# 2.12] What is a Strange Attractor? (nonlinear science)

## Description

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

# 2.12] What is a Strange Attractor? (nonlinear science)

Before Chaos (BC?), the only known attractors (see [2.8]) were fixed

points, periodic orbits (limit cycles), and invariant tori (quasiperiodic

orbits). In fact the famous PoincarĂ©-Bendixson theorem states that for a pair

of first order differential equations, only fixed points and limit cycles can

occur (there is no chaos in 2D flows).

In a famous paper in 1963, Ed Lorenz discovered that simple systems of

three differential equations can have complicated attractors. The Lorenz

attractor (with its butterfly wings reminding us of sensitive dependence (see

[2.10])) is the "icon" of chaos

http://kong.apmaths.uwo.ca/~bfraser/version1/lorenzintro.html. Lorenz showed

that his attractor was chaotic, since it exhibited sensitive dependence.

Moreover, his attractor is also "strange," which means that it is a fractal

(see [3.2]).

The term strange attractor was introduced by Ruelle and Takens in 1970

in their discussion of a scenario for the onset of turbulence in fluid flow.

They noted that when periodic motion goes unstable (with three or more modes),

the typical (see [2.14]) result will be a geometrically strange object.

Unfortunately, the term strange attractor is often used for any chaotic

attractor. However, the term should be reserved for attractors that are

"geometrically" strange, e.g. fractal. One can have chaotic attractors that

are not strange (a trivial example would be to take a system like the cat map,

which has the whole plane as a chaotic set, and add a third dimension which is

simply contracting onto the plane). There are also strange, nonchaotic

attractors (see Grebogi, C., et al. (1984). "Strange Attractors that are not

Chaotic." Physica D 13: 261-268).

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