Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit

The purpose of thiswork is to give precise estimates for the size of the remainder of the normalized Hamiltonian around a non-semi-simple 1 : −1 resonant periodic orbit, as a function of the distance to the orbit. We consider a periodic orbit of a real analytic three-degrees of freedom Hamiltonian s...

Descripción completa

Detalles Bibliográficos
Autores: Ollé Torner, Mercè|||0000-0002-8050-9055, Pacha Andújar, Juan Ramón|||0000-0003-4599-3141, Villanueva Castelltort, Jordi|||0000-0001-8725-2785
Tipo de recurso: artículo
Fecha de publicación:2003
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:2117/912
Acceso en línea:https://hdl.handle.net/2117/912
Access Level:acceso abierto
Palabra clave:Differential equations
Bifurcation theory
Hamiltonian systems
Quantitative estimates
periodic orbit
Equacions diferencials ordinàries
Bifurcació, Teoria de la
Hamilton, Sistemes de
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
id ES_3dfd8f68d73d751f5ed2cd7a40d95705
oai_identifier_str oai:upcommons.upc.edu:2117/912
network_acronym_str ES
network_name_str España
repository_id_str
spelling Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbitOllé Torner, Mercè|||0000-0002-8050-9055Pacha Andújar, Juan Ramón|||0000-0003-4599-3141Villanueva Castelltort, Jordi|||0000-0001-8725-2785Differential equationsBifurcation theoryHamiltonian systemsQuantitative estimatesperiodic orbitEquacions diferencials ordinàriesBifurcació, Teoria de laHamilton, Sistemes deClassificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systemsClassificació AMS::34 Ordinary differential equations::34C Qualitative theoryClassificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theoryThe purpose of thiswork is to give precise estimates for the size of the remainder of the normalized Hamiltonian around a non-semi-simple 1 : −1 resonant periodic orbit, as a function of the distance to the orbit. We consider a periodic orbit of a real analytic three-degrees of freedom Hamiltonian system having a pairwise collision of its non-trivial characteristic multipliers on the unit circle. Under generic hypotheses of non-resonance and non-degeneracy of the collision, we present a constructive methodology to reduce the Hamiltonian around the orbit to its (integrable) normal form, up to any given order. This constructive process allows to obtain quantitative estimates for the size of the remainder of the normal form, as a function of the normalizing order. By selecting appropriately this order in terms of the distance R to the resonant orbit (measured using suitable coordinates), r = ropt(R) := 2 + ?exp(W(log(1/R1/(τ+1+ε))))?, we have proved that the size of the remainder can be bounded (for small R) by Rropt(R)/2. Here, W(·) stands for Lambert’s W function and verifies that W(z) exp(W(z)) = z, τ ? 1 is the exponent of the required Diophantine condition and ε > 0 is any small quantity. The reasons leading to this bound instead of classical exponentially small estimates are also discussed.20032003-01-0120072007-05-07journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/912reponame: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:2117/9122026-05-27T15:37:01Z
dc.title.none.fl_str_mv Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
title Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
spellingShingle Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
Ollé Torner, Mercè|||0000-0002-8050-9055
Differential equations
Bifurcation theory
Hamiltonian systems
Quantitative estimates
periodic orbit
Equacions diferencials ordinàries
Bifurcació, Teoria de la
Hamilton, Sistemes de
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
title_short Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
title_full Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
title_fullStr Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
title_full_unstemmed Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
title_sort Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
dc.creator.none.fl_str_mv Ollé Torner, Mercè|||0000-0002-8050-9055
Pacha Andújar, Juan Ramón|||0000-0003-4599-3141
Villanueva Castelltort, Jordi|||0000-0001-8725-2785
author Ollé Torner, Mercè|||0000-0002-8050-9055
author_facet Ollé Torner, Mercè|||0000-0002-8050-9055
Pacha Andújar, Juan Ramón|||0000-0003-4599-3141
Villanueva Castelltort, Jordi|||0000-0001-8725-2785
author_role author
author2 Pacha Andújar, Juan Ramón|||0000-0003-4599-3141
Villanueva Castelltort, Jordi|||0000-0001-8725-2785
author2_role author
author
dc.subject.none.fl_str_mv Differential equations
Bifurcation theory
Hamiltonian systems
Quantitative estimates
periodic orbit
Equacions diferencials ordinàries
Bifurcació, Teoria de la
Hamilton, Sistemes de
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
topic Differential equations
Bifurcation theory
Hamiltonian systems
Quantitative estimates
periodic orbit
Equacions diferencials ordinàries
Bifurcació, Teoria de la
Hamilton, Sistemes de
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
description The purpose of thiswork is to give precise estimates for the size of the remainder of the normalized Hamiltonian around a non-semi-simple 1 : −1 resonant periodic orbit, as a function of the distance to the orbit. We consider a periodic orbit of a real analytic three-degrees of freedom Hamiltonian system having a pairwise collision of its non-trivial characteristic multipliers on the unit circle. Under generic hypotheses of non-resonance and non-degeneracy of the collision, we present a constructive methodology to reduce the Hamiltonian around the orbit to its (integrable) normal form, up to any given order. This constructive process allows to obtain quantitative estimates for the size of the remainder of the normal form, as a function of the normalizing order. By selecting appropriately this order in terms of the distance R to the resonant orbit (measured using suitable coordinates), r = ropt(R) := 2 + ?exp(W(log(1/R1/(τ+1+ε))))?, we have proved that the size of the remainder can be bounded (for small R) by Rropt(R)/2. Here, W(·) stands for Lambert’s W function and verifies that W(z) exp(W(z)) = z, τ ? 1 is the exponent of the required Diophantine condition and ε > 0 is any small quantity. The reasons leading to this bound instead of classical exponentially small estimates are also discussed.
publishDate 2003
dc.date.none.fl_str_mv 2003
2003-01-01
2007
2007-05-07
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/2117/912
url https://hdl.handle.net/2117/912
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
_version_ 1869406482292801536
score 15.300724

Similares en LA Referencia