Number of limit cycles for planar systems with invariant algebraic curves

For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric fa...

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Autores: Gasull, Armengol|||0000-0002-1719-8231, Giacomini, Hector
Tipo de recurso: artículo
Fecha de publicación:2023
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:274790
Acceso en línea:https://ddd.uab.cat/record/274790
https://dx.doi.org/urn:doi:10.1007/s12346-023-00746-7
Access Level:acceso abierto
Palabra clave:Invariant algebraic curve
Limit cycle
Periodic orbit
Quadratic system
Cubic system
Liénard system
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spelling Number of limit cycles for planar systems with invariant algebraic curvesGasull, Armengol|||0000-0002-1719-8231Giacomini, HectorInvariant algebraic curveLimit cyclePeriodic orbitQuadratic systemCubic systemLiénard systemFor planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non-existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Liénard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line. 22023-01-0120232023-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/274790https://dx.doi.org/urn:doi:10.1007/s12346-023-00746-7reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2017/SGR-1617Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 CEX2020-001084-MAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104658GB-I00open 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:2747902026-06-06T12:50:31Z
dc.title.none.fl_str_mv Number of limit cycles for planar systems with invariant algebraic curves
title Number of limit cycles for planar systems with invariant algebraic curves
spellingShingle Number of limit cycles for planar systems with invariant algebraic curves
Gasull, Armengol|||0000-0002-1719-8231
Invariant algebraic curve
Limit cycle
Periodic orbit
Quadratic system
Cubic system
Liénard system
title_short Number of limit cycles for planar systems with invariant algebraic curves
title_full Number of limit cycles for planar systems with invariant algebraic curves
title_fullStr Number of limit cycles for planar systems with invariant algebraic curves
title_full_unstemmed Number of limit cycles for planar systems with invariant algebraic curves
title_sort Number of limit cycles for planar systems with invariant algebraic curves
dc.creator.none.fl_str_mv Gasull, Armengol|||0000-0002-1719-8231
Giacomini, Hector
author Gasull, Armengol|||0000-0002-1719-8231
author_facet Gasull, Armengol|||0000-0002-1719-8231
Giacomini, Hector
author_role author
author2 Giacomini, Hector
author2_role author
dc.subject.none.fl_str_mv Invariant algebraic curve
Limit cycle
Periodic orbit
Quadratic system
Cubic system
Liénard system
topic Invariant algebraic curve
Limit cycle
Periodic orbit
Quadratic system
Cubic system
Liénard system
description For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non-existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Liénard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.
publishDate 2023
dc.date.none.fl_str_mv 2
2023-01-01
2023
2023-01-01
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/274790
https://dx.doi.org/urn:doi:10.1007/s12346-023-00746-7
url https://ddd.uab.cat/record/274790
https://dx.doi.org/urn:doi:10.1007/s12346-023-00746-7
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2017/SGR-1617
Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 CEX2020-001084-M
Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104658GB-I00
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
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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repository.mail.fl_str_mv
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