Generic perturbations of linear integrable Hamiltonian systems
| Autor: | Abed Bounemoura |
|---|---|
| Přispěvatelé: | UMI CNRS-IMPA (UCI), Institut National de Mathématiques Pures-Centre National de la Recherche Scientifique (CNRS), Institut de Mécanique Céleste et de Calcul des Ephémérides (IMCCE), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de Paris, PSL Research University (PSL)-PSL Research University (PSL)-Université de Lille-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Lille-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL) |
| Rok vydání: | 2016 |
| Předmět: |
[PHYS]Physics [physics]
Mathematics::Dynamical Systems Integrable system Kolmogorov–Arnold–Moser theorem [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] 010102 general mathematics Perturbation (astronomy) Torus Dynamical Systems (math.DS) 01 natural sciences Instability Hamiltonian system 010101 applied mathematics Mathematics (miscellaneous) FOS: Mathematics Mathematics - Dynamical Systems 0101 mathematics [PHYS.ASTR]Physics [physics]/Astrophysics [astro-ph] Arnold diffusion ComputingMilieux_MISCELLANEOUS Mathematical physics Mathematics |
| Zdroj: | Regular and Chaotic Dynamics Regular and Chaotic Dynamics, MAIK Nauka/Interperiodica, 2016, 21 (6), pp.665-681. ⟨10.1134/S1560354716060071⟩ Regular and Chaotic Dynamics, 2016, 21 (6), pp.665-681. ⟨10.1134/S1560354716060071⟩ |
| ISSN: | 1468-4845 1560-3547 |
| Popis: | International audience; 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 which 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 non-resonant 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 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). |
| Databáze: | OpenAIRE |
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