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Generic perturbations of linear integrable Hamiltonian systems

  • On the 70th Birthday of Nikolai N. Nekhoroshev Special Memorial Issue. Part 1
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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 ε −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).

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Authors and Affiliations

  1. CNRS – CEREMADE, Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Abed Bounemoura

  2. IMCCE, Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014, Paris, France

    Abed Bounemoura

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  1. Abed Bounemoura
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Correspondence to Abed Bounemoura.

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To the memory of N.N. Nekhoroshev

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Bounemoura, A. Generic perturbations of linear integrable Hamiltonian systems. Regul. Chaot. Dyn. 21, 665–681 (2016). https://doi.org/10.1134/S1560354716060071

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  • DOI: https://doi.org/10.1134/S1560354716060071

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