Horseshoe map Guide, Meaning , Facts, Information and Description

In dynamical systems theory, the horseshoe map was introduced by Stephen Smale as a simple model of complex behavior. It is given on the unit square by the formula:

(1)     ,

where:

(2)    

This map serves as a model for general behavior at transverse homoclinic points, and can be fairly easily shown to have an invariant compact set on which it acts as a shift map.

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