Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion

Abstract. We consider models given by Hamiltonians of the form H ( I;';p;q;t ; " ) = h ( I )+ n X j =1 1 2 p 2 j + V j ( q j ) + "Q ( I;';p;q;t ; " ) where I 2I R d ;' 2 T d , p;q 2 R n , t 2 T 1 . These are higher di- mensional analogues, both in the center and hyperbo...

Descripción completa

Detalles Bibliográficos
Autores: Delshams Valdés, Amadeu|||0000-0003-4134-8882, de la Llave, Rafael, Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
Tipo de recurso: informe técnico
Fecha de publicación:2013
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:2117/20671
Acceso en línea:https://hdl.handle.net/2117/20671
Access Level:acceso abierto
Palabra clave:Hamiltonian systems
Sistemes hamiltonians
Sistemes dinàmics diferenciables
Àrees temàtiques de la UPC::Matemàtiques i estadística
id ES_42961afbecfae9abcdcba0b804c71e49
oai_identifier_str oai:upcommons.upc.edu:2117/20671
network_acronym_str ES
network_name_str España
repository_id_str
spelling Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusionDelshams Valdés, Amadeu|||0000-0003-4134-8882de la Llave, RafaelMartínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717Hamiltonian systemsSistemes hamiltoniansSistemes dinàmics diferenciablesÀrees temàtiques de la UPC::Matemàtiques i estadísticaAbstract. We consider models given by Hamiltonians of the form H ( I;';p;q;t ; " ) = h ( I )+ n X j =1 1 2 p 2 j + V j ( q j ) + "Q ( I;';p;q;t ; " ) where I 2I R d ;' 2 T d , p;q 2 R n , t 2 T 1 . These are higher di- mensional analogues, both in the center and hyperbolic directions, of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All these models present the large gap problem . We show that, for 0 < " 1, under regularity and explicit non- degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O (1). This domain includes resonance lines and, hence, large gaps among d -dimensional KAM tori. The method of proof follows closely the strategy of [DLS03, DLS06a]. The main new phenomenon that appears when the di- mension d of the center directions is larger than one, is the exis- tence of multiple resonances. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I , they can be contoured. This corresponds to the mechanism called di usion across resonances in the Physics literature. The present paper, however, di ers substantially from [DLS03, DLS06a]. On the technical details of the proofs, we have taken advantage of the theory of the scattering map [DLS08], not avail- able when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the res- onances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [DLS03, DLS06a]20132013-06-0120132013-11-20reporthttp://purl.org/coar/resource_type/c_93fcAOhttp://purl.org/coar/version/c_b1a7d7d4d402bcceinfo:eu-repo/semantics/reportapplication/pdfhttps://hdl.handle.net/2117/20671reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/206712026-05-27T15:37:01Z
dc.title.none.fl_str_mv Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
title Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
spellingShingle Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
Delshams Valdés, Amadeu|||0000-0003-4134-8882
Hamiltonian systems
Sistemes hamiltonians
Sistemes dinàmics diferenciables
Àrees temàtiques de la UPC::Matemàtiques i estadística
title_short Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
title_full Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
title_fullStr Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
title_full_unstemmed Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
title_sort Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
dc.creator.none.fl_str_mv Delshams Valdés, Amadeu|||0000-0003-4134-8882
de la Llave, Rafael
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
author Delshams Valdés, Amadeu|||0000-0003-4134-8882
author_facet Delshams Valdés, Amadeu|||0000-0003-4134-8882
de la Llave, Rafael
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
author_role author
author2 de la Llave, Rafael
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
author2_role author
author
dc.subject.none.fl_str_mv Hamiltonian systems
Sistemes hamiltonians
Sistemes dinàmics diferenciables
Àrees temàtiques de la UPC::Matemàtiques i estadística
topic Hamiltonian systems
Sistemes hamiltonians
Sistemes dinàmics diferenciables
Àrees temàtiques de la UPC::Matemàtiques i estadística
description Abstract. We consider models given by Hamiltonians of the form H ( I;';p;q;t ; " ) = h ( I )+ n X j =1 1 2 p 2 j + V j ( q j ) + "Q ( I;';p;q;t ; " ) where I 2I R d ;' 2 T d , p;q 2 R n , t 2 T 1 . These are higher di- mensional analogues, both in the center and hyperbolic directions, of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All these models present the large gap problem . We show that, for 0 < " 1, under regularity and explicit non- degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O (1). This domain includes resonance lines and, hence, large gaps among d -dimensional KAM tori. The method of proof follows closely the strategy of [DLS03, DLS06a]. The main new phenomenon that appears when the di- mension d of the center directions is larger than one, is the exis- tence of multiple resonances. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I , they can be contoured. This corresponds to the mechanism called di usion across resonances in the Physics literature. The present paper, however, di ers substantially from [DLS03, DLS06a]. On the technical details of the proofs, we have taken advantage of the theory of the scattering map [DLS08], not avail- able when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the res- onances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [DLS03, DLS06a]
publishDate 2013
dc.date.none.fl_str_mv 2013
2013-06-01
2013
2013-11-20
dc.type.none.fl_str_mv report
http://purl.org/coar/resource_type/c_93fc
AO
http://purl.org/coar/version/c_b1a7d7d4d402bcce
dc.type.openaire.fl_str_mv info:eu-repo/semantics/report
format report
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/20671
url https://hdl.handle.net/2117/20671
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 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/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 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/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_ 1869406949680873472
score 15.300719

Similares en LA Referencia