Articles producció científica> Enginyeria Informàtica i Matemàtiques

The period of the limit cycle bifurcating from a persistent polycycle

  • Dades identificatives

    Identificador: imarina:9391714
    Handle: https://hdl.handle.net/20.500.11797/imarina9391714
    Autors:
    Marín, DavidQueiroz, LucasVilladelprat, Jordi
    Resum:
    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.

  • Altres:

    Autor segons l'article: Marín, David; Queiroz, Lucas; Villadelprat, Jordi
    Departament: Enginyeria Informàtica i Matemàtiques
    Codi de projecte: PID2020-118281GB-C33
    Resum: 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.
    Accès a la llicència d'ús: https://creativecommons.org/licenses/by/3.0/es/
    Adreça de correu electrònic de l'autor: jordi.villadelprat@urv.cat
    ISSN: 0214-1493
    Versió de l'article dipositat: info:eu-repo/semantics/acceptedVersion
    Programa de finançament: Herramientas para el análisis de diagramas de bifurcación en sistemas dinámicos
    Acrònim: ATBiD
    Any de publicació de la revista: 2024
    Acció del programa de finançament: Proyectos I+D Generación de Conocimiento
    Tipus de publicació: info:eu-repo/semantics/article