Nekhoroshev estimates for steep real-analytic elliptic equilibrium points

Bounemoura, Abed; Fayad, Bassam; Niederman, Laurent

doi:10.1088/1361-6544/ab4c89

Cited by 5 publications

(3 citation statements)

References 22 publications

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“…where K(ε) is an ultraviolet cutoff which has to be properly chosen 9 . Both terms N Λ and R Λ of course depend on ε.…”

Section: General Setting and Classical Methods: A Geometric Frameworkmentioning

confidence: 99%

“…8 The smallness depends on the regularity of the system. 9 This choice is indeed a main issue in the theory.…”

Section: General Setting and Classical Methods: A Geometric Frameworkmentioning

confidence: 99%

“…[29], [8], [13]) but sharp estimates of stability under the two latter constraints seem to be reached only by Nekhoroshev's original construction 15 (see [23] and [35], [38]). On the other hand, the Lochak method turns out to be more adapted when trying to extend Nekhoroshev results to other contexts (for example to elliptic points [19], [34], [9], [10] or to infinite-dimensional systems [1], [40]) since it involves much easier computations.…”

Section: Figurementioning

confidence: 99%

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Flexibility and analytic smoothing in averaging theory

Barbieri,

Marco,

Massetti

2021

Preprint

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Using a new strategy, we extend the classical Nekhoroshev's estimates to the case of Hölder regular steep near-integrable hamiltonian systems, the stability times being polynomially long in the inverse of the size of the perturbation. We prove that the stability exponents can be taken to be ( − 1)/(2nα 1 ...α n−2 ) for the time of stability and 1/(2nα 1 ...α n−1 ) for the radius of stability, > n + 1 being the regularity and the α i 's being the indices of steepness. Our strategy consists in deriving a perturbation theory which exploits a sharp analytic smoothing theorem to approximate any Hölder function by an analytic one. In addition, an appropriate choice of the free parameters in the problem enables us to have a first grasp on the relation connecting the time and radius of stability to the threshold that the size of the perturbation must satisfy in order for the theorem to apply. Particular attention is payed to a geometric presentation of the construction of the so-called resonant blocks, in order to shed a definitive light on the nature of the steepness condition. We also investigate the convex setting, using a similar approach. Contents 1. Introduction and main results 1 2. General setting and classical methods: a geometric framework 6 3. Functional setting 13 4. Analytic smoothing 15 5. Stability estimates in the convex case with the patchwork method 20 6. Stability estimates in the convex case with the periodic averaging method 27 7. Stability estimates in the steep case 32 Appendix A. Smoothing estimates 37 Appendix B. Analytic and arithmetic tools for the convex case 38 References 40

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“…where K(ε) is an ultraviolet cutoff which has to be properly chosen 9 . Both terms N Λ and R Λ of course depend on ε.…”

Section: General Setting and Classical Methods: A Geometric Frameworkmentioning

confidence: 99%

“…8 The smallness depends on the regularity of the system. 9 This choice is indeed a main issue in the theory.…”

Section: General Setting and Classical Methods: A Geometric Frameworkmentioning

confidence: 99%

Section: Figurementioning

confidence: 99%

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Flexibility and analytic smoothing in averaging theory

Barbieri,

Marco,

Massetti

2021

Preprint

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Super-exponential stability for generic real-analytic elliptic equilibrium points

Bounemoura

Fayad²,

Niederman

2020

Advances in Mathematics

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We consider the dynamics in a neighborhood of an elliptic equilibrium point with a Diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability. Generically, both in a topological and measure-theoretical sense, any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point. We actually prove a more general statement assuming the frequency is only non-resonant. This improves previous results where much stronger non-generic assumptions were required.

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Sub-exponential stability for the beam equation

Feola

Massetti

2023

Journal of Differential Equations

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Nekhoroshev estimates for steep real-analytic elliptic equilibrium points

Cited by 5 publications

References 22 publications

Flexibility and analytic smoothing in averaging theory

Flexibility and analytic smoothing in averaging theory

Super-exponential stability for generic real-analytic elliptic equilibrium points

Sub-exponential stability for the beam equation

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