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.

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 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.

For a horseshoe map:

• 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.

The horseshoe map

The horseshoe map <math>f<math> is a diffeomorphism defined from a region <math>S<math> of the plane into itself. The region <math>S<math> is a square capped by two semi-disks. The action of <math>f<math> is defined through the composition of three geometrically defined transformations. First the square is contracted along the vertical direction by a factor <math>a<1/2<math>. 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 by a factor of <math>1/a<math>; the caps remain unchanged. Finally the resulting strip is folded into a horseshoe-shape and placed back into <math>S<math>.

The interesting part of the dynamics is the image of the square into itself. Once that part is defined, the map can be extended to a diffeomorphism by defining its action on the caps. The caps are made to contract and eventually map inside one of the caps (the left one in the figure). The extension of f to the caps adds a fixed point to the non-wandering set of the map. The curved region of the horseshoe should not map back into the square.

By folding the contracted and stretched square in different ways, other types of horseshoe maps are possible.

When the action on the square is extended to a diffeomorphism, the extension cannot always be done on the plane. For example, the map on the right needs to be extended to a diffeomorphism of the sphere by using a cap that wraps around the equator.

Invariant set

A set of points that maps to itself is an invariant set. Any periodic orbit (including fixed points) of the horseshoe map forms an invariant set. But there is a maximal invariant set <math>\Lambda<math> that includes more than all the periodic orbits. Points in the caps do not contribute to the invariant set. The only invariant subset in the caps—the single fixed point in the left cap—is excluded by convention. The other points, if any, must be in the square.

Under forward iterations of the horseshoe map, the original square gets mapped into a series of horizontal strips. The points in these horizontal strips come from vertical strips in the original square. Any point that is part of an invariant set must lie in the intersection of the vertical and horizontal strips. Let <math>S_0<math> be the original square, map it forward n times, and consider only the points that fall back into the square S₀, which is a set of horizontal stripes

<math>H_n = f^n(S_0)<math> ∩<math>S_0<math>.

The points in the horizontal stripes came from the vertical stripes

<math>V_n = f^{-n}(H_n)<math>,

which are the horizontal strips <math>H_n<math> mapped backwards n times. That is, a point in V_n will, under n iterations of the horseshoe, end up in the set <math>H_n<math> of vertical strips.

The intersection of the horizontal and vertical stripes, <math>H_n<math> ∩ <math>V_n<math>, are squares that converge in the limit <math>n<math> → ∞ to the invariant set <math>\Lambda<math>. The horseshoe map acts on the compact set <math>\Lambda<math> as a shift map. The set <math>\Lambda<math> is homeomorphic to the Cartesian square of the Cantor set.

The horseshoe map is an Anosov diffeomorphism that serves as a model for general behavior at transverse homoclinic points.

References

• Stephen Smale (1967). "Differentiable dynamical systems". Bulletin of the American Mathematical Society 73: 747–817.

• P. Cvitanović, G. Gunaratne, and I. Procaccia (1988). "Topological and metric properties of Hénon-type strange atractors". Physical Review A 38: 1503–1520.

• André de Carvalho (1999). "Pruning fronts and the formation of horseshoes". Ergodic theory and dynamical systems 19: 851–894.

• André de Carvalho and Toby Hall (2002). "How to prune a horseshoe". Nonlinearity 15: R19-R68.