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2.12] What is a Strange Attractor? (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.

2.12] 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 Poincaré-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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