Arnold diffusion for smooth convex systems of two and a half degrees of freedom

Abstract

In the present note we announce a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. Let be a 2-dimensional torus and B2 be the unit ball around the origin in . Fix ρ > 0. Our main result says that for a ‘generic’ time-periodic perturbation of an integrable system of two degrees of freedom , with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in , namely, a ρ-neighborhood of the orbit contains .Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are the usage of crumpled normally hyperbolic invariant cylinders from [], flower and simple normally hyperbolic invariant manifolds from [] as well as their kissing property at a strong double resonance. This allows us to build a ‘connected’ net of three-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of the Mather variational method [] equipped with weak KAM theory [], proposed by Bernard in [].