The period of the limit cycle bifurcating from a persistent polycycle

We consider smooth families of planar polynomial vector fields {X mu}mu is an element of Lambda, where Lambda is an open subset of RN, for which there is a hyperbolic polycycle Gamma that is persistent (i.e., such that none of the separatrix connections is broken along the family). It is well known...

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Detalles Bibliográficos
Autores: David, Marín, Queiroz, L., Villadelprat, J.
Tipo de recurso: artículo
Fecha de publicación:2025
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2072/489254
Acceso en línea:https://hdl.handle.net/2072/489254
Access Level:acceso abierto
Palabra clave:Limit cycle
Polycycle
Cyclicity
Period
Asymptotic expansion
Dulac map
51
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spelling The period of the limit cycle bifurcating from a persistent polycycleDavid, MarínQueiroz, L.Villadelprat, J.Limit cyclePolycycleCyclicityPeriodAsymptotic expansionDulac map51We consider smooth families of planar polynomial vector fields {X mu}mu is an element of Lambda, where Lambda is an open subset of RN, for which there is a hyperbolic polycycle Gamma that is persistent (i.e., such that none of the separatrix connections is broken along the family). It is well known that in this case the cyclicity of Gamma at mu 0 is zero unless its graphic number r(mu 0) is equal to one. It is also well known that if r(mu 0) = 1 (and some generic conditions on the return map are verified), then the cyclicity of Gamma at mu 0 is one, i.e., exactly one limit cycle bifurcates from Gamma. In this paper we prove that this limit cycle approaches Gamma exponentially fast and that its period goes to infinity as 1/|r(mu)-1| when mu -> mu 0. Moreover, we prove that if those generic conditions are not satisfied, although the cyclicity may be exactly 1, the behavior of the period of the limit cycle is not determined.This work is financially supported by the Spanish Ministry of Science, Innovation and Universities, through grants PID2021-125625NB-I00 and PID2020-118281GB-C33 and by the Agency for Management of University and Research Grants of Catalonia through grants 2021SGR01015 and 2021SGR00113. This work is also supported by the Spanish State Research Agency, through the Severo Ochoa and Mar & imath;a de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). The second author is supported by Sao Paulo Research Foundation (FAPESP) grants 21/14450-4 and 19/13040-7.info:eu-repo/semantics/publishedVersionUniversitat Autònoma de Barcelona2025info:eu-repo/semantics/article20 p.application/pdfhttps://hdl.handle.net/2072/489254RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésPublicacions MatemàtiquesAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/4892542026-05-29T05:05:01Z
dc.title.none.fl_str_mv The period of the limit cycle bifurcating from a persistent polycycle
title The period of the limit cycle bifurcating from a persistent polycycle
spellingShingle The period of the limit cycle bifurcating from a persistent polycycle
David, Marín
Limit cycle
Polycycle
Cyclicity
Period
Asymptotic expansion
Dulac map
51
title_short The period of the limit cycle bifurcating from a persistent polycycle
title_full The period of the limit cycle bifurcating from a persistent polycycle
title_fullStr The period of the limit cycle bifurcating from a persistent polycycle
title_full_unstemmed The period of the limit cycle bifurcating from a persistent polycycle
title_sort The period of the limit cycle bifurcating from a persistent polycycle
dc.creator.none.fl_str_mv David, Marín
Queiroz, L.
Villadelprat, J.
author David, Marín
author_facet David, Marín
Queiroz, L.
Villadelprat, J.
author_role author
author2 Queiroz, L.
Villadelprat, J.
author2_role author
author
dc.subject.none.fl_str_mv Limit cycle
Polycycle
Cyclicity
Period
Asymptotic expansion
Dulac map
51
topic Limit cycle
Polycycle
Cyclicity
Period
Asymptotic expansion
Dulac map
51
description We consider smooth families of planar polynomial vector fields {X mu}mu is an element of Lambda, where Lambda is an open subset of RN, for which there is a hyperbolic polycycle Gamma that is persistent (i.e., such that none of the separatrix connections is broken along the family). It is well known that in this case the cyclicity of Gamma at mu 0 is zero unless its graphic number r(mu 0) is equal to one. It is also well known that if r(mu 0) = 1 (and some generic conditions on the return map are verified), then the cyclicity of Gamma at mu 0 is one, i.e., exactly one limit cycle bifurcates from Gamma. In this paper we prove that this limit cycle approaches Gamma exponentially fast and that its period goes to infinity as 1/|r(mu)-1| when mu -> mu 0. Moreover, we prove that if those generic conditions are not satisfied, although the cyclicity may be exactly 1, the behavior of the period of the limit cycle is not determined.
publishDate 2025
dc.date.none.fl_str_mv 2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2072/489254
url https://hdl.handle.net/2072/489254
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Publicacions Matemàtiques
dc.rights.none.fl_str_mv Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 20 p.
application/pdf
dc.publisher.none.fl_str_mv Universitat Autònoma de Barcelona
publisher.none.fl_str_mv Universitat Autònoma de Barcelona
dc.source.none.fl_str_mv RECERCAT (Dipòsit de la Recerca de Catalunya)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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