Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities

In this paper we study the limit cycles of the planar polynomial differential systems * x=ax-y P_n(x,y),\\ y=x ay Q_n(x,y), * where P_n and Q_n are homogeneous polynomials of degree n2, and a R. Consider the functions * &()=P_n() Q_n()\\ &()=Q_n()-P_n()\\ &_1()=a()-(),\\ &_2()=(n-1)(...

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Autores: Huang, Jianfeng, Liang, Haihua, Llibre, Jaume|||0000-0002-9511-5999
Tipo de recurso: artículo
Fecha de publicación:2018
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:199335
Acceso en línea:https://ddd.uab.cat/record/199335
https://dx.doi.org/urn:doi:10.1016/j.jde.2018.05.019
Access Level:acceso abierto
Palabra clave:Homogeneous nonlinearities
Limit cycles
Non-existence and uniqueness
Polynomial differential systems
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spelling Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearitiesHuang, JianfengLiang, HaihuaLlibre, Jaume|||0000-0002-9511-5999Homogeneous nonlinearitiesLimit cyclesNon-existence and uniquenessPolynomial differential systemsIn this paper we study the limit cycles of the planar polynomial differential systems * x=ax-y P_n(x,y),\\ y=x ay Q_n(x,y), * where P_n and Q_n are homogeneous polynomials of degree n2, and a R. Consider the functions * &()=P_n() Q_n()\\ &()=Q_n()-P_n()\\ &_1()=a()-(),\\ &_2()=(n-1)(2a()-()) '(). *First we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of _1 and _2 with definite sign. This result improves previous knwon results. Furthermore, if _1(_1a-_2)0 for some _1,_20, we provide necessary and sufficient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these differential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin. 22018-01-0120182018-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/199335https://dx.doi.org/urn:doi:10.1016/j.jde.2018.05.019reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2016-77278-PMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2013-40998-PAgència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2014/SGR-568open accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:1993352026-06-06T12:50:31Z
dc.title.none.fl_str_mv Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
title Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
spellingShingle Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
Huang, Jianfeng
Homogeneous nonlinearities
Limit cycles
Non-existence and uniqueness
Polynomial differential systems
title_short Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
title_full Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
title_fullStr Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
title_full_unstemmed Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
title_sort Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
dc.creator.none.fl_str_mv Huang, Jianfeng
Liang, Haihua
Llibre, Jaume|||0000-0002-9511-5999
author Huang, Jianfeng
author_facet Huang, Jianfeng
Liang, Haihua
Llibre, Jaume|||0000-0002-9511-5999
author_role author
author2 Liang, Haihua
Llibre, Jaume|||0000-0002-9511-5999
author2_role author
author
dc.subject.none.fl_str_mv Homogeneous nonlinearities
Limit cycles
Non-existence and uniqueness
Polynomial differential systems
topic Homogeneous nonlinearities
Limit cycles
Non-existence and uniqueness
Polynomial differential systems
description In this paper we study the limit cycles of the planar polynomial differential systems * x=ax-y P_n(x,y),\\ y=x ay Q_n(x,y), * where P_n and Q_n are homogeneous polynomials of degree n2, and a R. Consider the functions * &()=P_n() Q_n()\\ &()=Q_n()-P_n()\\ &_1()=a()-(),\\ &_2()=(n-1)(2a()-()) '(). *First we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of _1 and _2 with definite sign. This result improves previous knwon results. Furthermore, if _1(_1a-_2)0 for some _1,_20, we provide necessary and sufficient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these differential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin.
publishDate 2018
dc.date.none.fl_str_mv 2
2018-01-01
2018
2018-01-01
dc.type.none.fl_str_mv 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://ddd.uab.cat/record/199335
https://dx.doi.org/urn:doi:10.1016/j.jde.2018.05.019
url https://ddd.uab.cat/record/199335
https://dx.doi.org/urn:doi:10.1016/j.jde.2018.05.019
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 https://doi.org/10.13039/501100003329 MTM2016-77278-P
Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2013-40998-P
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2014/SGR-568
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
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://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
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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