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

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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.

Manuscrit reĉu le 30 décembre 2001.
In memoriam Michael R. Herman
The present article is the result of a collaboration with Michael Herman, which started in October 1999. He had had the idea of studying the Nekhoroshev theory in the Gevrey category and, using a lemma of his, of producing new examples of unstable orbits for which the instability time could be compared with the distance of the system to integrability. Together we improved both the stability and instability results which he had already obtained, in view of making them match. Michael Herman’s sudden death in November 2000 prevented him from participating to the last developments and to the final writing of a work the main contributor of which he was.