Generic Nekhoroshev theory without small divisors

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href="/articles/10.5802/aif.2705/" class="btn btn-light border"><i class="fa fa-chevron-left"></i></a> <a href="/item/AIF_2012__62_1/" class="btn btn-light border">no. 1</a> <a href="/articles/10.5802/aif.2707/" class="btn btn-light border"><i class="fa fa-chevron-right"></i></a> </div> <div class="clearfix"> </div> </div> </div> <div id="article-div" class="article-div"> <div > <div class="col-12"> <div style="font-size:18px;margin-bottom:10px"> <span class="common-title article-title"> Generic Nekhoroshev theory without small divisors </span> <br> <span class="article-trans-title">[Théorie de Nekhoroshev générique sans petits diviseurs]</span> </div> <div class="article-author" style="display:inline-block"> <div style="margin-bottom:5px; font-size: 16px"> <a class="author_link" href="/search?"Bounemoura, Abed"-c">Bounemoura, Abed<span href=#aff_1></span></a><a class="author_link" href="#affiliation" data-bs-toggle="collapse" aria-controls="affiliation"><sup><span href=#aff_1>1</span></sup></a> ; <a class="author_link" href="/search?"Niederman, Laurent"-c">Niederman, Laurent<span href=#aff_1></span></a><a class="author_link" href="#affiliation" data-bs-toggle="collapse" aria-controls="affiliation"><sup><span href=#aff_1>1</span></sup></a> <a data-bs-toggle="collapse" href="#affiliation" aria-controls="affiliation"> <button id='house-affiliation' type="button" class="btn btn-link btn-xs" title="Affiliations des auteurs"> <span class="fa fa-home"></span><span id='house-caret' class="caret-icon fa fa-caret-down" style="margin-left:3px"></span> </button> </a> <div class="collapse" id="affiliation"> <div class="card card-body"> <div class="affiliation-authors"> <span id=aff_1> <sup>1</sup> Université Paris-Sud 11 Faculté des Sciences d’Orsay Département de Mathématiques Bâtiment 425 91405 Orsay cedex (France) </span> <br> </div> </div> </div> </div> </div> <div style="margin-bottom:10px"> Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 277-324. </div> <div class="binary-file-icon" style="margin-top:20px;margin-bottom:10px"> <div class="wrapper-float"> <div class="d-flex flex-nowrap" style="margin-top:0px"> <a href="/item/10.5802/aif.2706.pdf" target="_blank"> <img src="/static/ptf/img/pdf.svg"> </a> </div> </div> </div> </div> </div> <div> <div class="col-12"> <div style="margin-top:20px;margin-bottom:30px"> <ul class="nav nav-tabs article-tab"> <li class="nav-item active"> <a class="nav-link active" lang="fr" data-bs-toggle="tab" data-bs-target="#abstract-fr" href="#abstract-fr"><span title='Français '>Résumé</span></a> </li> <li class="nav-item"> <a class="nav-link" lang="en" data-bs-toggle="tab" data-bs-target="#abstract-en" href="#abstract-en"> <span title='Anglais '>Abstract</span> </a> </li> </ul> <div class="tab-content" style="margin-top:10px"> <div lang="fr" id="abstract-fr" class="tab-pane fade show active"> <p>Dans cet article, nous présentons une nouvelle approche de la théorie de Nekhoroshev pour un hamiltonien intégrable générique, qui évite complètement le problème des petits diviseurs. La preuve est une extension d’une méthode introduite par Lochak, elle n’utilise que des moyennisations périodiques et de l’approximation diophantienne simultanée, ainsi que des arguments géométriques introduit par le second auteur. Notre méthode permet également d’obtenir des résultats de stabilité pour des hamiltoniens génériques non-analytiques, ainsi que de nouveaux résultats de stabilité au voisinage des tores invariants linéairement stables.</p> </div> <div lang="en" id="abstract-en" class="tab-pane fade"> <p>In this article, we present a new approach of Nekhoroshev’s theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak, it combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.</p> </div> </div> </div> </div> </div> <div> <div class="col-12"> <div style="margin-top:10px;margin-bottom:10px"> <ul class="nav nav-tabs article-tab"> <li class="nav-item"><a class="nav-link active" data-bs-toggle="tab" data-bs-target="#info-tab" href="#info-tab">Détail</a></li> <li class="nav-item"><a class="nav-link" data-bs-toggle="tab"data-bs-target="#citation-tab" href="#citation-tab">Export</a></li> <li class="nav-item"> <a class="nav-link " data-bs-toggle="tab" data-toggle="tab" data-bs-target="#how-to-cite-tab" href="#how-to-cite-tab">Comment citer</a> </li> </ul> <div class="tab-content" style="margin-top:10px"> <div id="info-tab" class="tab-pane fade show active">  <div style="margin-top:5px"> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2986272" class="badge bg-secondary rounded-pill ref-badge extid-badge"> MR </a>   <a href="https://zbmath.org/?q=an:1257.37036" class="badge bg-secondary rounded-pill ref-badge extid-badge"> Zbl </a> </div> DOI : <a href="https://doi.org/10.5802/aif.2706">10.5802/aif.2706</a> <br> <div style="margin-top:5px"> Classification : <span class="mathjax-formula"> 37J25, 37J40, 70H08, 70H09, 70K45, 70K60, 70K65 </span> <br/> Keywords: <span class="mathjax-formula"> Hamiltonian systems, perturbation of integrable systems, effective stability </span> <br/> Mots-clés : <span class="mathjax-formula"> systèmes hamiltoniens, perturbation de systèmes intégrables, stabilité effective </span> <br/> </div> <div style="margin-top:5px"> </div> <div style="margin-top:5px"> <span id="affiliation-toggle" data-bs-toggle="collapse" href="#affiliation_details" role="button" aria-expanded="false" aria-controls="affiliation_details">Affiliations des auteurs : <span id="details-caret" class="caret-icon fa fa-caret-down" style="margin-left:3px"></span></span> <div class="collapse" id="affiliation_details" style="margin-top:10px;margin-left:10px"> <div class="card card-body"> <p> Bounemoura, Abed&hairsp;<sup class="d-inline-block">1</sup> ; Niederman, Laurent&hairsp;<sup class="d-inline-block">1</sup> </p> <div class="affiliation-authors"> <span id=aff_1> <sup>1</sup> Université Paris-Sud 11 Faculté des Sciences d’Orsay Département de Mathématiques Bâtiment 425 91405 Orsay cedex (France) </span> <br> </div> </div> </div> </div> </div> <div id="citation-tab" class="tab-pane fade"> <div class="upper"> <ul class="nav nav-pills"> <li role="presentation" class="nav-item"><a class="nav-link active" data-bs-toggle="tab" data-bs-target="#bibtex" href="#bibtex">BibTeX</a></li> <li role="presentation" class="nav-item"><a class="nav-link" data-bs-toggle="tab" data-bs-target="#ris" href="#ris">RIS</a></li> <li role="presentation" class="nav-item"><a class="nav-link" data-bs-toggle="tab" data-bs-target="#enw" href="#enw">EndNote</a></li> </ul> <div class="tab-content"> <div id="bibtex" class="tab-pane fade show active"> <pre class="upper-right-content" id="bibtex-div">@article{AIF_2012__62_1_277_0, author = {Bounemoura, Abed and Niederman, Laurent}, title = {Generic {Nekhoroshev} theory without small divisors}, journal = {Annales de l'Institut Fourier}, pages = {277--324}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {62}, number = {1}, year = {2012}, doi = {10.5802/aif.2706}, mrnumber = {2986272}, zbl = {1257.37036}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2706/} }</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#bibtex-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="Télécharger" href="/export_citation/AIF_2012__62_1_277_0/bib/"></a> </div> <div id="ris" class="tab-pane fade"> <pre class="upper-right-content" id="ris-div">TY - JOUR AU - Bounemoura, Abed AU - Niederman, Laurent TI - Generic Nekhoroshev theory without small divisors JO - Annales de l'Institut Fourier PY - 2012 SP - 277 EP - 324 VL - 62 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2706/ DO - 10.5802/aif.2706 LA - en ID - AIF_2012__62_1_277_0 ER - </pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#ris-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="Télécharger" href="/export_citation/AIF_2012__62_1_277_0/ris/"></a> </div> <div id="enw" class="tab-pane fade"> <pre class="upper-right-content" id="enw-div">%0 Journal Article %A Bounemoura, Abed %A Niederman, Laurent %T Generic Nekhoroshev theory without small divisors %J Annales de l'Institut Fourier %D 2012 %P 277-324 %V 62 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2706/ %R 10.5802/aif.2706 %G en %F AIF_2012__62_1_277_0</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#enw-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="Télécharger" href="/export_citation/AIF_2012__62_1_277_0/enw/"></a> </div> </div> </div> </div> <div id="how-to-cite-tab" class="tab-pane fade "> <div class="upper-right"> <pre class="upper-right-content" id="how-to-cite-div">Bounemoura, Abed; Niederman, Laurent. Generic Nekhoroshev theory without small divisors. Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 277-324. doi : 10.5802/aif.2706. https://aif.centre-mersenne.org/articles/10.5802/aif.2706/</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#how-to-cite-div" aria-hidden="true" title="Copier dans le presse-papier"></button> </div> </div> </div> </div> </div> </div> <div> <div class="col-12"> <div style="margin-top:30px;margin-bottom:10px"> <ul class="nav nav-tabs article-tab" > <li class="nav-item" id="main-citations-li" > <a class="nav-link active" data-bs-toggle="tab" data-bs-target="#references-tab" href="#references-tab">Bibliographie</a> </li> <li class="nav-item" id="citations-html" style="display:none"> <a class="nav-link" data-bs-toggle="tab" data-bs-target="#citations-tab" href="#citations-tab">Cité par</a> </li> </ul> <div class="tab-content" style="margin-top:10px"> <div id="references-tab" class="tab-pane fade show active"> <div style="margin-top:11px;margin-bottom: 0px"> <p class="bibitemcls " > [1] <span class="citation-author">Arnold, Vladimir I.</span><span class="citation-document-title"> Instability of dynamical systems with many degrees of freedom</span><span class="citation-publication-title">, Dokl. 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Generic Nekhoroshev theory without small divisors