Chaotic Attractor

A chaotic attractor is a set of states in a system's state space with very special properties. First of all, the set is an attracting set. So that the system, starting with its initial condition in the appropriate basin, eventually ends up in the set. Even if the system is perturbed off the attractor, it eventually returns. Second, and most important, once the system is on the attractor nearby states diverge from each other exponentially fast.

Due to this second property, small amounts of noise are amplified. Once sufficiently amplified the noise determines the system's large-scale behavior and the system is then unpredictable.

Chaotic attractors themselves are quite orderly---their often elegant, geometric structures are fixed and unchanging---despite the fact that the trajectories moving within them appear unpredictable. In this sense, the chaotic attractor's geometric shape is the order underlying the apparent chaos.

The geometric shape of a chaotic attractor implements a kind of dough kneading. The local separation of trajectories corresponds to stretching the dough and the global attraction property corresponds to folding the stretched dough back onto itself. One result of the stretch-and-fold aspect of chaotic attractors is that they are fractals---some cross-section of them reveals similar structure on all scales.

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