Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders

TL;DR

This paper proves the existence of Arnold diffusion in nearly integrable Hamiltonian systems with arbitrary degrees of freedom, using geometric and variational methods to construct invariant cylinders and demonstrate diffusive orbits.

Contribution

It introduces a new proof of Arnold diffusion for systems with any number of degrees of freedom, constructing large normally hyperbolic invariant cylinders and applying variational techniques.

Findings

Existence of diffusive orbits in generic perturbations of nearly integrable systems.

Construction of large, normally hyperbolic invariant cylinders of limited regularity.

Application of Mather variational methods to demonstrate Arnold diffusion.

Abstract

In the present paper we prove a form of Arnold diffusion. The main result says that for a "generic" perturbation of a nearly integrable system of arbitrary degrees of freedom $n\ge 2$ \[ H_0(p)+\eps H_1(\th,p,t),\quad \th\in \T^n,\ p\in B^n,\ t\in \T=\R/\T, \] with strictly convex $H_0$ there exists an orbit $(\th_{\e},p_{e})(t)$ exhibiting Arnold diffusion in the sens that [\sup_{t>0}\|p(t)-p(0) \| >l(H_1)>0] where $l(H_1)$ is a positive constant independant of $\e$. Our proof is a combination of geometric and variational methods. We first build 3-dimensional normally hyperbolic invariant cylinders of limited regularity, but of large size, extrapolating on \cite{Be3} and \cite{KZZ}. Once these cylinders are constructed we use versions of Mather variational method developed in Bernard \cite{Be1}, Cheng-Yan \cite{CY1, CY2}.