Double exponential stability of quasi-periodic motion in Hamiltonian systems

Abed Bounemoura (1, 2) , Bassam Fayad (3) , Laurent Niederman (4, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
4 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is doubly exponentially stable in the sense that nearby solutions remain close to the torus for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the torus. We also prove that for an arbitrary small perturbation of a generic integrable Hamiltonian system, there is a set of almost full positive Lebesgue measure of KAM tori which are doubly exponentially stable. Our results hold true for real-analytic but more generally for Gevrey smooth systems.

Domaines: Systèmes dynamiques [math.DS]

Abed Bounemoura, Bassam Fayad, Laurent Niederman. Double exponential stability of quasi-periodic motion in Hamiltonian systems. Communications in Mathematical Physics, 2017, 350 (1), pp.361-386. ⟨10.1007/s00220-016-2782-9⟩. ⟨hal-01252300⟩