Abed Bounemoura
08193, Bellaterra, Barcelona, Spain
Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C
Publications:
Bounemoura A.
Generic Perturbations of Linear Integrable Hamiltonian Systems
2016, vol. 21, no. 6, pp. 665681
Abstract
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 $\varepsilon^{1}$) 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).

Bounemoura A., Fischler S.
The Classical KAM Theorem for Hamiltonian Systems via Rational Approximations
2014, vol. 19, no. 2, pp. 251265
Abstract
In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasiperiodic torus, whose frequency vector satisfies the Bruno–Rüssmann condition, in realanalytic nondegenerate 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.

Bounemoura A.
Normal Forms, Stability and Splitting of Invariant Manifolds I. Gevrey Hamiltonians
2013, vol. 18, no. 3, pp. 237260
Abstract
In this paper, we give a new construction of resonant normal forms with a small remainder for nearintegrable Hamiltonians at a quasiperiodic 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 nonanalytic 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.

Bounemoura A.
Normal Forms, Stability and Splitting of Invariant Manifolds II. Finitely Differentiable Hamiltonians
2013, vol. 18, no. 3, pp. 261276
Abstract
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 nearintegrable Gevrey 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.
