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

We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+¿j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I¿I¿Rd,f¿TdI¿I¿Rd,f¿Td, p,q¿Rnp,q¿Rn, t¿T1t¿T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the model...

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Autores: Delshams Valdés, Amadeu|||0000-0003-4134-8882, Llave Canosa, Rafael de la, Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
Tipo de recurso: artículo
Fecha de publicación:2016
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:2117/89859
Acceso en línea:https://hdl.handle.net/2117/89859
https://dx.doi.org/10.1016/j.aim.2015.11.010
Access Level:acceso abierto
Palabra clave:Hamiltonian systems
Arnold diffusion
Hamiltonian
Instability
Resonances
Scattering map
Sistemes hamiltonians
Àrees temàtiques de la UPC::Matemàtiques i estadística
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spelling Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusionDelshams Valdés, Amadeu|||0000-0003-4134-8882Llave Canosa, Rafael de laMartínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717Hamiltonian systemsArnold diffusionHamiltonianInstabilityResonancesScattering mapSistemes hamiltoniansÀrees temàtiques de la UPC::Matemàtiques i estadísticaWe consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+¿j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I¿I¿Rd,f¿TdI¿I¿Rd,f¿Td, p,q¿Rnp,q¿Rn, t¿T1t¿T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0<e«10<e«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)O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of and . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I¿I¿RdI¿I¿Rd. 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 diffusion across resonances in the Physics literature. The present paper, however, differs substantially from and . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available 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 resonances 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 andPeer Reviewed20162016-05-1420162016-09-13journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/89859https://dx.doi.org/10.1016/j.aim.2015.11.010reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengMinisterio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2012-31714 DINAMICA ASOCIADA A CONEXIONES ENTRE OBJETOS INVARIANTES. APLICACIONES A ASTRODINAMICA, NEUROCIENCIA Y OTROS CAMPOSopen accesshttp://purl.org/coar/access_right/c_abf2http://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/898592026-05-27T15:37:01Z
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
Arnold diffusion
Hamiltonian
Instability
Resonances
Scattering map
Sistemes hamiltonians
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
Llave Canosa, Rafael de la
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
Llave Canosa, Rafael de la
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
author_role author
author2 Llave Canosa, Rafael de la
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
author2_role author
author
dc.subject.none.fl_str_mv Hamiltonian systems
Arnold diffusion
Hamiltonian
Instability
Resonances
Scattering map
Sistemes hamiltonians
Àrees temàtiques de la UPC::Matemàtiques i estadística
topic Hamiltonian systems
Arnold diffusion
Hamiltonian
Instability
Resonances
Scattering map
Sistemes hamiltonians
Àrees temàtiques de la UPC::Matemàtiques i estadística
description We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+¿j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I¿I¿Rd,f¿TdI¿I¿Rd,f¿Td, p,q¿Rnp,q¿Rn, t¿T1t¿T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0<e«10<e«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)O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of and . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I¿I¿RdI¿I¿Rd. 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 diffusion across resonances in the Physics literature. The present paper, however, differs substantially from and . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available 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 resonances 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 and
publishDate 2016
dc.date.none.fl_str_mv 2016
2016-05-14
2016
2016-09-13
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/89859
https://dx.doi.org/10.1016/j.aim.2015.11.010
url https://hdl.handle.net/2117/89859
https://dx.doi.org/10.1016/j.aim.2015.11.010
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2012-31714 DINAMICA ASOCIADA A CONEXIONES ENTRE OBJETOS INVARIANTES. APLICACIONES A ASTRODINAMICA, NEUROCIENCIA Y OTROS CAMPOS
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2

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

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
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