Semiclassical limit for perturbations of non-resonant rotators
Slow and fast systems with Hamiltonian reduced problems.
Small divisors and large multipliers
We study germs of singular holomorphic vector fields at the origin of of which the linear part is -resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms....
Smooth Gevrey normal forms of vector fields near a fixed point
We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth -Gevrey vector field with an hyperbolic linear part admits a smooth -Gevrey transformation to a smooth -Gevrey normal form. The Gevrey order depends on...
Soluzioni periodiche di PDEs Hamiltoniane
Presentiamo nuovi risultati di esistenza e molteplicità di soluzioni periodiche di piccola ampiezza per equazioni alle derivate parziali Hamiltoniane. Otteniamo soluzioni periodiche di equazioni «completamente risonanti» aventi nonlinearità generali grazie ad una riduzione di tipo Lyapunov-Schmidt variazionale ed usando argomenti di min-max. Per equazioni «non risonanti» dimostriamo l'esistenza di soluzioni periodiche di tipo Birkhoff-Lewis, mediante un'opportuna forma normale di Birkhoff e realizzando...
Some perturbation results for non-linear problems
We discuss the existence of closed geodesic on a Riemannian manifold and the existence of periodic solution of second order Hamiltonian systems.
Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom.
Stabilité ou instabilité des points fixes elliptiques
Stability and instability for Gevrey quasi-convex near-integrable hamiltonian systems
Stability in a ball-partition problem.
Stochastic Stability in Some Chaotic Dynamical Systems.
Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles
This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density,...
Strong stochastic stability and rate of mixing for unimodal maps
Sur la persistance des courbes invariantes pour les dynamiques holomorphes fibrées lisses
En s’appuyant sur un théorème des fonctions implicites de Hamilton, nous montrons la persistance d’une courbe invariante indifférente pour une dynamique holomorphe fibrée de classe . Une condition diophantienne sur la paire de nombres de rotation est demandée. On montre également que cette condition est optimale.
Systèmes bihamiltoniens en dimension impaire