Arnold diffusion in multidimensional a priori unstable Hamiltonian systems

TL;DR

This paper demonstrates that in certain multidimensional Hamiltonian systems, generic small perturbations induce trajectories that cross resonances, confirming the occurrence of Arnold diffusion with quantifiable velocity in a broad class of near-integrable systems.

Contribution

It proves that Arnold diffusion occurs for generic perturbations in a priori unstable systems near low-order resonances, with trajectories crossing the resonance manifold and moving along any smooth action space curve.

Findings

Diffusion trajectories cross the resonance manifold.

Diffusion occurs for generic perturbations.

Average velocity of diffusion is of order ε/|log ε|.

Abstract

We study the Arnold diffusion in a priori unstable near-integrable systems in a neighbourhood of a resonance of low order. We consider a non-autonomous near-integrable Hamiltonian system with $n+1/2$ degrees of freedom, $n\ge 2$. Let the Hamilton function $H$ of depend on the parameter $\varepsilon$, for $\varepsilon=0$ the system is integrable and has a homoclinic asymptotic manifold $\Gamma$. Our main result is that for small generic perturbation in an $\varepsilon$-neighborhood of $\Gamma$ there exist trajectories the projections of which on the space of actions cross the resonance. By ``generic perturbations'' we mean an open dense set in the space of $C^r$-smooth functions $\frac{d}{d\varepsilon}\big|_{\varepsilon=0} H$, $r=r_0,r_0+1,\ldots,\infty,\omega$. Combination of this result with results of \cite{DT} answers the main questions on the Arnold diffusion in a priori unstable case: the diffusion takes place for generic perturbation, diffusion trajectories can go along any smooth curve in the action space with average velocity of order $\varepsilon/|\log \varepsilon|$.