Summary
The text discusses attractors in dynamical systems, which are sets of states that systems tend to evolve towards. Attractors can be points, curves, manifolds, or fractal structures like strange attractors. They play a key role in chaos theory and can be periodic or chaotic. Different types of attractors include fixed points, limit cycles, and strange attractors with fractal structures.
Highlights
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In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve,[2] for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed
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If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).
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Some attractors are known to be chaotic (see strange attractor), in which case the evolution of any two distinct points of the attractor result in exponentially diverging trajectories, which complicates prediction when even the smallest noise is present in the system
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An attractor’s basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor