Smale Horseshoe

  In section 4.6.2 we stressed the importance of analyzing the orbit structure arising within a homoclinic tangle. From a topological and physical point of view, analyzing the orbit structure primarily means answering two questions:

1. What are the relative locations of the periodic orbits?
2. How are the stable and unstable manifolds interwoven within a homoclinic tangle?

By studying the horseshoe example we will see that these questions are intimately connected.

In sections 2.11 and 2.12 we answered the first question for the one-dimensional quadratic map by using symbolic dynamics. For the special case of a chaotic hyperbolic invariant set (to be discussed in section 4.9), Smale  found an answer to both of the above questions for maps of any dimension. Again, the solution involves the use of symbolic dynamics.

The prototypical example of a chaotic hyperbolic invariant set is the Smale horseshoe   [11]. A detailed knowledge of this example is essential for understanding chaos. The Smale horseshoe (like the quadratic map for ) is an example of a chaotic repeller. It is not an attractor. Physical applications properly focus on attractors since these are directly observable. It is, therefore, sometimes believed that the chaotic horseshoe has little use in physical applications. In Chapter 5 we will show that such a belief could not be further from the truth. Remnants of a horseshoe (sometimes called the proto-horseshoe [11]) are buried within a chaotic attractor. The horseshoe (or some other variant of a hyperbolic invariant set) acts as the skeleton on which chaotic and periodic orbits are organized. To quote Holmes , ``Horseshoes in a sense provide the `backbone' for the attractors [12].'' Therefore, horseshoes are essential to both the mathematical and physical analysis of a chaotic system.

• From Tangles to Horseshoes
• Horseshoe Map
• Symbolic Dynamics
• From Horseshoes to Tangles