A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

TL;DR

This paper introduces a new, broadly applicable mechanism for proving the existence of diffusing orbits in nearly integrable Hamiltonian systems, relying solely on outer dynamics and topological methods, without assumptions on inner dynamics.

Contribution

The authors develop a novel diffusion mechanism based on the scattering map and recurrence, applicable to systems of arbitrary degrees of freedom without convexity assumptions.

Findings

Applicable to Hamiltonians of any degree of freedom

Can establish diffusion in non-convex systems

Easy to verify in concrete examples

Abstract

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the `scattering map'. We find pseudo-orbits of the scattering map that keep advancing in some privileged direction. Then we use the recurrence property of the `inner dynamics', restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics. This method differs, in several crucial aspects, from earlier works. Unlike the well known `two-dynamics' approach, the method we present relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems. We include several applications, such as bridging large gaps in a priori unstable models in any dimension, and establishing diffusion in cases when the inner dynamics is a non-twist map.