Strange attractors are a popular subject in dynamics. They are attractors with a complicated geometric structure, in particular attractors that are not simple curves or surfaces. Before looking at strange attractors it is appropriate to look at some geometrically simple ones. From there we will get to strange attractors via an important explicit model situation.

Hyperbolicity of some kind is a characteristic feature of strange attractors, but the study of the most popular examples is difficult because the hyperbolicity of those attractors is of a weaker form than the uniform hyperbolicity discussed in Chapter 10. There is enough of it to produce a great deal of complexity, but not enough to apply the tools from Chapter 10 directly. It is often similar to the nonuniformly hyperbolic behavior that appears in stochastic quadratic maps (the Jakobson–Collet–Eckmann case; see Section 11.4.3.6), and, in fact, those are used as both models and as the basis for perturbative constructions for some popular strange attractors. However, we consider only the Lorenz attractor, where the difficulties are of a different sort and may by described as “uniform hyperbolicity with singularities.” The proof of existence of the Lorenz attractor is one of the most spectacular examples of computer-assisted proofs in continuous mathematics.

FAMILIAR ATTRACTORS

Not all attractors are strangers to us at this point. The simplest ones are attracting fixed points, which were formally introduced in Definition 2.2.22.