Diffusion and stability in perturbed non-convex integrable systems

The Nekhoroshev theorem has become an important tool for explaining the long-term stability of many quasi-integrable systems of interest in physics. The action variables of systems that satisfy the hypotheses of the Nekhoroshev theorem remain close to their initial value up to very long times, which grow exponentially as an inverse power of the perturbation's norm. In this paper we study some of the simplest systems that do not satisfy the hypotheses of the Nekhoroshev theorem. These systems can be represented by a perturbed Hamiltonian whose integrable part is a quadratic non-convex function of the action variables. We study numerically the possibility of action diffusion over short times for these systems (continuous or maps) and we compare it with the so-called Arnold diffusion. More precisely we find that, except for very special non-convex functions, for which the effect of non-convexity concerns low-order resonances, the diffusion coefficient decreases faster than a power law (and possibly exponentially) of the perturbation's norm. According to theory, we find that the diffusion coefficient as a function of the perturbation's norm decreases more slowly than in the convex case.