On the number of limit cycles in piecewise planar quadratic differential systems

We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals i...

Descripción completa

Detalles Bibliográficos
Autores: Braun, Francisco|||0000-0003-3594-9809, Da Cruz, Leonardo Pereira Costa|||0000-0002-2853-4974, Torregrosa, Joan|||0000-0002-2753-1827
Tipo de recurso: artículo
Fecha de publicación:2024
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:299739
Acceso en línea:https://ddd.uab.cat/record/299739
https://dx.doi.org/urn:doi:10.1016/j.nonrwa.2024.104124
Access Level:acceso embargado
Palabra clave:Periodic solution
Averaging method
Non-smooth differential system
id ES_dfdf8cfb0f48e695cdebd0d555b3fb27
oai_identifier_str oai:ddd.uab.cat:299739
network_acronym_str ES
network_name_str España
repository_id_str
spelling On the number of limit cycles in piecewise planar quadratic differential systemsBraun, Francisco|||0000-0003-3594-9809Da Cruz, Leonardo Pereira Costa|||0000-0002-2853-4974Torregrosa, Joan|||0000-0002-2753-1827Periodic solutionAveraging methodNon-smooth differential systemWe consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S1, S2, S3, and S4, as well as to non-smooth perturbations of non-smooth centers given by putting different Si's in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper. 220242024-01-0120262026-10-31Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articlehttps://ddd.uab.cat/record/299739https://dx.doi.org/urn:doi:10.1016/j.nonrwa.2024.104124reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104658GB-I00Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 CEX2020-001084-MEuropean Commission https://doi.org/10.13039/501100000780 777911Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00113embargoed accesshttp://purl.org/coar/access_right/c_f1cfAquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.https://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/embargoedAccessoai:ddd.uab.cat:2997392026-06-06T12:50:31Z
dc.title.none.fl_str_mv On the number of limit cycles in piecewise planar quadratic differential systems
title On the number of limit cycles in piecewise planar quadratic differential systems
spellingShingle On the number of limit cycles in piecewise planar quadratic differential systems
Braun, Francisco|||0000-0003-3594-9809
Periodic solution
Averaging method
Non-smooth differential system
title_short On the number of limit cycles in piecewise planar quadratic differential systems
title_full On the number of limit cycles in piecewise planar quadratic differential systems
title_fullStr On the number of limit cycles in piecewise planar quadratic differential systems
title_full_unstemmed On the number of limit cycles in piecewise planar quadratic differential systems
title_sort On the number of limit cycles in piecewise planar quadratic differential systems
dc.creator.none.fl_str_mv Braun, Francisco|||0000-0003-3594-9809
Da Cruz, Leonardo Pereira Costa|||0000-0002-2853-4974
Torregrosa, Joan|||0000-0002-2753-1827
author Braun, Francisco|||0000-0003-3594-9809
author_facet Braun, Francisco|||0000-0003-3594-9809
Da Cruz, Leonardo Pereira Costa|||0000-0002-2853-4974
Torregrosa, Joan|||0000-0002-2753-1827
author_role author
author2 Da Cruz, Leonardo Pereira Costa|||0000-0002-2853-4974
Torregrosa, Joan|||0000-0002-2753-1827
author2_role author
author
dc.subject.none.fl_str_mv Periodic solution
Averaging method
Non-smooth differential system
topic Periodic solution
Averaging method
Non-smooth differential system
description We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S1, S2, S3, and S4, as well as to non-smooth perturbations of non-smooth centers given by putting different Si's in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.
publishDate 2024
dc.date.none.fl_str_mv
2
2024
2024-01-01
2026
2026-10-31
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/299739
https://dx.doi.org/urn:doi:10.1016/j.nonrwa.2024.104124
url https://ddd.uab.cat/record/299739
https://dx.doi.org/urn:doi:10.1016/j.nonrwa.2024.104124
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104658GB-I00
Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 CEX2020-001084-M
European Commission https://doi.org/10.13039/501100000780 777911
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2021/SGR-00113
dc.rights.none.fl_str_mv embargoed access
http://purl.org/coar/access_right/c_f1cf
https://creativecommons.org/licenses/by-nc-nd/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/embargoedAccess
rights_invalid_str_mv embargoed access
http://purl.org/coar/access_right/c_f1cf
https://creativecommons.org/licenses/by-nc-nd/4.0/
eu_rights_str_mv embargoedAccess
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
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869422103233560576
score 15.812429

Similares en LA Referencia