The local cyclicity problem: Melnikov method using Lyapunov constants

Detalhes bibliográficos
Autor(a) principal: Gouveia, Luiz F. S. [UNESP]
Data de Publicação: 2022
Outros Autores: Torregrosa, Joan
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Institucional da UNESP
Texto Completo: http://dx.doi.org/10.1017/S0013091522000128
http://hdl.handle.net/11449/239965
Resumo: In 1991, Chicone and Jacobs showed the equivalence between the computation of the firstorder Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, soM(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp(4) ≥ 43 and Mcp(5) ≥ 65.
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spelling The local cyclicity problem: Melnikov method using Lyapunov constantslocal cyclicityLyapunov constantsMelnikov theoryIn 1991, Chicone and Jacobs showed the equivalence between the computation of the firstorder Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, soM(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp(4) ≥ 43 and Mcp(5) ≥ 65.Departament de Matemàtiques Universitat Autònoma de Barcelona, BellaterraDepartamento de Matemática Universidade Estadual PaulistaCentre de Recerca Matemàtica, Campus de Bellaterra, BellaterraDepartamento de Matemática Universidade Estadual PaulistaUniversitat Autònoma de BarcelonaUniversidade Estadual Paulista (UNESP)Centre de Recerca MatemàticaGouveia, Luiz F. S. [UNESP]Torregrosa, Joan2023-03-01T19:55:25Z2023-03-01T19:55:25Z2022-05-19info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article356-375http://dx.doi.org/10.1017/S0013091522000128Proceedings of the Edinburgh Mathematical Society, v. 65, n. 2, p. 356-375, 2022.1464-38390013-0915http://hdl.handle.net/11449/23996510.1017/S00130915220001282-s2.0-85129202595Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengProceedings of the Edinburgh Mathematical Societyinfo:eu-repo/semantics/openAccess2023-03-01T19:55:25Zoai:repositorio.unesp.br:11449/239965Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T23:38:23.387988Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false
dc.title.none.fl_str_mv The local cyclicity problem: Melnikov method using Lyapunov constants
title The local cyclicity problem: Melnikov method using Lyapunov constants
spellingShingle The local cyclicity problem: Melnikov method using Lyapunov constants
Gouveia, Luiz F. S. [UNESP]
local cyclicity
Lyapunov constants
Melnikov theory
title_short The local cyclicity problem: Melnikov method using Lyapunov constants
title_full The local cyclicity problem: Melnikov method using Lyapunov constants
title_fullStr The local cyclicity problem: Melnikov method using Lyapunov constants
title_full_unstemmed The local cyclicity problem: Melnikov method using Lyapunov constants
title_sort The local cyclicity problem: Melnikov method using Lyapunov constants
author Gouveia, Luiz F. S. [UNESP]
author_facet Gouveia, Luiz F. S. [UNESP]
Torregrosa, Joan
author_role author
author2 Torregrosa, Joan
author2_role author
dc.contributor.none.fl_str_mv Universitat Autònoma de Barcelona
Universidade Estadual Paulista (UNESP)
Centre de Recerca Matemàtica
dc.contributor.author.fl_str_mv Gouveia, Luiz F. S. [UNESP]
Torregrosa, Joan
dc.subject.por.fl_str_mv local cyclicity
Lyapunov constants
Melnikov theory
topic local cyclicity
Lyapunov constants
Melnikov theory
description In 1991, Chicone and Jacobs showed the equivalence between the computation of the firstorder Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, soM(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp(4) ≥ 43 and Mcp(5) ≥ 65.
publishDate 2022
dc.date.none.fl_str_mv 2022-05-19
2023-03-01T19:55:25Z
2023-03-01T19:55:25Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://dx.doi.org/10.1017/S0013091522000128
Proceedings of the Edinburgh Mathematical Society, v. 65, n. 2, p. 356-375, 2022.
1464-3839
0013-0915
http://hdl.handle.net/11449/239965
10.1017/S0013091522000128
2-s2.0-85129202595
url http://dx.doi.org/10.1017/S0013091522000128
http://hdl.handle.net/11449/239965
identifier_str_mv Proceedings of the Edinburgh Mathematical Society, v. 65, n. 2, p. 356-375, 2022.
1464-3839
0013-0915
10.1017/S0013091522000128
2-s2.0-85129202595
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Proceedings of the Edinburgh Mathematical Society
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 356-375
dc.source.none.fl_str_mv Scopus
reponame:Repositório Institucional da UNESP
instname:Universidade Estadual Paulista (UNESP)
instacron:UNESP
instname_str Universidade Estadual Paulista (UNESP)
instacron_str UNESP
institution UNESP
reponame_str Repositório Institucional da UNESP
collection Repositório Institucional da UNESP
repository.name.fl_str_mv Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)
repository.mail.fl_str_mv
_version_ 1808129539609460736

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