Arnold diffusion for a complete family of perturbations

TL;DR

This paper demonstrates Arnold diffusion in a specific Hamiltonian system with a family of small periodic perturbations, using geometric methods involving scattering maps to prove global instability of the action variable.

Contribution

It introduces a geometric approach using scattering maps to prove Arnold diffusion for a complete family of perturbations in a concrete Hamiltonian system.

Findings

Global instability of the action variable for all small perturbations of a specified form.

Existence of simple diffusion paths called highways in the restricted case.

Estimate of diffusion time primarily based on scattering map dynamics.

Abstract

In this work we illustrate the Arnold diffusion in a concrete example---the \emph{a priori} unstable Hamiltonian system of $2+1/2$ degrees of freedom $H(p,q,I,\varphi,s) = p^{2}/2+\cos q -1 +I^{2}/2 + h(q,\varphi,s;\varepsilon)$---proving that for \emph{any} small periodic perturbation of the form $h(q,\varphi,s;\varepsilon) = \varepsilon\cos q\left( a_{00} + a_{10}\cos\varphi + a_{01}\cos s \right)$ ($a_{10}a_{01} \neq 0$) there is global instability for the action. For the proof we apply a geometrical mechanism based in the so-called Scattering map. This work has the following structure: In a first stage, for a more restricted case ($I^*\thicksim\pi/2\mu$, $\mu = a_{10}/a_{01}$), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of the instability for any $\mu$). The bifurcations of the scattering map are also studied as a function of $\mu$. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.