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Strange Attractors

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.