Articles producció científica> Enginyeria Informàtica i Matemàtiques

The period of the limit cycle bifurcating from a persistent polycycle

  • Identification data

    Identifier: imarina:9391714
    Handle: https://hdl.handle.net/20.500.11797/imarina9391714
    Authors:
    Marín, DavidQueiroz, LucasVilladelprat, Jordi
    Abstract:
    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.

  • Others:

    Author, as appears in the article.: Marín, David; Queiroz, Lucas; Villadelprat, Jordi
    Department: Enginyeria Informàtica i Matemàtiques
    Project code: PID2020-118281GB-C33
    Abstract: 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.
    licence for use: https://creativecommons.org/licenses/by/3.0/es/
    Author's mail: jordi.villadelprat@urv.cat
    ISSN: 0214-1493
    Papper version: info:eu-repo/semantics/acceptedVersion
    Funding program: Herramientas para el análisis de diagramas de bifurcación en sistemas dinámicos
    Acronym: ATBiD
    Journal publication year: 2024
    Funding program action: Proyectos I+D Generación de Conocimiento
    Publication Type: info:eu-repo/semantics/article