Articles producció científica> Enginyeria Informàtica i Matemàtiques

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

  • Dades identificatives

    Identificador: imarina:9382519
    Handle: https://hdl.handle.net/20.500.11797/imarina9382519
    Autors:
    Fontich, ErnestGarijo, AntonioJarque, Xavier
    Resum:
    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).

  • Altres:

    Autor segons l'article: Fontich, Ernest; Garijo, Antonio; Jarque, Xavier
    Departament: Enginyeria Informàtica i Matemàtiques
    Codi de projecte: PID2020-118281GB-C33
    Resum: 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).
    Accès a la llicència d'ús: https://creativecommons.org/licenses/by/3.0/es/
    Adreça de correu electrònic de l'autor: antonio.garijo@urv.cat
    Versió de l'article dipositat: info:eu-repo/semantics/publishedVersion
    Enllaç font original: https://www.aimsciences.org//article/doi/10.3934/dcds.2024122
    Programa de finançament: Herramientas para el análisis de diagramas de bifurcación en sistemas dinámicos
    Acrònim: ATBiD
    DOI de l'article: 10.3934/dcds.2024122
    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