Our study of complicated dynamics in the course developed mostly by looking at examples. One reason is that this is an effective way to develop important concepts in a natural fashion. But another reason is that those examples almost fully represent the range of phenomena responsible for chaotic behavior. We now return to the horseshoe and explain why this seemingly particular example is an important mechanism that gives rise to chaotic dynamics for some orbits. Specifically, we show how it appears in real systems and that it does so often, and we describe how it has been used as an important tool in deciding fundamental questions in dynamics. We first present the principal mechanism that produces horseshoes and then give an account of the various ways this scenario arises in real problems.

NONLINEAR HORSESHOES

For the discussion of the horseshoe in Section 7.4.4 it was convenient to assume linearity, but as we mentioned in Section 10.2.6, it is not essential. We now introduce nonlinear horseshoes.

It is easy and useful to define horseshoes in arbitrary dimension, but there are several reasons to restrict ourselves to the planar situation. It makes it easier to picture the arguments, and, accordingly, in the development of the theory of dynamical systems the planar case played the leading role. Finally, it is in dimension two that the full topological entropy of a dynamical system can be accounted for by looking only at horseshoes in it.