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This article is cited in 1 scientific paper (total in 1 paper)
Generic Perturbations of Linear Integrable Hamiltonian Systems
Abed Bounemouraab a CNRS – CEREMADE, Université Paris Dauphine
Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
b IMCCE, Observatoire de Paris,
77 avenue Denfert-Rochereau, 75014 Paris, France
Abstract:
In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of $\varepsilon^{-1}$) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).
Keywords:
Hamiltonian perturbation theory, KAM theory, Nekhoroshev theory, Arnold diffusion.
Received: 02.09.2016 Accepted: 02.10.2016
Citation:
Abed Bounemoura, “Generic Perturbations of Linear Integrable Hamiltonian Systems”, Regul. Chaotic Dyn., 21:6 (2016), 665–681
Linking options:
https://www.mathnet.ru/eng/rcd217 https://www.mathnet.ru/eng/rcd/v21/i6/p665
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Abstract page: | 179 | References: | 49 |
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