Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem. Our stability result generalizes those by Lochak-Neishtadt and Poschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-alpha Hamiltonians, alpha greater than or equal to 1, we prove that one can choose a = 1/2nalpha as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting alpha = 1). On the other hand, for alpha > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n < 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and-where the speed of drift is characterized by the exponent 1/2(n - 2)alpha. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold's mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.