Horseshoe map

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)

(x_{n+1},y_{n+1}) = H((x_n,y_n))

, where: :(2)

H(x,y)=\\left\\{\\begin{matrix} (2x,\\frac {y} {2}), & \\mbox{if }x< \\frac {1} {2} \\\\ (2x-1,1-\\frac {y} {2}), & \\mbox{if }x\\ge\\frac {1} {2} \\end{matrix}\ight.

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. Using a few hundred mirrors, one can build an optical universal Turing machine in one's backyard, using the Horseshoe map. Category:Dynamical systems

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Horseshoe map

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