Limit cycles via higher order perturbations for some piecewise differential systems

A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, (x',y')=(-y + εf(x,y,ε),x εg(x,y,ε)). In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line....

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Autores: Buzzi, Claudio.|||0000-0003-2037-8417, Lima, Mauricio Firmino Silva, Torregrosa, Joan|||0000-0002-2753-1827
Tipo de documento: artigo
Data de publicação:2018
País:España
Recursos:Universitat Autònoma de Barcelona
Repositório:Dipòsit Digital de Documents de la UAB
Idioma:inglês
OAI Identifier:oai:ddd.uab.cat:199355
Acesso em linha:https://ddd.uab.cat/record/199355
https://dx.doi.org/urn:doi:10.1016/j.physd.2018.01.007
Access Level:Acceso aberto
Palavra-chave:Liénard piecewise differential system
Limit cycle in Melnikov higher order perturbation
Non-smooth differential system
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spelling Limit cycles via higher order perturbations for some piecewise differential systemsBuzzi, Claudio.|||0000-0003-2037-8417Lima, Mauricio Firmino SilvaTorregrosa, Joan|||0000-0002-2753-1827Liénard piecewise differential systemLimit cycle in Melnikov higher order perturbationNon-smooth differential systemA classical perturbation problem is the polynomial perturbation of the harmonic oscillator, (x',y')=(-y + εf(x,y,ε),x εg(x,y,ε)). In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n, no more than Nn-1 limit cycles appear up to a study of order N. We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Li\'enard differential systems. When we restrict the analysis to some special class this upper bound never is attained and we show which is this upper bound for higher order perturbation in . The Poincar\'e--Pontryagin--Melnikov theory is the main technique used to prove all the results. 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/199355https://dx.doi.org/urn:doi:10.1016/j.physd.2018.01.007reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengEuropean Commission https://doi.org/10.13039/501100000780 318999European Commission https://doi.org/10.13039/501100000780 316338Ministerio 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:1993552026-06-06T12:50:31Z
dc.title.none.fl_str_mv Limit cycles via higher order perturbations for some piecewise differential systems
title Limit cycles via higher order perturbations for some piecewise differential systems
spellingShingle Limit cycles via higher order perturbations for some piecewise differential systems
Buzzi, Claudio.|||0000-0003-2037-8417
Liénard piecewise differential system
Limit cycle in Melnikov higher order perturbation
Non-smooth differential system
title_short Limit cycles via higher order perturbations for some piecewise differential systems
title_full Limit cycles via higher order perturbations for some piecewise differential systems
title_fullStr Limit cycles via higher order perturbations for some piecewise differential systems
title_full_unstemmed Limit cycles via higher order perturbations for some piecewise differential systems
title_sort Limit cycles via higher order perturbations for some piecewise differential systems
dc.creator.none.fl_str_mv Buzzi, Claudio.|||0000-0003-2037-8417
Lima, Mauricio Firmino Silva
Torregrosa, Joan|||0000-0002-2753-1827
author Buzzi, Claudio.|||0000-0003-2037-8417
author_facet Buzzi, Claudio.|||0000-0003-2037-8417
Lima, Mauricio Firmino Silva
Torregrosa, Joan|||0000-0002-2753-1827
author_role author
author2 Lima, Mauricio Firmino Silva
Torregrosa, Joan|||0000-0002-2753-1827
author2_role author
author
dc.subject.none.fl_str_mv Liénard piecewise differential system
Limit cycle in Melnikov higher order perturbation
Non-smooth differential system
topic Liénard piecewise differential system
Limit cycle in Melnikov higher order perturbation
Non-smooth differential system
description A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, (x',y')=(-y + εf(x,y,ε),x εg(x,y,ε)). In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n, no more than Nn-1 limit cycles appear up to a study of order N. We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Li\'enard differential systems. When we restrict the analysis to some special class this upper bound never is attained and we show which is this upper bound for higher order perturbation in . The Poincar\'e--Pontryagin--Melnikov theory is the main technique used to prove all the results.
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/199355
https://dx.doi.org/urn:doi:10.1016/j.physd.2018.01.007
url https://ddd.uab.cat/record/199355
https://dx.doi.org/urn:doi:10.1016/j.physd.2018.01.007
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv European Commission https://doi.org/10.13039/501100000780 318999
European Commission https://doi.org/10.13039/501100000780 316338
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
repository.name.fl_str_mv
repository.mail.fl_str_mv
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