Generic global diffusion for analytic uncoupled a priori unstable systems

TL;DR

This paper proves that in certain uncoupled Hamiltonian systems with a small perturbation, there exist orbits where the momentum variable can change by any amount, demonstrating a form of global diffusion independent of the perturbation size.

Contribution

It introduces a new geometric method to establish global diffusion in uncoupled a priori unstable Hamiltonian systems with generic analytic perturbations.

Findings

Existence of orbits with arbitrarily large momentum change

Diffusion occurs independently of perturbation size and specific perturbation form

Constructive geometric approach based on Poincaré functions and scattering maps

Abstract

We show that given a general uncoupled a priori unstable Hamiltonian \[ \frac12 p^2 + V(q) + G(I) + \epsilon h(p, q, I, \varphi, t), \] where $h$ is a generic Ma\~n\'e analytic function and $\epsilon$ is small enough, there is an orbit for which the momentum $I$ changes by any arbitrarily prescribed value. We call this phenomenon as global diffusion since the size of the change in $I$ is independent of both $\epsilon$ and $h$. The fact that the pendulum and rotor variables are uncoupled is used essentially in our proof. The proof is based on simple and constructive geometrical methods, carefully studying the reduced Poincar\'e functions of the problem which generate the corresponding scattering maps.