Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory

Recently there is increasing interest in studying the limit cycles of the piecewise differential systems due to their many applications. In this paper we prove that the linear system ẋ = y, ẏ = -x, can produce at most seven crossing limit cycles for n ≥ 4 using the averaging theory of first order, w...

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Autores: Guo, Zhifei|||0000-0002-8092-7986, Llibre, Jaume|||0000-0002-9511-5999
Tipo de recurso: artículo
Fecha de publicación:2022
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:268645
Acceso en línea:https://ddd.uab.cat/record/268645
https://dx.doi.org/urn:doi:10.1142/S0218127422501875
Access Level:acceso abierto
Palabra clave:Limit cycles
The method of averaging
Discontinuous piecewise differential systems
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spelling Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theoryLimit cycles of discontinuous piecewise quadratic perturbations of a linear center separated by the curve y = xnGuo, Zhifei|||0000-0002-8092-7986Llibre, Jaume|||0000-0002-9511-5999Limit cyclesThe method of averagingDiscontinuous piecewise differential systemsRecently there is increasing interest in studying the limit cycles of the piecewise differential systems due to their many applications. In this paper we prove that the linear system ẋ = y, ẏ = -x, can produce at most seven crossing limit cycles for n ≥ 4 using the averaging theory of first order, where the bounds ≤ 4 for n ≥ 4 even and the bounds ≤ 7 for n ≥ 5 odd are reachable, when it is perturbed by discontinuous piecewise polynomials formed by two pieces separated by the curve y = xn (n ≥ 4), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has 0, 1, 2, 3, 4, 5 or 6 crossing limit cycles for n ≥ 4 even and, furthermore, under a particular condition we prove that the number of crossing limit cycles does not exceed 9 (resp., 11) for 4 ≤ n ≤ 74 even (resp., n ≥ 76 even). The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if n ≥ 5 is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems.The study of the limit cycles of planar differential systems is one of the main problems in the qualitative theory of differential systems. These last years a big interest appeared for studying the limit cycles of the piecewise differential systems due to their many applications. Here we prove that the linear center x˙ = y, y˙ = -x, can produce at most 6 crossing limit cycles for n ≥ 4 even and at most 7 crossing limit cycles for n ≥ 5 odd using the averaging theory of first order, when it is perturbed by discontinuous piecewise differential systems formed by two pieces separated by the curve y = xn (n ≥ 4), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has 0, 1, 2, 3, 4, 5, 6, 7 or 9 crossing limit cycles if 4 ≤ n ≤ 74 is even and 0, 1, 2, 3, 4, 5, 6, 7, 9 or 11 crossing limit cycles if n ≥ 76 is even. The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if n ≥ 5 is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems. 22022-01-0120222022-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/268645https://dx.doi.org/urn:doi:10.1142/S0218127422501875reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104658GB-I00European 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:2686452026-06-06T12:50:31Z
dc.title.none.fl_str_mv Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory
Limit cycles of discontinuous piecewise quadratic perturbations of a linear center separated by the curve y = xn
title Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory
spellingShingle Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory
Guo, Zhifei|||0000-0002-8092-7986
Limit cycles
The method of averaging
Discontinuous piecewise differential systems
title_short Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory
title_full Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory
title_fullStr Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory
title_full_unstemmed Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory
title_sort Limit cycles of a class of discontinuous piecewise differential systems separated by the curve y = xn via averaging theory
dc.creator.none.fl_str_mv Guo, Zhifei|||0000-0002-8092-7986
Llibre, Jaume|||0000-0002-9511-5999
author Guo, Zhifei|||0000-0002-8092-7986
author_facet Guo, Zhifei|||0000-0002-8092-7986
Llibre, Jaume|||0000-0002-9511-5999
author_role author
author2 Llibre, Jaume|||0000-0002-9511-5999
author2_role author
dc.subject.none.fl_str_mv Limit cycles
The method of averaging
Discontinuous piecewise differential systems
topic Limit cycles
The method of averaging
Discontinuous piecewise differential systems
description Recently there is increasing interest in studying the limit cycles of the piecewise differential systems due to their many applications. In this paper we prove that the linear system ẋ = y, ẏ = -x, can produce at most seven crossing limit cycles for n ≥ 4 using the averaging theory of first order, where the bounds ≤ 4 for n ≥ 4 even and the bounds ≤ 7 for n ≥ 5 odd are reachable, when it is perturbed by discontinuous piecewise polynomials formed by two pieces separated by the curve y = xn (n ≥ 4), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has 0, 1, 2, 3, 4, 5 or 6 crossing limit cycles for n ≥ 4 even and, furthermore, under a particular condition we prove that the number of crossing limit cycles does not exceed 9 (resp., 11) for 4 ≤ n ≤ 74 even (resp., n ≥ 76 even). The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if n ≥ 5 is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems.
publishDate 2022
dc.date.none.fl_str_mv 2
2022-01-01
2022
2022-01-01
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/268645
https://dx.doi.org/urn:doi:10.1142/S0218127422501875
url https://ddd.uab.cat/record/268645
https://dx.doi.org/urn:doi:10.1142/S0218127422501875
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
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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