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. Instituto de Biociências, Letras e Ciências Exatas)
Villadelprat Yagüe, Jordi (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2024
Description:	27 pàg.
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
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Published in:	Publicacions Matemàtiques, (February 2024) , ISSN 0214-1493

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