Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems

Abstract

By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative Henon map, both of which exhibit strong homoclinic chaos. In terms of the asymptotic expansion, a simple formulation is presented to give the first homoclinic point in the double- well map. Even a truncated expansion of the unstable manifold is shown to reproduce the well-known many-leaved (fractal) structure of the strange attractor in the Henon map.