Arnold diffusion in nearly integrable Hamiltonian systems

TL;DR

This paper proves that Arnold diffusion, a phenomenon where trajectories in nearly integrable Hamiltonian systems can drift across large regions of phase space, is a generic feature in three-degree-of-freedom convex systems with typical small perturbations.

Contribution

It establishes the genericity of Arnold diffusion in nearly integrable convex Hamiltonian systems with three degrees of freedom under typical perturbations.

Findings

Existence of connecting orbits between prescribed small neighborhoods within the same energy level.

Diffusion occurs for energy levels above the minimal average action.

The result applies to a broad class of convex Hamiltonian systems with small perturbations.

Abstract

In this paper, Arnold diffusion is proved to be generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$ H(x,y)=h(y)+\epsilon P(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3. $$ Under typical perturbation $\epsilon P$, the system admits "connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\min\alpha$.