Tracking Invariant Manifolds up to Exponentially Small Errors

Abstract

This work establishes a new tool for proving the existence of multiple-pulse homoclinic orbits in perturbed Hamiltonian systems and general multidimensional singular-perturbation problems. The center-stable and center-unstable manifolds of slow manifolds in these problems intersect transversely at angles that are of the same order as the asymptotically small parameter in the problem, which can be either an amplitude or a frequency. To deal with the difficulties associated with small angles of intersection, we develop the exchange lemma with exponentially small error (ELESE), which is the main technical result of this work. This lemma enables highly accurate tracking of invariant manifolds while orbits on them spend long intervals of time near slow manifolds.