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

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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