On the cyclicity of Kolmogorov polycycles

In this paper we study planar polynomial Kolmogorov's differential systems Xμ{x˙=f(x,y;μ),y˙=g(x,y;μ), with the parameter μ varying in an open subset Λ⊂RN. Compactifying Xμ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic pol...

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Autores: Marín, David|||0000-0003-4422-6418, Villadelprat Yagüe, Jordi|||0000-0002-1168-9750
Tipo de recurso: artículo
Fecha de publicación:2022
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:265184
Acceso en línea:https://ddd.uab.cat/record/265184
https://dx.doi.org/urn:doi:10.14232/ejqtde.2022.1.35
Access Level:acceso abierto
Palabra clave:Limit cycle
Polycycle
Cyclicity
Asymptotic expansion
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spelling On the cyclicity of Kolmogorov polycyclesMarín, David|||0000-0003-4422-6418Villadelprat Yagüe, Jordi|||0000-0002-1168-9750Limit cyclePolycycleCyclicityAsymptotic expansionIn this paper we study planar polynomial Kolmogorov's differential systems Xμ{x˙=f(x,y;μ),y˙=g(x,y;μ), with the parameter μ varying in an open subset Λ⊂RN. Compactifying Xμ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all μ∈Λ. We are interested in the cyclicity of Γ inside the family {Xμ}μ∈Λ, i.e., the number of limit cycles that bifurcate from Γ as we perturb μ. In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with N=3 and N=5, and in both cases we are able to determine the cyclicity of the polycycle for all μ∈Λ, including those parameters for which the return map along Γ is the identity.Centre de Recerca Matemàtica 22022-01-0120222022-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/265184https://dx.doi.org/urn:doi:10.14232/ejqtde.2022.1.35reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2651842026-06-06T12:50:31Z
dc.title.none.fl_str_mv On the cyclicity of Kolmogorov polycycles
title On the cyclicity of Kolmogorov polycycles
spellingShingle On the cyclicity of Kolmogorov polycycles
Marín, David|||0000-0003-4422-6418
Limit cycle
Polycycle
Cyclicity
Asymptotic expansion
title_short On the cyclicity of Kolmogorov polycycles
title_full On the cyclicity of Kolmogorov polycycles
title_fullStr On the cyclicity of Kolmogorov polycycles
title_full_unstemmed On the cyclicity of Kolmogorov polycycles
title_sort On the cyclicity of Kolmogorov polycycles
dc.creator.none.fl_str_mv Marín, David|||0000-0003-4422-6418
Villadelprat Yagüe, Jordi|||0000-0002-1168-9750
author Marín, David|||0000-0003-4422-6418
author_facet Marín, David|||0000-0003-4422-6418
Villadelprat Yagüe, Jordi|||0000-0002-1168-9750
author_role author
author2 Villadelprat Yagüe, Jordi|||0000-0002-1168-9750
author2_role author
dc.contributor.none.fl_str_mv Centre de Recerca Matemàtica
dc.subject.none.fl_str_mv Limit cycle
Polycycle
Cyclicity
Asymptotic expansion
topic Limit cycle
Polycycle
Cyclicity
Asymptotic expansion
description In this paper we study planar polynomial Kolmogorov's differential systems Xμ{x˙=f(x,y;μ),y˙=g(x,y;μ), with the parameter μ varying in an open subset Λ⊂RN. Compactifying Xμ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all μ∈Λ. We are interested in the cyclicity of Γ inside the family {Xμ}μ∈Λ, i.e., the number of limit cycles that bifurcate from Γ as we perturb μ. In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with N=3 and N=5, and in both cases we are able to determine the cyclicity of the polycycle for all μ∈Λ, including those parameters for which the return map along Γ is the identity.
publishDate 2022
dc.date.none.fl_str_mv 2
2022-01-01
2022
2022-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/265184
https://dx.doi.org/urn:doi:10.14232/ejqtde.2022.1.35
url https://ddd.uab.cat/record/265184
https://dx.doi.org/urn:doi:10.14232/ejqtde.2022.1.35
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
https://creativecommons.org/licenses/by/4.0/
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
https://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
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dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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