A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Abstract

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three‐body problem and the spatial circular restricted three‐body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well‐known “two‐dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower‐dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry‐Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.

Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

“…in [31] that the finite-length diffusing orbits of the iterated function system on Λ correspond to Arnold diffusion in the original diffeomorphism near Λ ∪ Γ, under the assumption of strong transversality of homoclinic intersections. This result was generalised by Gidea, de la Llave and Seara [32] to the orbits of semi-infinite length.…”

“…In [31] it is proved (Lemma 4.4) that if two points on a normally-hyperbolic invariant manifold of a symplectic diffeomorphism are connected by an orbit of the iterated function system (IFS) formed by the inner and scattering maps, then there exists a trajectory of the original diffeomorphism that connects arbitrarily small neighbourhoods of those two points. Lemmas 3.11 and 3.12 of [32] show that the same is true when orbits of an IFS are infinite (in one direction). In this section we prove Theorem 1.1 using these facts.…”

“…Since unperturbed W s,u 1 (Λ) and W s,u 2 (Λ) are symmetric to each other, we will study W s,u 2 (Λ) only. From (32), the unperturbed invariant manifolds W s,u 2 (Λ) coincide and form a 3-dimensional (unperturbed) homoclinic manifold W 2 given by…”

“…The shadowing Lemmas 3.11, 3.12 of [32] imply the existence of a true orbit of the map B that shadows the orbit of the IFS {Φ, S Γ }, so the energy E tends to infinity along this true orbit too (the shadowing lemmas of [32] require compactness of Λ, but it is easy to see that the result remains valid also in the situation where every orbit of the inner map is bounded -so the Poincare recurrence theorem can be used, and this property holds true in our case, as the KAM curves bound every orbit of Φ).…”

Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations

“…With the goal in mind of Arnol'd diffusion, obtaining the information on the effect of homoclinic excursions in the fast variables is a useful information. See [GdlLS14].…”