Abstract
For a given a normally hyperbolic invariant manifold, whose stable and unstablemanifolds 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 a Hamiltonian system. This consists in the following steps: (i) computation of the scattering map and of the transition map for the Hamiltonian 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.
Highlights
Consider a normally hyperbolic invariant manifold for a flow or a map, and assume that the stable and unstable manifolds of the normally hyperbolic invariant manifold have a transverse intersection along a homoclinic manifold
There exist pseudo-orbits obtained by alternatively following the inner dynamics and the outer dynamics for some finite periods of time
We develop a toolkit of instruments and techniques to detect true orbits near a normally hyperbolic invariant manifold, that alternatively follow the inner dynamics and the outer dynamics, for all time
Summary
Consider a normally hyperbolic invariant manifold for a flow or a map, and assume that the stable and unstable manifolds of the normally hyperbolic invariant manifold have a transverse intersection along a homoclinic manifold. There exists a true orbit that follows closely these windows, in the prescribed order To apply this lemma for a normally hyperbolic invariant manifold for a map, one needs to reduce the dynamics from the continuous case to the discrete case by considering the return map to a surface of section, and construct the sequence of correctly aligned windows for the resulting normally hyperbolic invariant manifold for the return map. The shadowing lemma plays a key role in reducing the dimensionality of the problem: it requires the verification of topological conditions in the normally hyperbolic invariant manifold for the return map to conclude the existence of trajectories in the phase space of the flow. By making a choice on how closely to follow the homoclinic orbits to the normally hyperbolic invariant manifold, one computes the corresponding transition map.
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