Generic Nekhoroshev theory without small divisors

Bounemoura, Abed; Niederman, Laurent

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 277-324.

Dans cet article, nous présentons une nouvelle approche de la théorie de Nekhoroshev pour un hamiltonien intégrable générique, qui évite complètement le problème des petits diviseurs. La preuve est une extension d’une méthode introduite par Lochak, elle n’utilise que des moyennisations périodiques et de l’approximation diophantienne simultanée, ainsi que des arguments géométriques introduit par le second auteur. Notre méthode permet également d’obtenir des résultats de stabilité pour des hamiltoniens génériques non-analytiques, ainsi que de nouveaux résultats de stabilité au voisinage des tores invariants linéairement stables.

In this article, we present a new approach of Nekhoroshev’s theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak, it combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.

Classification : 37J25, 37J40, 70H08, 70H09, 70K45, 70K60, 70K65
Keywords: Hamiltonian systems, perturbation of integrable systems, effective stability
Mot clés : systèmes hamiltoniens, perturbation de systèmes intégrables, stabilité effective

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