Stable Manifolds and Crises

Abstract

WE INTRODUCED the subject of stable and unstable manifolds for saddles of planar maps in Chapter 2. There we emphasized that Poincaré used properties of these sets to predict when systems would contain complicated dynamics. He showed that if the stable and unstable manifolds crossed, there was behavior that we now call chaos. For a saddle fixed point in the plane, these “manifolds” are curves that can be highly convoluted. In general, we cannot hope to describe the manifolds with simple formulas, and we need to investigate properties that do not depend on this knowledge. Recall that for an invertible map of the plane and a fixed point saddle p, the stable manifold of p is the set of initial points whose forward orbits (under iteration by the map) converge to p, and the unstable manifold of p is the set whose backward orbits (under iteration by the inverse of the map) converge to p.