Transition map and shadowing lemma for normally hyperbolic invariant manifolds

TL;DR

This paper develops a comprehensive method using the scattering map, transition map, and shadowing lemma to detect trajectories with specific itineraries near normally hyperbolic invariant manifolds, with applications to Hamiltonian systems.

Contribution

It provides a practical guide for applying these tools to identify unstable orbits in Hamiltonian systems, including step-by-step procedures and illustrative examples.

Findings

Effective detection of unstable orbits in Hamiltonian systems.

Application to large gap problem and three-body problem models.

Method demonstrates how to construct sequences of correctly aligned windows.

Abstract

For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in Hamiltonian systems. This consists in the following steps: (i) computation of the scattering map and of the transition map for a flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows. We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem.