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

Abstract

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 Pöschel, 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-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα 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 α = 1).On the other hand, for α > 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)α. 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.

KAM theorem for Gevrey Hamiltonians

“…These inequalities yield effective stability of the quasiperiodic motion near the invariant tori as in [10]. Effective stability of the action along all the trajectories for Gevrey smooth Hamiltonians has been obtained recently in in [7]. The importance of the Gevrey category for that kind of problems is indicated by Lochak [6].…”

An example of Arnold diffusion for near-integrable Hamiltonians

“…Our primary goal is to present the geometrical essence of Arnold diffusion in what seems to be the simplest possible setting. A rich literature (see, e.g., [3], [8], [19]) on Arnold diffusion goes far beyond the example discussed here. We outline our construction in this Introduction, giving details in later sections.…”

“…Marco and Sauzin [19], in a collaboration started with Herman, presented such a family in the class of Gevrey 4 near-integrable Hamiltonians.…”

“…The famous Nekhoroshev stability estimates [27] show that for analytic near-integrable Hamiltonian systems with quasiconvex unperturbed Hamiltonian, diffusion time is at least exponentially long. Marco and Sausin [19] extend Nekhoroshev estimates to the Gevrey class and construct a family of examples in this class with optimal, i.e., maximal possible, speed of diffusion. In [9] Bourgain and Kaloshin give an open family of analytic examples which possess trajectories diffusing with optimal speed, thus proving optimality of the Nekhoroshev stability estimates [27].…”

“…To prove the estimate (19), notice that the minimum with respect to τ for |τ −T | < 4 √ ε is bounded from below by the one for |τ −T | < 25 √ ε. Thus it suffices to prove (19) T 2 .…”