Abed Bounemoura

Optimal Stability and Instability for Near-Linear Hamiltonians

In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover, in the same spirit as the notion of KAM stable integrable Hamiltonians, we will introduce a notion of effectively stable integrable Hamiltonians, conjecture a characterization of these Hamiltonians and show that our result proves this conjecture in the linear case.

A geometrical proof of the persistence of normally hyperbolic submanifolds

We present a simple, computation-free and geometrical proof of the following classical result: for a diffeomorphism of a manifold, any compact submanifold that is invariant and normally hyperbolic persists under small perturbations of the diffeomorphism. The persistence of a Lipschitz invariant submanifold follows from an application of the Schauder fixed point theorem to a graph transform, while smoothness and uniqueness of the invariant submanifold are obtained through geometrical arguments. Moreover, we also prove a new result on the persistence and regularity of ‘topologically’ normally hyperbolic submanifolds, but without any uniqueness statement.

Effective Stability for Gevrey and Finitely Differentiable Prevalent Hamiltonians

For perturbations of integrable Hamiltonian systems, the Nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time, provided the integrable part satisfies a steepness condition and the system is analytic. This fundamental result has been extended in two distinct directions. The first one is due to Niederman, who showed that under the analyticity assumption, the result holds true for a prevalent class of integrable systems which is much wider than the steep systems. The second one is due to Marco-Sauzin but it is limited to quasi-convex integrable systems, for which they showed exponential stability if the system is assumed to be only Gevrey regular. If the system is finitely differentiable, we showed polynomial stability, still in the quasi-convex case. The goal of this work is to generalize all these results in a unified way, by proving exponential or polynomial stability for Gevrey or finitely differentiable perturbations of prevalent integrable Hamiltonian systems.

Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians

In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging. It enables us to deal with non-analytic Hamiltonians, and in this first part we will focus on Gevrey Hamiltonians and derive normal forms with an exponentially small remainder. This extends a result which was known for analytic Hamiltonians, and only in the periodic case for Gevrey Hamiltonians. As applications, we obtain an exponentially large upper bound on the stability time for the evolution of the action variables and an exponentially small upper bound on the splitting of invariant manifolds for hyperbolic tori, generalizing corresponding results for analytic Hamiltonians.

Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians

This paper is a sequel to “Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians”, in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrableGevrey Hamiltonians at a quasiperiodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable Hamiltonians, and we derive normal forms with a polynomially small remainder. As applications, we obtain a polynomially large upper bound on the stability time for the evolution of the action variables and a polynomially small upper bound on the splitting of invariant manifolds for hyperbolic tori.

The Classical KAM Theorem for Hamiltonian Systems via Rational Approximations

In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in [4] for perturbations of constant vector fields on the torus.

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ε−1) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).