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Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: General setting

  • Dades identificatives

    Identificador: imarina:9138883
    Autors:
    Marín DVilladelprat J
    Resum:
    © 2020 Elsevier Inc. Given a C∞ family of planar vector fields {Xμˆ}μˆ∈Wˆ having a hyperbolic saddle, we study the Dulac map D(s;μˆ) and the Dulac time T(s;μˆ) between two transverse sections located in the separatrices at arbitrary distance from the saddle. We show (Theorems A and B, respectively) that, for any μˆ0∈Wˆ and L>0, the functions T(s;μˆ) and D(s;μˆ) have an asymptotic expansion at s=0 for μˆ≈μˆ0 with the remainder being uniformly L-flat with respect to the parameters. The principal part of both asymptotic expansions is given in a monomial scale containing a deformation of the logarithm, the so-called Roussarie-Ecalle compensator. The coefficients of these monomials are C∞ functions “universally” defined, meaning that their existence is established before fixing the flatness L of the remainder and the unfolded parameter μˆ0. Moreover the flatness L of the remainder is preserved after any derivation with respect to the parameters. We also provide (Theorem C) an explicit upper bound for the number of zeros of T′(s;μˆ) bifurcating from s=0 as μˆ≈μˆ0. This result enables to tackle finiteness problems for the number of critical periodic orbits along the lines of those theorems on finite cyclicity around Hilbert's 16th problem. As an application we prove two finiteness results (Corollaries D and E) about the number of critical periodic orbits of polynomial vector fields.
  • Altres:

    Autor segons l'article: Marín D; Villadelprat J
    Departament: Enginyeria Informàtica i Matemàtiques
    Autor/s de la URV: Villadelprat Yagüe, Jordi
    Codi de projecte: PID2020-118281GB-C33
    Paraules clau: asymptotic expansion critical periods criticality cyclicity dulac time families uniform flatness Asymptotic expansion Criticality Dulac map Dulac time Hilberts 16th problem Uniform flatness
    Resum: © 2020 Elsevier Inc. Given a C∞ family of planar vector fields {Xμˆ}μˆ∈Wˆ having a hyperbolic saddle, we study the Dulac map D(s;μˆ) and the Dulac time T(s;μˆ) between two transverse sections located in the separatrices at arbitrary distance from the saddle. We show (Theorems A and B, respectively) that, for any μˆ0∈Wˆ and L>0, the functions T(s;μˆ) and D(s;μˆ) have an asymptotic expansion at s=0 for μˆ≈μˆ0 with the remainder being uniformly L-flat with respect to the parameters. The principal part of both asymptotic expansions is given in a monomial scale containing a deformation of the logarithm, the so-called Roussarie-Ecalle compensator. The coefficients of these monomials are C∞ functions “universally” defined, meaning that their existence is established before fixing the flatness L of the remainder and the unfolded parameter μˆ0. Moreover the flatness L of the remainder is preserved after any derivation with respect to the parameters. We also provide (Theorem C) an explicit upper bound for the number of zeros of T′(s;μˆ) bifurcating from s=0 as μˆ≈μˆ0. This result enables to tackle finiteness problems for the number of critical periodic orbits along the lines of those theorems on finite cyclicity around Hilbert's 16th problem. As an application we prove two finiteness results (Corollaries D and E) about the number of critical periodic orbits of polynomial vector fields.
    Àrees temàtiques: Analysis Applied mathematics Astronomia / física Ciência da computação Ciências agrárias i Engenharias iii Interdisciplinar Matemática / probabilidade e estatística Mathematics
    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
    Identificador de l'autor: 0000-0002-1168-9750
    Data d'alta del registre: 2023-02-19
    Versió de l'article dipositat: info:eu-repo/semantics/acceptedVersion
    Enllaç font original: https://www.sciencedirect.com/science/article/abs/pii/S0022039620306021
    Programa de finançament: Herramientas para el análisis de diagramas de bifurcación en sistemas dinámicos
    Referència a l'article segons font original: Journal Of Differential Equations. 275 684-732
    Referència de l'ítem segons les normes APA: Marín D; Villadelprat J (2021). Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: General setting. Journal Of Differential Equations, 275(), 684-732. DOI: 10.1016/j.jde.2020.11.020
    URL Document de llicència: https://repositori.urv.cat/ca/proteccio-de-dades/
    Acrònim: ATBiD
    DOI de l'article: 10.1016/j.jde.2020.11.020
    Entitat: Universitat Rovira i Virgili
    Any de publicació de la revista: 2021
    Acció del programa de finançament: Proyectos I+D Generación de Conocimiento
    Tipus de publicació: Journal Publications
  • Paraules clau:

    Analysis,Applied Mathematics,Mathematics
    asymptotic expansion
    critical periods
    criticality
    cyclicity
    dulac time
    families
    uniform flatness
    Asymptotic expansion
    Criticality
    Dulac map
    Dulac time
    Hilberts 16th problem
    Uniform flatness
    Analysis
    Applied mathematics
    Astronomia / física
    Ciência da computação
    Ciências agrárias i
    Engenharias iii
    Interdisciplinar
    Matemática / probabilidade e estatística
    Mathematics
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