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On the basin of attraction of a critical three-cycle of a model for the secant map

  • Datos identificativos

    Identificador: imarina:9382519
    Handle: https://hdl.handle.net/20.500.11797/imarina9382519
    Autores:
    Fontich, ErnestGarijo, AntonioJarque, Xavier
    Resumen:
    We consider the secant method Sp applied to a real polynomial p of degree d + 1 as a discrete dynamical system on R2 . If the polynomial p has a local extremum at a point α then the discrete dynamical system generated by the iterates of the secant map exhibits a critical periodic orbit of period 3 or three-cycle at the point (α, α). We propose a simple model map Ta,d having a unique fixed point at the origin which encodes the dynamical behaviour of S 3 p at the critical three-cycle. The main goal of the paper is to describe the geometry and topology of the basin of attraction of the origin of Ta,d as well as its boundary. Our results concern global, rather than local, dynamical behaviour. They include that the boundary of the basin of attraction is the stable manifold of a fixed point or contains the stable manifold of a two-cycle, depending on the values of the parameters of d (even or odd) and a ∈ R (positive or negative).

  • Otros:

    Autor según el artículo: Fontich, Ernest; Garijo, Antonio; Jarque, Xavier
    Departamento: Enginyeria Informàtica i Matemàtiques
    Código de proyecto: PID2020-118281GB-C33
    Resumen: We consider the secant method Sp applied to a real polynomial p of degree d + 1 as a discrete dynamical system on R2 . If the polynomial p has a local extremum at a point α then the discrete dynamical system generated by the iterates of the secant map exhibits a critical periodic orbit of period 3 or three-cycle at the point (α, α). We propose a simple model map Ta,d having a unique fixed point at the origin which encodes the dynamical behaviour of S 3 p at the critical three-cycle. The main goal of the paper is to describe the geometry and topology of the basin of attraction of the origin of Ta,d as well as its boundary. Our results concern global, rather than local, dynamical behaviour. They include that the boundary of the basin of attraction is the stable manifold of a fixed point or contains the stable manifold of a two-cycle, depending on the values of the parameters of d (even or odd) and a ∈ R (positive or negative).
    Acceso a la licencia de uso: https://creativecommons.org/licenses/by/3.0/es/
    Direcció de correo del autor: antonio.garijo@urv.cat
    Versión del articulo depositado: info:eu-repo/semantics/publishedVersion
    Enlace a la fuente original: https://www.aimsciences.org//article/doi/10.3934/dcds.2024122
    Programa de financiación: Herramientas para el análisis de diagramas de bifurcación en sistemas dinámicos
    Acrónimo: ATBiD
    DOI del artículo: 10.3934/dcds.2024122
    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