On the number of limit cycles in generalized abel equations

Given p, q ∊ Z ≥ 2 with p ≠ q, we study generalized Abel differential equations (Equation presented), where A and B are trigonometric polynomials of degrees n, m ≥ 1, respectively, and we are interested in the number of limit cycles (i.e., isolated periodic orbits) that they can have. More concretel...

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Autores: Huang, Jianfeng, Torregrosa, Joan|||0000-0002-2753-1827, Villadelprat Yagüe, Jordi|||0000-0002-1168-9750
Tipo de recurso: artículo
Fecha de publicación:2020
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:236658
Acceso en línea:https://ddd.uab.cat/record/236658
https://dx.doi.org/urn:doi:10.1137/20M1340083
Access Level:acceso abierto
Palabra clave:Generalized Abel equations
Melnikov theory
Second order perturbation
Limit cycles
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spelling On the number of limit cycles in generalized abel equationsHuang, JianfengTorregrosa, Joan|||0000-0002-2753-1827Villadelprat Yagüe, Jordi|||0000-0002-1168-9750Generalized Abel equationsMelnikov theorySecond order perturbationLimit cyclesGiven p, q ∊ Z ≥ 2 with p ≠ q, we study generalized Abel differential equations (Equation presented), where A and B are trigonometric polynomials of degrees n, m ≥ 1, respectively, and we are interested in the number of limit cycles (i.e., isolated periodic orbits) that they can have. More concretely, in this context, an open problem is to prove the existence of an integer, depending only on p, q, m, and n and that we denote by H p,q(n, m), such that the above differential equation has at most H p,q(n, m) limit cycles. In the present paper, by means of a second order analysis using Melnikov functions, we provide lower bounds of H p,q(n, m) that, to the best of our knowledge, are larger than the previous ones appearing in the literature. In particular, for classical Abel differential equations (i.e., p = 3 and q = 2), we prove that H 3,2(n, m) ≥ 2(n + m) - 1. 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/236658https://dx.doi.org/urn:doi:10.1137/20M1340083reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2017/SGR-1617Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2016-77278-PAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 MTM2017-86795-C3-2-PEuropean 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:2366582026-06-06T12:50:31Z
dc.title.none.fl_str_mv On the number of limit cycles in generalized abel equations
title On the number of limit cycles in generalized abel equations
spellingShingle On the number of limit cycles in generalized abel equations
Huang, Jianfeng
Generalized Abel equations
Melnikov theory
Second order perturbation
Limit cycles
title_short On the number of limit cycles in generalized abel equations
title_full On the number of limit cycles in generalized abel equations
title_fullStr On the number of limit cycles in generalized abel equations
title_full_unstemmed On the number of limit cycles in generalized abel equations
title_sort On the number of limit cycles in generalized abel equations
dc.creator.none.fl_str_mv Huang, Jianfeng
Torregrosa, Joan|||0000-0002-2753-1827
Villadelprat Yagüe, Jordi|||0000-0002-1168-9750
author Huang, Jianfeng
author_facet Huang, Jianfeng
Torregrosa, Joan|||0000-0002-2753-1827
Villadelprat Yagüe, Jordi|||0000-0002-1168-9750
author_role author
author2 Torregrosa, Joan|||0000-0002-2753-1827
Villadelprat Yagüe, Jordi|||0000-0002-1168-9750
author2_role author
author
dc.subject.none.fl_str_mv Generalized Abel equations
Melnikov theory
Second order perturbation
Limit cycles
topic Generalized Abel equations
Melnikov theory
Second order perturbation
Limit cycles
description Given p, q ∊ Z ≥ 2 with p ≠ q, we study generalized Abel differential equations (Equation presented), where A and B are trigonometric polynomials of degrees n, m ≥ 1, respectively, and we are interested in the number of limit cycles (i.e., isolated periodic orbits) that they can have. More concretely, in this context, an open problem is to prove the existence of an integer, depending only on p, q, m, and n and that we denote by H p,q(n, m), such that the above differential equation has at most H p,q(n, m) limit cycles. In the present paper, by means of a second order analysis using Melnikov functions, we provide lower bounds of H p,q(n, m) that, to the best of our knowledge, are larger than the previous ones appearing in the literature. In particular, for classical Abel differential equations (i.e., p = 3 and q = 2), we prove that H 3,2(n, m) ≥ 2(n + m) - 1.
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/236658
https://dx.doi.org/urn:doi:10.1137/20M1340083
url https://ddd.uab.cat/record/236658
https://dx.doi.org/urn:doi:10.1137/20M1340083
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2017/SGR-1617
Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2016-77278-P
Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 MTM2017-86795-C3-2-P
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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