Articles producció científica> Enginyeria Informàtica i Matemàtiques

The period of the limit cycle bifurcating from a persistent polycycle

  • Datos identificativos

    Identificador: imarina:9391714
    Handle: https://hdl.handle.net/20.500.11797/imarina9391714
    Autores:
    Marín, DavidQueiroz, LucasVilladelprat, Jordi
    Resumen:
    We consider smooth families of planar polynomial vector fields {Xμ}μ∈Λ, where Λ is an open subset of RN, 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.

  • Otros:

    Autor según el artículo: Marín, David; Queiroz, Lucas; Villadelprat, Jordi
    Departamento: Enginyeria Informàtica i Matemàtiques
    Código de proyecto: PID2020-118281GB-C33
    Resumen: We consider smooth families of planar polynomial vector fields {Xμ}μ∈Λ, where Λ is an open subset of RN, 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.
    Acceso a la licencia de uso: https://creativecommons.org/licenses/by/3.0/es/
    Direcció de correo del autor: jordi.villadelprat@urv.cat
    ISSN: 0214-1493
    Versión del articulo depositado: info:eu-repo/semantics/acceptedVersion
    Programa de financiación: Herramientas para el análisis de diagramas de bifurcación en sistemas dinámicos
    Acrónimo: ATBiD
    Año de publicación de la revista: 2024
    Acción del progama de financiación: Proyectos I+D Generación de Conocimiento
    Tipo de publicación: info:eu-repo/semantics/article