CALCULATING STABLE AND UNSTABLE MANIFOLDS

Abstract

A numerical procedure is described for computing the successive images of a curve under a diffeomorphism of R^N. Given a tolerance ε, we show how to rigorously guarantee that each point on the computed curve lies no further than a distance ε from the "true" image curve. In particular, if ε is the distance between adjacent points (pixels) on a computer screen, then a plot of the computed curve coincides with the true curve within the resolution of the display. A second procedure is described to minimize the amount of computation of parts of the curve that lie outside a region of interest. We apply the method to compute the one-dimensional stable and unstable manifolds of the Hénon and Ikeda maps, as well as a Poincaré map for the forced damped pendulum.