Attractor

An attractor is a Set of states (points in the Phase Space), invariant under the dynamics, towards which neighboring states in a given Basin of Attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct Basins of Attraction. This restriction is necessary since a Dynamical System may have multiple attractors, each with its own Basin of Attraction.

Conservative systems do not have attractors, since the motion is periodic. For dissipative Dynamical Systems, however, volumes shrink exponentially so attractors have 0 volume in $n$-D phase space.

A stable Fixed Point surrounded by a dissipative region is an attractor known as a Sink. Regular attractors (corresponding to 0 Lyapunov Characteristic Exponents) act as Limit Cycles, in which trajectories circle around a limiting trajectory which they asymptotically approach, but never reach. Strange Attractors are bounded regions of Phase Space (corresponding to Positive Lyapunov Characteristic Exponents) having zero Measure in the embedding Phase Space and a Fractal Dimension. Trajectories within a Strange Attractor appear to skip around randomly.

See also Barnsley's Fern, Basin of Attraction, Chaos Game, Fractal Dimension, Limit Cycle, Lyapunov Characteristic Exponent, Measure, Sink (Map), Strange Attractor