The period of the limit cycle bifurcating from a persistent polycycle
Marín Pérez, David (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Queiroz, Lucas (Universidade Estadual Paulista "Júlio de Mesquita Filho". Instituto de Biociências, Letras e Ciências Exatas)
Villadelprat Yagüe, Jordi (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2025
Abstract:	We consider smooth families of planar polynomial vector fields {Xµ}µ∈Λ, where Λ is an open subset of R N , for which there is a hyperbolic polycycle Γ that is persistent (i. e. , such that none of the separatrix connections is broken along the family). It is well known that in this case the cyclicity of Γ at µ0 is zero unless its graphic number r(µ0) is equal to one. It is also well known that if r(µ0) = 1 (and some generic conditions on the return map are verified) then the cyclicity of Γ at µ0 is one, i. e. , exactly one limit cycle bifurcates from Γ. In this paper we prove that this limit cycle approaches Γ exponentially fast and that its period goes to infinity as 1/|r(µ) - 1| when µ → µ0. Moreover, we prove that if those generic conditions are not satisfied, although the cyclicity may be exactly 1, the behavior of the period of the limit cycle is not determined.
Grants:	Agencia Estatal de Investigación PID2021-125625NB-I00 Agencia Estatal de Investigación PID2020-118281GB-C33 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-01015 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113 Agencia Estatal de Investigación CEX2020-001084-M
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Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Limit cycle ; Polycycle ; Cyclicity ; Period ; Asymptotic expansion ; Dulac map
Published in:	Publicacions matemàtiques, Vol. 69 Núm. 2 (2025) , p. 299-318 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/10000006089
DOI: 10.5565/PUBLMAT6922502

20 p, 567.8 KB

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Articles > Published articles > Publicacions matemàtiques
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