Hamiltonian perturbation theory for ultra-differentiable functions

Author:

Bounemoura Abed,Féjoz Jacques

Abstract

Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M M bounding the growth of derivatives.

In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BR M _M , and which generalizes the Bruno-Rüssmann condition; and Nekhoroshev’s theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M M . Our proof uses periodic averaging, while a substitute for the analyticity width allows us to bypass analytic smoothing.

We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco-Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it matching), we manage to narrow the gap between stability hypotheses (e.g. the BR M _M condition) and instability hypotheses, thus circumbscribing the stability threshold.

The formulas relating the growth M M of derivatives of the perturbation on the one hand, and the arithmetics of robust frequencies or the stability time on the other hand, bring light to the competition between stability properties of nearly integrable systems and the distance to integrability. Due to our method of proof using width of regularity as a regularizing parameter, these formulas are closer to optimal as the the regularity tends to analyticity.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference50 articles.

1. [Arn64] V.I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady 5 (1964), 581–585.

2. An approach to Arnol′d’s diffusion through the calculus of variations;Bessi, Ugo;Nonlinear Anal.,1996

3. Arnold’s example with three rotators;Bessi, Ugo;Nonlinearity,1997

4. An analytic counterexample to the KAM theorem;Bessi, Ugo;Ergodic Theory Dynam. Systems,2000

5. A Diophantine duality applied to the KAM and Nekhoroshev theorems;Bounemoura, Abed;Math. Z.,2013

Cited by 9 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3