Chapter 22

The sets generated as the long time attractors of “physical” dynamical systems described by ordinary differential equations or discrete evolution equations in the case of maps such as the Hénon map appear to be very complicated, often with unpleasant properties such as the lack of robustness (the parameters leading to chaotic and nonchaotic solution may be intertwined on an arbitrarily small scale). Few proofs are available for the conjectured properties of these sets. To make mathematical progress it is necessary to make restrictions on the type of dynamical systems considered, that go far beyond standard smoothness assumptions. Often the assumptions needed to construct proofs are so strict that it is known that all common “physical” attractors violate the assumptions. (However, numerical tests of the results proven for the restricted systems often show that they apply, at least as an excellent approximation, to the physical systems.) In this chapter the goal is to present some of the flavor of this mathematical approach, since many of our ideas on chaotic systems have arisen in this context. A key assumption in much of the mathematical development is the notion of “hyperbolicity”. To define this concept it we first introduce the general idea of stable and unstable manifolds. Useful references are: Eckmann and Ruelle [1], sections IIIE, F; Ott [2], sections 4.3 and 9.5; Aligood et al. [3], section 2.6 and chapter 10; and Guckenheimer and Holmes [4] section 5.2.