Lindstedt series for lower dimensional tori

We consider the existence and effective computation of low-dimensional (less independent frequencies than degrees of freedom) invariant tori of a near-integrable system. Lindstedt method is a systematic procedure to compute formal power series expansions of quasi-periodic solutions. This procedure i...

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Detalles Bibliográficos
Autores: Jorba, Angel, Llave Canosa, Rafael de la, Zou, Maorong
Tipo de recurso: artículo
Fecha de publicación:1995
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/859
Acceso en línea:https://hdl.handle.net/2117/859
Access Level:acceso abierto
Palabra clave:Dynamical systems
Low-dimensional invariant tori
KAM theory
Lindstedt series
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
Classificació AMS::70 Mechanics of particles and systems::70K Nonlinear dynamics
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spelling Lindstedt series for lower dimensional toriJorba, AngelLlave Canosa, Rafael de laZou, MaorongDynamical systemsLow-dimensional invariant toriKAM theoryLindstedt seriesSistemes dinàmicsClassificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systemsClassificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanicsClassificació AMS::70 Mechanics of particles and systems::70K Nonlinear dynamicsWe consider the existence and effective computation of low-dimensional (less independent frequencies than degrees of freedom) invariant tori of a near-integrable system. Lindstedt method is a systematic procedure to compute formal power series expansions of quasi-periodic solutions. This procedure is very suitable for numerical computations. Under some non-degeneracy assumptions it is possible to show that a finite number of this low dimensional tori persist in the sense of formal power series expansions of the perturbation parameter ($\varepsilon$). Contrary to the series for full dimensional tori, whose convergence is established by KAM theory, the convergence of the expansions for low-dimensional tori is not settled -- even if its reasonable to suspect they diverge for typical systems --. Nevertheless, we show that these tori are analytic functions in $\varepsilon$ in a complex disk minus a thin wedge ending at the origin. The formal power series obtained in the Lindstedt method are an asymptotic expansion to them on this set. The main technical tool is a KAM theorem that shows that near a torus which is approximately invariant and approximately reducible (the variation equations can be reduced to constant up to some small error) there is a truly invariant torus. We point out that this KAM theorem presents small divisors involving the normal and intrinsic frequencies of the torus whereas the Linsdedt procedure only presents small divisors coming from the intrinsic frequencies. Note also that the quasi-invariant, quasi reducible tori that are the input for the KAM procedure may have been produced by other methods than Lindstedt series, notably numerical computations or other perturbative expansions.19951995-01-0120072007-05-03journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/859reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 2.5 Spainhttp://creativecommons.org/licenses/by-nc-nd/2.5/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/8592026-05-27T15:37:01Z
dc.title.none.fl_str_mv Lindstedt series for lower dimensional tori
title Lindstedt series for lower dimensional tori
spellingShingle Lindstedt series for lower dimensional tori
Jorba, Angel
Dynamical systems
Low-dimensional invariant tori
KAM theory
Lindstedt series
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
Classificació AMS::70 Mechanics of particles and systems::70K Nonlinear dynamics
title_short Lindstedt series for lower dimensional tori
title_full Lindstedt series for lower dimensional tori
title_fullStr Lindstedt series for lower dimensional tori
title_full_unstemmed Lindstedt series for lower dimensional tori
title_sort Lindstedt series for lower dimensional tori
dc.creator.none.fl_str_mv Jorba, Angel
Llave Canosa, Rafael de la
Zou, Maorong
author Jorba, Angel
author_facet Jorba, Angel
Llave Canosa, Rafael de la
Zou, Maorong
author_role author
author2 Llave Canosa, Rafael de la
Zou, Maorong
author2_role author
author
dc.subject.none.fl_str_mv Dynamical systems
Low-dimensional invariant tori
KAM theory
Lindstedt series
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
Classificació AMS::70 Mechanics of particles and systems::70K Nonlinear dynamics
topic Dynamical systems
Low-dimensional invariant tori
KAM theory
Lindstedt series
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
Classificació AMS::70 Mechanics of particles and systems::70K Nonlinear dynamics
description We consider the existence and effective computation of low-dimensional (less independent frequencies than degrees of freedom) invariant tori of a near-integrable system. Lindstedt method is a systematic procedure to compute formal power series expansions of quasi-periodic solutions. This procedure is very suitable for numerical computations. Under some non-degeneracy assumptions it is possible to show that a finite number of this low dimensional tori persist in the sense of formal power series expansions of the perturbation parameter ($\varepsilon$). Contrary to the series for full dimensional tori, whose convergence is established by KAM theory, the convergence of the expansions for low-dimensional tori is not settled -- even if its reasonable to suspect they diverge for typical systems --. Nevertheless, we show that these tori are analytic functions in $\varepsilon$ in a complex disk minus a thin wedge ending at the origin. The formal power series obtained in the Lindstedt method are an asymptotic expansion to them on this set. The main technical tool is a KAM theorem that shows that near a torus which is approximately invariant and approximately reducible (the variation equations can be reduced to constant up to some small error) there is a truly invariant torus. We point out that this KAM theorem presents small divisors involving the normal and intrinsic frequencies of the torus whereas the Linsdedt procedure only presents small divisors coming from the intrinsic frequencies. Note also that the quasi-invariant, quasi reducible tori that are the input for the KAM procedure may have been produced by other methods than Lindstedt series, notably numerical computations or other perturbative expansions.
publishDate 1995
dc.date.none.fl_str_mv 1995
1995-01-01
2007
2007-05-03
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/859
url https://hdl.handle.net/2117/859
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.300724

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