331-1392.
   D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics, 
      Springer-Verlag http://www.cnd.mcgill.ca/books_understanding.html
   E. Peters (1994) Fractal Market Analysis : Applying Chaos Theory to 
      Investment and Economics, Wiley
      http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471585246.html


[3.13] Is there chaos in the stock market?
(Thanks to Bruce Stewart for Contributions to this answer)

In order to address this question, we must first agree what we mean by chaos, 
see [2.9].

In dynamical systems theory, chaos means irregular fluctuations in a 
deterministic system (see [2.3] and [3.7]). This means the system behaves 
irregularly because of its own internal logic, not because of random forces 
acting from outside. Of course, if you define your dynamical system to be the 
socio-economic behavior of the entire planet, nothing acts randomly from 
outside (except perhaps the occasional meteor), so you have a dynamical 
system. But its dimension (number of state variables--see [2.4]) is vast, and 
there is no hope of exploiting the determinism. This is high-dimensional 
chaos, which might just as well be truly random behavior. In this sense, the 
stock market is chaotic, but who cares?

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

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

A number of people and groups have tried to find these patterns. So far the 
published results are negative. There are also commercial ventures involving 
prominent researchers in the field of chaos; we have no idea how well they are 
succeeding, or indeed whether they are looking for low-dimensional chaos. In 
fact it seems unlikely that markets remain stationary long enough to identify 
a chaotic attractor (see [2.12]). If you know chaos theory and would like to 
devote yourself to the rhythms of market trading, you might find a trading 
firm which will give you a chance to try your ideas. But don't expect them to 
give you a share of any profits you may make for them :-) !

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

On the other hand, one might ask a more general question: is market behavior 
adequately described by linear models, or are there signs of nonlinearity in 
financial market data? Here the prospect is more favorable. Time series 
analysis (see [3.14]) has been applied these tests to financial data; the 
results often indicate that nonlinear structure is present. See e.g. the book 
by Brock, Hsieh, LeBaron, "Nonlinear Dynamics, Chaos, and Instability", MIT 
Press, 1991; and an update by B. LeBaron, "Chaos and nonlinear forecastability 
in economics and finance," Philosophical Transactions of the Royal Society, 
Series A, vol 348, Sept 1994, pp 397-404. This approach does not provide a 
formula for making money, but it is stimulating some rethinking of economic 
modeling. A book by Richard M. Goodwin, "Chaotic Economic Dynamics," Oxford 
UP, 1990, begins to explore the implications for business cycles.


[3.14] What are solitons?

The process of obtaining a solution of a linear (constant coefficient) 
differential equations is simplified by the Fourier transform (it converts 
such an equation to an algebraic equation, and we all know that algebra is 
easier than calculus!); is there a counterpart which similarly simplifies 
nonlinear equations? The answer is No. Nonlinear equations are qualitatively 
more complex than linear equations, and a procedure which gives the dynamics 
as simply as for linear equations must contain a mistake. There are, however, 
exceptions to any rule.

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

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

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

The soliton literature is by now vast. Two books which contain clear 
discussions of solitons as well as references to original papers are
   A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia,
      Penn. (1985)
   M.J. Ablowitz and P.A.Clarkson, Solitons, nonlinear evolution equations and 
inverse
      scattering, Cambridge (1991). 
http://www.cup.org/titles/catalogue.asp?isbn=0521387302
   See  http://www.ma.hw.ac.uk/solitons/


[3.15] What is spatio-temporal chaos?

   Spatio-temporal chaos occurs when system of coupled dynamical systems 
gives rise to dynamical behavior that exhibits both spatial disorder (as in 
rapid decay of spatial correlations) and temporal disorder (as in nonzero 
Lyapunov exponents). This is an extremely active, and rather unsettled area of 
research. For an introduction see:
   Cross, M. C. and P. C. Hohenberg (1993). "Pattern Formation outside of
       Equilibrium."  Rev. Mod. Phys. 65: 851-1112.
   http://www.cmp.caltech.edu/~mcc/st_chaos.html Spatio-Temporal Chaos

An interesting application which exhibits pattern formation and spatio-
temporal chaos is to excitable media in biological or chemical systems. See

   Chaos, Solitions and Fractals 5 #3&4 (1995) Nonlinear Phenomena in Excitable 
      Physiological System,  http://www.elsevier.nl/locate/chaos
   http://ojps.aip.org/journal_cgi/dbt?KEY=CHAOEH&Volume=8&Issue=1 
      Chaos focus issue on Fibrillation


[3.16] What are cellular automata?
(Thanks to Pavel Pokorny for Contributions to this answer)

   A Cellular automaton (CA) is a dynamical system with discrete time (like 
a map, see [2.6]), discrete state space and discrete geometrical space (like 
an ODE), see [2.7]). Thus they can be represented by a state s(i,j) for 
spatial state i, at time j, where s is taken from some finite set. The update 
rule is that the new state is some function of the old states, s(i,j+1) = 
f(s). The following table shows the distinctions between PDE's, ODE's, coupled 
map lattices (CML) and CA in taking time, state space or geometrical space 
either continuous (C) of discrete (D):
        time   state space    geometrical space
 PDE      C          C              C
 ODE      C          C              D
 CML      D          C              D
 CA       D          D              D

   Perhaps the most famous CA is Conway's game "life." This CA evolves 
according to a deterministic rule which gives the state of a site in the next 
generation as a function of the states of neighboring sites in the present 
generation. This rule is applied to all sites.

For further reading see

   S. Wolfram (1986) Theory and Application of Cellular Automata, World 
Scientific Singapore.
   Physica 10D (1984)--the entire volume

Some programs that do CA, as well as more generally "artificial life" are 
available at 
   http://www.alife.org/links.html
   http://www.kasprzyk.demon.co.uk/www/ALHome.html


[3.17] What is a Bifurcation?
(Thanks to Zhen Mei for Contributions to this answer)

A bifurcation is a qualitative change in dynamics upon a small variation in 
the parameters of a system.

Many dynamical systems depend on parameters, e.g. Reynolds number, catalyst 

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