On the persistence of lower dimensional invariant tori under quasiperiodic perturbations

In this work we consider time dependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the effect that this kind of perturbations has on lower dimensional invariant tori. Our results show that, under standard conditions of analyticity, nondegeneracy and nonresonance, most...

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Detalles Bibliográficos
Autores: Jorba, Angel, Villanueva Castelltort, Jordi|||0000-0001-8725-2785
Tipo de recurso: artículo
Fecha de publicación:1996
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:2117/1218
Acceso en línea:https://hdl.handle.net/2117/1218
Access Level:acceso abierto
Palabra clave:Differential equations
Dynamics
Hamiltonian dynamical systems
Lagrangian functions
quasiperiodic perturbations
Equacions diferencials ordinàries
Partícules (Física nuclear)
Hamilton, Sistemes de
Lagrange, Funcions de
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::70 Mechanics of particles and systems::70F Dynamics of a system of particles, including celestial mechanics
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
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spelling On the persistence of lower dimensional invariant tori under quasiperiodic perturbationsJorba, AngelVillanueva Castelltort, Jordi|||0000-0001-8725-2785Differential equationsDynamicsHamiltonian dynamical systemsLagrangian functionsquasiperiodic perturbationsEquacions diferencials ordinàriesPartícules (Física nuclear)Hamilton, Sistemes deLagrange, Funcions deClassificació AMS::34 Ordinary differential equations::34C Qualitative theoryClassificació AMS::70 Mechanics of particles and systems::70F Dynamics of a system of particles, including celestial mechanicsClassificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanicsIn this work we consider time dependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the effect that this kind of perturbations has on lower dimensional invariant tori. Our results show that, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the perturbation to the ones they already have. The paper also contains estimates on the amount of surviving tori. The worst situation happens when the initial tori are normally elliptic. In this case, a torus (identified by the vector of intrinsic frequencies) can be continued with respect to a perturbative parameter $\epsilon\in[0,\epsilon_0]$, except for a set of $\epsilon$ of measure exponentially small with $\epsilon_0$. In case that $\epsilon$ is fixed (and sufficiently small), we prove the existence of invariant tori for every vector of frequencies close to the one of the initial torus, except for a set of frequencies of measure exponentially small with the distance to the unperturbed torus. As a particular case, if the perturbation is autonomous, these results also give the same kind of estimates on the measure of destroyed tori. Finally, these results are applied to some problems of celestial mechanics, in order to help in the description of the phase space of some concrete models.19961996-01-0120072007-10-03journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/1218reponame: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:2117/12182026-05-27T15:37:01Z
dc.title.none.fl_str_mv On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
title On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
spellingShingle On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
Jorba, Angel
Differential equations
Dynamics
Hamiltonian dynamical systems
Lagrangian functions
quasiperiodic perturbations
Equacions diferencials ordinàries
Partícules (Física nuclear)
Hamilton, Sistemes de
Lagrange, Funcions de
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::70 Mechanics of particles and systems::70F Dynamics of a system of particles, including celestial mechanics
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
title_short On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
title_full On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
title_fullStr On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
title_full_unstemmed On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
title_sort On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
dc.creator.none.fl_str_mv Jorba, Angel
Villanueva Castelltort, Jordi|||0000-0001-8725-2785
author Jorba, Angel
author_facet Jorba, Angel
Villanueva Castelltort, Jordi|||0000-0001-8725-2785
author_role author
author2 Villanueva Castelltort, Jordi|||0000-0001-8725-2785
author2_role author
dc.subject.none.fl_str_mv Differential equations
Dynamics
Hamiltonian dynamical systems
Lagrangian functions
quasiperiodic perturbations
Equacions diferencials ordinàries
Partícules (Física nuclear)
Hamilton, Sistemes de
Lagrange, Funcions de
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::70 Mechanics of particles and systems::70F Dynamics of a system of particles, including celestial mechanics
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
topic Differential equations
Dynamics
Hamiltonian dynamical systems
Lagrangian functions
quasiperiodic perturbations
Equacions diferencials ordinàries
Partícules (Física nuclear)
Hamilton, Sistemes de
Lagrange, Funcions de
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::70 Mechanics of particles and systems::70F Dynamics of a system of particles, including celestial mechanics
Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics
description In this work we consider time dependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the effect that this kind of perturbations has on lower dimensional invariant tori. Our results show that, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the perturbation to the ones they already have. The paper also contains estimates on the amount of surviving tori. The worst situation happens when the initial tori are normally elliptic. In this case, a torus (identified by the vector of intrinsic frequencies) can be continued with respect to a perturbative parameter $\epsilon\in[0,\epsilon_0]$, except for a set of $\epsilon$ of measure exponentially small with $\epsilon_0$. In case that $\epsilon$ is fixed (and sufficiently small), we prove the existence of invariant tori for every vector of frequencies close to the one of the initial torus, except for a set of frequencies of measure exponentially small with the distance to the unperturbed torus. As a particular case, if the perturbation is autonomous, these results also give the same kind of estimates on the measure of destroyed tori. Finally, these results are applied to some problems of celestial mechanics, in order to help in the description of the phase space of some concrete models.
publishDate 1996
dc.date.none.fl_str_mv 1996
1996-01-01
2007
2007-10-03
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/2117/1218
url https://hdl.handle.net/2117/1218
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 15,300719

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