The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum nu...

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Authors: Li, Shimin|||0000-0003-1695-0097, Liu, Changjian, Llibre, Jaume|||0000-0002-9511-5999
Format: article
Publication Date:2021
Country:España
Institution:Universitat Autònoma de Barcelona
Repository:Dipòsit Digital de Documents de la UAB
Language:English
OAI Identifier:oai:ddd.uab.cat:257117
Online Access:https://ddd.uab.cat/record/257117
https://dx.doi.org/urn:doi:10.1016/j.nahs.2021.101045
Access Level:Open access
Keyword:Piecewise linear systems
Refracting systems
Limit cycle
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spelling The planar discontinuous piecewise linear refracting systems have at most one limit cycleLi, Shimin|||0000-0003-1695-0097Liu, ChangjianLlibre, Jaume|||0000-0002-9511-5999Piecewise linear systemsRefracting systemsLimit cycleIn this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles. The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus-focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle. 22021-01-0120212021-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/257117https://dx.doi.org/urn:doi:10.1016/j.nahs.2021.101045reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2013-40998-PAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 MTM2016-77278-PAgència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2014/SGR-568Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MDM-2014-0445open 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, 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/openAccessoai:ddd.uab.cat:2571172026-06-06T12:50:31Z
dc.title.none.fl_str_mv The planar discontinuous piecewise linear refracting systems have at most one limit cycle
title The planar discontinuous piecewise linear refracting systems have at most one limit cycle
spellingShingle The planar discontinuous piecewise linear refracting systems have at most one limit cycle
Li, Shimin|||0000-0003-1695-0097
Piecewise linear systems
Refracting systems
Limit cycle
title_short The planar discontinuous piecewise linear refracting systems have at most one limit cycle
title_full The planar discontinuous piecewise linear refracting systems have at most one limit cycle
title_fullStr The planar discontinuous piecewise linear refracting systems have at most one limit cycle
title_full_unstemmed The planar discontinuous piecewise linear refracting systems have at most one limit cycle
title_sort The planar discontinuous piecewise linear refracting systems have at most one limit cycle
dc.creator.none.fl_str_mv Li, Shimin|||0000-0003-1695-0097
Liu, Changjian
Llibre, Jaume|||0000-0002-9511-5999
author Li, Shimin|||0000-0003-1695-0097
author_facet Li, Shimin|||0000-0003-1695-0097
Liu, Changjian
Llibre, Jaume|||0000-0002-9511-5999
author_role author
author2 Liu, Changjian
Llibre, Jaume|||0000-0002-9511-5999
author2_role author
author
dc.subject.none.fl_str_mv Piecewise linear systems
Refracting systems
Limit cycle
topic Piecewise linear systems
Refracting systems
Limit cycle
description In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles. The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus-focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.
publishDate 2021
dc.date.none.fl_str_mv 2
2021-01-01
2021
2021-01-01
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/257117
https://dx.doi.org/urn:doi:10.1016/j.nahs.2021.101045
url https://ddd.uab.cat/record/257117
https://dx.doi.org/urn:doi:10.1016/j.nahs.2021.101045
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 MTM2013-40998-P
Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 MTM2016-77278-P
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2014/SGR-568
Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MDM-2014-0445
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://creativecommons.org/licenses/by-nc-nd/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-nc-nd/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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