Hyperbolicity

  The horseshoe map is just one of an infinity of possible return maps (chaotic forms) that can be successfully analyzed using symbolic dynamics. A few other possibilities are shown in Figure 4.23.

  
Figure 4.23: Examples of horseshoe-like maps that generate hyperbolic invariant sets.

Each different return map generates a different homoclinic tangle, but all these tangles can be dissected using symbolic dynamics. All these maps are similar to the horseshoe because they are topologically conjugate to an appropriate symbol space with a shift map. All these maps possess an invariant Cantor set . These invariant sets all possess a special property that ensures their successful analysis using symbolic dynamics, namely, hyperbolicity.

Recall our definition of a hyperbolic point. A fixed point of a map is hyperbolic  if none of the moduli of its eigenvalues exactly equals one. The notion of a hyperbolic invariant set is a generalization of a hyperbolic fixed point. Informally, to define a hyperbolic set we extend this property of ``no eigenvalues on the unit circle'' to each point of the invariant set. In other words, there is no center manifold for any point of the invariant set. Technically, a set arising in a diffeomorphism is a hyperbolic set  if

1. there exists a pair of tangent lines and for each which are preserved by Df(x);
2. and vary smoothly with x;
3. there is a constant such that for all and for all .

A more complete mathematical discussion of hyperbolicity can be found in Devaney [13] or Wiggins [14]. A general mathematical theory for the symbolic analysis of chaotic hyperbolic invariant sets is described by Devaney [13] and goes under the rubric of ``subshifts and transition matrices.''

Like the horseshoe, the chaotic hyperbolic invariant sets encountered in the mathematics literature are often chaotic repellers. In physical applications, on the other hand, we are more commonly faced with the analysis of nonhyperbolic chaotic attractors. The extension of symbolic analysis from the (mathematical) hyperbolic regime to the (more physical) nonhyperbolic regime is still an active research question [15].