Horseshoe map

In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe.

Related Topics:
Mathematics - Chaos theory - Core example - Dynamical systems - Stephen Smale - Van der Pol oscillator

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Most points eventually leave the square under the action of the map. They go to the side caps where they will, under iteration, converge to a fixed point in one of the caps. The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map.

Related Topics:
Fixed point - Fractal - Invariant set

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The squishing, stretching and folding of the horseshoe map are the essential elements that must be present in a chaotic system. In the horseshoe map the squeezing and stretching are uniform. They compensate each other so that the area of the square does not change. The folding is done neatly, so that the orbits that remain forever in the square can be simply described.

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For a horseshoe map:

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• there is an infinite number of periodic orbits;
• periodic orbits of arbitrarily long period exist;
• the number or periodic orbits grows exponentially with the period; and
• close to any point of the fractal invariant set there is a point of a periodic orbit.

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Horseshoe map: The horseshoe map ►