Inverse Jacobi multiplier as a link between conservative systems and Poisson structures

Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the Flow-Box Theorem we restrict ourselves to neighborhoods of sin...

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Detalles Bibliográficos
Autores: García, I. A. (Isaac A.), Hernández Bermejo, Benito
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2017
País:España
Institución:Universitat de Lleida (UdL)
Repositorio:Repositori Obert UdL
OAI Identifier:oai:repositori.udl.cat:10459.1/60207
Acceso en línea:https://doi.org/10.1088/1751-8121/aa7bda
http://hdl.handle.net/10459.1/60207
Access Level:acceso abierto
Palabra clave:Matemáticas
Inverse Jacobi multipliers
Conservative systems
Poisson systems
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spelling Inverse Jacobi multiplier as a link between conservative systems and Poisson structuresGarcía, I. A. (Isaac A.)Hernández Bermejo, BenitoMatemáticasInverse Jacobi multipliersConservative systemsPoisson systemsSome aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the Flow-Box Theorem we restrict ourselves to neighborhoods of singularities. In this sense, we characterize Poisson structures around the typical zero-Hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non-vanishing first jet by connecting with the classical Poincar\'e center problem. From the global point of view, we connect the property of being strictly conservative (the invariant measure must be positive) with the existence of a Poisson structure depending on the phase space dimension. Finally, weak conservativeness in dimension two is introduced by the extension of inverse Jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed. Examples including Lotka-Volterra systems, quadratic isochronous centers, and non-smooth oscillators are provided.Both authors would like to acknowledge Ministerio de Econom´ıa y Competitividad for Project Ref. MTM2014-53703-P. In addition, I.A.G. acknowledges AGAUR grant number 2014SGR 1204. B.H.-B. acknowledges Ministerio de Econom´ıa y Competitividad for Project Ref. MTM2016-80276-P as well as financial support from Universidad Rey Juan Carlos-Banco de Santander (Excellence Group QUINANOAP, grant number 30VCPIGI14). Finally, B.H.-B. is sincerely indebted to the members of Departament de Matemàtica, Universitat de Lleida, for their kind hospitalityIOP Publishing2017info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttps://doi.org/10.1088/1751-8121/aa7bdahttp://hdl.handle.net/10459.1/60207reponame:Repositori Obert UdL instname:Universitat de Lleida (UdL)InglésMINECO/PN2013-2016/MTM2014-53703-PMINECO/PN2013-2016/MTM2016-80276-PVersió postprint del document publicat a https://doi.org/10.1088/1751-8121/aa7bdaJournal of Physics A: Mathematical and Theoretical, 2017, vol. 50, p. 1-17(c) IOP Publishing, 2017info:eu-repo/semantics/openAccessoai:repositori.udl.cat:10459.1/602072026-06-24T12:42:17Z
dc.title.none.fl_str_mv Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
title Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
spellingShingle Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
García, I. A. (Isaac A.)
Matemáticas
Inverse Jacobi multipliers
Conservative systems
Poisson systems
title_short Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
title_full Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
title_fullStr Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
title_full_unstemmed Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
title_sort Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
dc.creator.none.fl_str_mv García, I. A. (Isaac A.)
Hernández Bermejo, Benito
author García, I. A. (Isaac A.)
author_facet García, I. A. (Isaac A.)
Hernández Bermejo, Benito
author_role author
author2 Hernández Bermejo, Benito
author2_role author
dc.subject.none.fl_str_mv Matemáticas
Inverse Jacobi multipliers
Conservative systems
Poisson systems
topic Matemáticas
Inverse Jacobi multipliers
Conservative systems
Poisson systems
description Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the Flow-Box Theorem we restrict ourselves to neighborhoods of singularities. In this sense, we characterize Poisson structures around the typical zero-Hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non-vanishing first jet by connecting with the classical Poincar\'e center problem. From the global point of view, we connect the property of being strictly conservative (the invariant measure must be positive) with the existence of a Poisson structure depending on the phase space dimension. Finally, weak conservativeness in dimension two is introduced by the extension of inverse Jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed. Examples including Lotka-Volterra systems, quadratic isochronous centers, and non-smooth oscillators are provided.
publishDate 2017
dc.date.none.fl_str_mv 2017
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://doi.org/10.1088/1751-8121/aa7bda
http://hdl.handle.net/10459.1/60207
url https://doi.org/10.1088/1751-8121/aa7bda
http://hdl.handle.net/10459.1/60207
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv MINECO/PN2013-2016/MTM2014-53703-P
MINECO/PN2013-2016/MTM2016-80276-P
Versió postprint del document publicat a https://doi.org/10.1088/1751-8121/aa7bda
Journal of Physics A: Mathematical and Theoretical, 2017, vol. 50, p. 1-17
dc.rights.none.fl_str_mv (c) IOP Publishing, 2017
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) IOP Publishing, 2017
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv IOP Publishing
publisher.none.fl_str_mv IOP Publishing
dc.source.none.fl_str_mv reponame:Repositori Obert UdL
instname:Universitat de Lleida (UdL)
instname_str Universitat de Lleida (UdL)
reponame_str Repositori Obert UdL
collection Repositori Obert UdL
repository.name.fl_str_mv
repository.mail.fl_str_mv
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