Arnold diffusion in Hamiltonian systems on infinite lattices

TL;DR

This paper demonstrates energy transfer along prescribed paths in an infinite lattice of weakly coupled penduli using Arnold diffusion, extending classical finite-dimensional methods to infinite-dimensional Hamiltonian systems.

Contribution

It extends Arnold diffusion techniques to infinite lattices, constructing orbits transferring energy along paths with decaying long-range couplings in infinite-dimensional Hamiltonian systems.

Findings

Constructed orbits transfer energy along prescribed lattice paths.

Extended invariant manifold theory to infinite-dimensional hyperbolic tori.

Developed a Lambda Lemma for infinite coupled map lattices.

Abstract

We consider a system of infinitely many penduli on an $m$-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasiperiodic tori with transverse heteroclinic orbits. We implement this approach in an infinite dimensional setting, both in the space of bounded $\mathbb{Z}^m$-sequences and in spaces of decaying $\mathbb{Z}^m$-sequences. Key steps in the proof are an invariant manifold theory for hyperbolic tori and a Lambda Lemma for infinite dimensional coupled map lattices with decaying interaction.