Types of Attractors
Attractors are parts of the phase space of the dynamical system. Until the 1960s, as evidenced by textbooks of that era, attractors were thought of as being simple geometrical subsets of the phase space: points, lines, surfaces, volumes. The (topologically) wild sets that had been observed were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.
Two simple attractors are the fixed point and the limit cycle. There can be many other geometrical sets that are attractors. When these sets (or the motions on them), are harder to describe than the classical geometric objects, then the attractor is a strange attractor, as described in the section below.
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