Smale's horseshoe

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.

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.

The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient.

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.

For a horseshoe map:

The horseshoe map  f  is a diffeomorphism defined from a region S of the plane into itself. The region S is a square capped by two semi-disks. The action of  f  is defined through the composition of three geometrically defined transformations. First the square is contracted along the vertical direction by a factor a < 1/2. The caps are contracted so as to remain semi-disks attached to the resulting rectangle. Contracting by a factor smaller than one half assures that there will be a gap between the branches of the horseshoe. Next the rectangle is stretched horizontally by a factor of 1/a; the caps remain unchanged. Finally the resulting strip is folded into a horseshoe-shape and placed back into S.

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