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09 What is a strange attractor?




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This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.

09 What is a strange attractor?

A strange attractor is the limit set of a chaotic trajectory. A
strange attractor is an attractor that is topologically distinct from
a periodic orbit or a limit cycle. A strange attractor can be
considered a fractal attractor. An example of a strange attractor is
the Henon attractor.

Consider a volume in phase space defined by all the initial conditions
a system may have. For a dissipative system, this volume will shrink
as the system evolves in time (Liouville's Theorem). If the system is
sensitive to initial conditions, the trajectories of the points
defining initial conditions will move apart in some directions, closer
in others, but there will be a net shrinkage in volume. Ultimately,
all points will lie along a fine line of zero volume. This is the
strange attractor. All initial points in phase space which ultimately
land on the attractor form a Basin of Attraction. A strange attractor
results if a system is sensitive to initial conditions and is not
conservative.

Note: While all chaotic attractors are strange, not all strange
attractors are chaotic.

Reference:

1. Grebogi, et al., Strange Attractors that are not Chaotic, "Physica
D" 13 (1984), pp. 261-268.

 

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