Symbolic Description of Orbits: The Stable Manifolds of C₁ and C₂

Abstract

At various points in the last four chapters we have found it useful to be able to describe periodic orbits and trajectories with sequences of two symbols. When we studied homoclinic explosions in Chapter 2, we saw that we were rigorously justified in describing the periodic orbits and trajectories born (or destroyed) in these bifurcations with sequences of two symbols; each symbol corresponded to one of the two tubes surrounding (at the critical r-value) the two branches of the homoclinic orbit, and trajectories were assigned symbolic sequences according to the order in which they journeyed through these tubes. These descriptions were only local. In Chapters 3 through 5, we found we had need for a global method of describing orbits and trajectories. In Chapters 3 and 5, we defined symbol sequences according to the order in which trajectories intersected two halves of some suitable return surface. In Chapter 4, we assigned symbol sequences to periodic orbits according to whether their successive local maxima in the z-variable lay in x > 0 or x < 0. These two methods appeared to be equivalent.