Large Normally Hyperbolic Cylinders in a priori Stable Hamiltonian Systems

Patrick Bernard (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DMA - Département de Mathématiques et Applications - ENS Paris

We prove the existence of normally hyperbolic cylinders in a pri-ori stable Hamiltonian systems the size of which is bounded from below independently of the size of the perturbation. This result should have applications to the study of Arnold's diffusion. A major problem in dynamical systems consists in studying the Hamiltonian systems on T n × R n of the form H(q, p) = h(p) − 2 G(t, q, p), (t, q, p) ∈ T × T n × R n. (H) Here should be considered as a small perturbation parameter, we put a square because the sign of the perturbation will play a role in our discussion. In the unperturbed system (= 0) the momentum variable p is constant. We want to study the dynamics of the perturbed system in the neighborhood of a torus {p = p 0 }, corresponding to a resonant frequency. There is no loss of generality in assuming that the frequency is of the form ∂h(p 0) = (ω, 0) ∈ R

Domaines: Systèmes dynamiques [math.DS]

Patrick Bernard. Large Normally Hyperbolic Cylinders in a priori Stable Hamiltonian Systems. Annales de l'Institut Henri Poincaré, 2010, 11 (5), pp.929-942. ⟨10.1007/s00023-010-0040-9⟩. ⟨hal-00438208v3⟩