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2.9] What is chaos? (nonlinear science)




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This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

2.9] What is chaos? (nonlinear science)

It has been said that "Chaos is a name for any order that produces confusion
in our minds." (George Santayana, thanks to Fred Klingener for finding this).
However, the mathematical definition is, roughly speaking,
Chaos: effectively unpredictable long time behavior arising in a deterministic
dynamical system because of sensitivity to initial conditions.
It must be emphasized that a deterministic dynamical system is perfectly
predictable given perfect knowledge of the initial condition, and is in
practice always predictable in the short term. The key to long-term
unpredictability is a property known as sensitivity to (or sensitive
dependence on) initial conditions.

For a dynamical system to be chaotic it must have a 'large' set of initial
conditions which are highly unstable. No matter how precisely you measure the
initial condition in these systems, your prediction of its subsequent motion
goes radically wrong after a short time. Typically (see [2.14] for one
definition of 'typical'), the predictability horizon grows only
logarithmically with the precision of measurement (for positive Lyapunov
exponents, see [2.11]). Thus for each increase in precision by a factor of 10,
say, you may only be able to predict two more time units (measured in units of
the Lyapunov time, i.e. the inverse of the Lyapunov exponent).

More precisely: A map f is chaotic on a compact invariant set S if
(i) f is transitive on S (there is a point x whose orbit is dense in S), and
(ii) f exhibits sensitive dependence on S (see [2.10]).
To these two requirements #DevaneyDevaney adds the requirement that periodic
points are dense in S, but this doesn't seem to be really in the spirit of the
notion, and is probably better treated as a theorem (very difficult and very
important), and not part of the definition.

Usually we would like the set S to be a large set. It is too much to hope for
except in special examples that S be the entire phase space. If the dynamical
system is dissipative then we hope that S is an attractor (see [2.8]) with a
large basin. However, this need not be the case--we can have a chaotic saddle,
an orbit that has some unstable directions as well as stable directions.

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

The notion of chaos seems to conflict with that attributed to Laplace: given
precise knowledge of the initial conditions, it should be possible to predict
the future of the universe. However, Laplace's dictum is certainly true for
any deterministic system, recall [2.3]. The main consequence of chaotic motion
is that given imperfect knowledge, the predictability horizon in a
deterministic system is much shorter than one might expect, due to the
exponential growth of errors. The belief that small errors should have small
consequences was perhaps engendered by the success of Newton's mechanics
applied to planetary motions. Though these happen to be regular on human
historic time scales, they are chaotic on the 5 million year time scale (see
e.g. "Newton's Clock", by Ivars Peterson (1993 W.H. Freeman).

 

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