Stability of Nearly Integrable Convex Hamiltonian Systems Over Exponentially Long Times

Abstract

In the present paper, we shall study the stability of a near integrable Hamil-tonian system over finite but very long intervals of times. So we look at the system governed by the Hamiltonian:

$$H(p,q) = h(p) + f(p,q)with(p,q) \in {R^n}x{T^n},T = R/Z$$

where (p, q) are action-angle variables of the integrable Hamiltonian h. We assume that H is analytic over some domain G × T ⁿ (G ⊂ R ⁿ a “nice” domain say convex open) and that h is a convex function ( ∇ ² h(p) is a sign definite symmetric matrix). The perturbation f is of size ε (see below).