Smale’s horseshoe map

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.

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.

It is special interesting because is able to show simply how too close points can have very different orbits.

For a horseshoe map:

• there are an infinite number of periodic orbits;
• periodic orbits of arbitrarily long period exist;
• the number of 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.

See also

Dynamical systems

Papers

• Rössler, O. E. (1977). Horseshoe-map chaos in the Lorenz equation. Physics Letters A, 60(5), 392-394.
• David Ruelle (2006). What is a strange attractor?. Notices of the American Mathematical Society 53 (7): 764-765.
• Stephen Smale (1967). Differentiable dynamical systems. Bulletin of the American Mathematical Society 73 (6): 747-817.