The local cyclicity problem: Melnikov method using Lyapunov constants
Autor(a) principal: | |
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Data de Publicação: | 2022 |
Outros Autores: | |
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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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 |