Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems as...

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Autores: Cardoso, Joäo L., Llibre, Jaume|||0000-0002-9511-5999, Novaes, Douglas D.|||0000-0002-9147-8442, Tonon, Durval José|||0000-0002-2733-1825
Tipo de documento: artigo
Data de publicação:2020
País:España
Recursos:Universitat Autònoma de Barcelona
Repositório:Dipòsit Digital de Documents de la UAB
Idioma:inglês
OAI Identifier:oai:ddd.uab.cat:221365
Acesso em linha:https://ddd.uab.cat/record/221365
https://dx.doi.org/urn:doi:10.1080/14689367.2020.1722064
Access Level:Acceso aberto
Palavra-chave:Non-smooth differential system
Limit cycle
Piecewise linear differential system
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spelling Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fieldsCardoso, Joäo L.Llibre, Jaume|||0000-0002-9511-5999Novaes, Douglas D.|||0000-0002-9147-8442Tonon, Durval José|||0000-0002-2733-1825Non-smooth differential systemLimit cyclePiecewise linear differential systemIn the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has centre, a real one for y<0 and a virtual one for y>0, and such that the real centre is a global centre. Then, working with a first-order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second-order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation. 22020-01-0120202020-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/221365https://dx.doi.org/urn:doi:10.1080/14689367.2020.1722064reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2016-77278-PAgència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2017/SGR-1617European Commission https://doi.org/10.13039/501100000780 777911open 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:2213652026-06-06T12:50:31Z
dc.title.none.fl_str_mv Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
title Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
spellingShingle Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
Cardoso, Joäo L.
Non-smooth differential system
Limit cycle
Piecewise linear differential system
title_short Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
title_full Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
title_fullStr Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
title_full_unstemmed Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
title_sort Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
dc.creator.none.fl_str_mv Cardoso, Joäo L.
Llibre, Jaume|||0000-0002-9511-5999
Novaes, Douglas D.|||0000-0002-9147-8442
Tonon, Durval José|||0000-0002-2733-1825
author Cardoso, Joäo L.
author_facet Cardoso, Joäo L.
Llibre, Jaume|||0000-0002-9511-5999
Novaes, Douglas D.|||0000-0002-9147-8442
Tonon, Durval José|||0000-0002-2733-1825
author_role author
author2 Llibre, Jaume|||0000-0002-9511-5999
Novaes, Douglas D.|||0000-0002-9147-8442
Tonon, Durval José|||0000-0002-2733-1825
author2_role author
author
author
dc.subject.none.fl_str_mv Non-smooth differential system
Limit cycle
Piecewise linear differential system
topic Non-smooth differential system
Limit cycle
Piecewise linear differential system
description In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has centre, a real one for y<0 and a virtual one for y>0, and such that the real centre is a global centre. Then, working with a first-order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second-order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.
publishDate 2020
dc.date.none.fl_str_mv 2
2020-01-01
2020
2020-01-01
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/221365
https://dx.doi.org/urn:doi:10.1080/14689367.2020.1722064
url https://ddd.uab.cat/record/221365
https://dx.doi.org/urn:doi:10.1080/14689367.2020.1722064
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
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2017/SGR-1617
European Commission https://doi.org/10.13039/501100000780 777911
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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repository.mail.fl_str_mv
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