Diffusion along chains of normally hyperbolic cylinders

TL;DR

This paper introduces a geometric method to prove the existence of drifting orbits along chains of invariant cylinders in Hamiltonian systems, extending results on Arnold diffusion to more general, non-convex cases.

Contribution

The paper develops a new geometric framework for establishing Arnold diffusion in Hamiltonian systems with invariant cylinders, applicable beyond convex cases and providing two proof strategies.

Findings

Existence of diffusing orbits along chains of cylinders under specific conditions.

Applicable to large families of perturbations of Tonelli Hamiltonians.

Provides both abstract and constructive proofs for the diffusion mechanism.

Abstract

We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on $\mathbb{A}^3$. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are $3$-dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections. Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders -- i.e., the existence of Poincar\'e sections with the return maps satisfying a tilt condition -- and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders -- i.e., certain compatibility conditions between the tilt map and the homoclinic maps attached to its essential invariant circles. We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.