RUS  ENG
Полная версия
ЖУРНАЛЫ // Regular and Chaotic Dynamics // Архив

Regul. Chaotic Dyn., 2013, том 18, выпуск 6, страницы 719–731 (Mi rcd166)

Эта публикация цитируется в 1 статье

Generic Super-exponential Stability of Elliptic Equilibrium Positions for Symplectic Vector Fields

Laurent Niedermanab

a Topologie et Dynamique, Laboratoire de Mathématiques d’Orsay (UMR-CNRS 8628), Université Paris Sud, 91405 Orsay Cedex, Paris, France
b Astronomie et Systèmes Dynamiques, Institut de Mécanique Céleste et de calcul des éphémérides (UMR-CNRS 8028), Observatoire de Paris, 75014 Paris, France

Аннотация: In this article, we consider linearly stable elliptic fixed points (equilibrium) for a symplectic vector field and prove generic results of super-exponential stability for nearby solutions. We will focus on the neighborhood of elliptic fixed points but the case of linearly stable isotropic reducible invariant tori in a Hamiltonian system should be similar.
More specifically, Morbidelli and Giorgilli have proved a result of stability over superexponentially long times if one considers an analytic Lagrangian torus, invariant for an analytic Hamiltonian system, with a diophantine translation vector which admits a sign-definite torsion. Then, the solutions of the system move very little over times which are super-exponentially long with respect to the inverse of the distance to the invariant torus.
The proof proceeds in two steps: first one constructs a high-order Birkhoff normal form, then one applies the Nekhoroshev theory. Bounemoura has shown that the second step of this construction remains valid if the Birkhoff normal form linked to the invariant torus or the elliptic fixed point belongs to a generic set among the formal series.
This is not sufficient to prove this kind of super-exponential stability results in a general setting. We should also establish that the most strongly non resonant elliptic fixed point or invariant torus in a Hamiltonian system admits Birkhoff normal forms fitted for the application of the Nekhoroshev theory. Actually, the set introduced by Bounemoura is already very large but not big enough to ensure that a typical Birkhoff normal form falls into this class. We show here that this property is satisfied generically in the sense of the measure (prevalence) through infinitedimensional probe spaces (that is, an infinite number of parameters chosen at random) with methods similar to those developed in a paper of Gorodetski, Kaloshin and Hunt in another setting.

Ключевые слова: Hamiltonian systems, perturbation of integrable systems, effective stability.

MSC: 37J25, 37J40, 70H20, 70K45, 70K60, 70K65

Поступила в редакцию: 16.09.2013
Принята в печать: 14.11.2013

Язык публикации: английский

DOI: 10.1134/S1560354713060117



Реферативные базы данных:


© МИАН, 2024