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

  • Identification data

    Identifier: imarina:9138883
    Authors:
    Marín DVilladelprat J
    Abstract:
    © 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.
  • Others:

    Author, as appears in the article.: Marín D; Villadelprat J
    Department: Enginyeria Informàtica i Matemàtiques
    URV's Author/s: Villadelprat Yagüe, Jordi
    Keywords: Uniform flatness Hilberts 16th problem Dulac time Dulac map Criticality Asymptotic expansion uniform flatness families dulac time cyclicity criticality critical periods asymptotic expansion
    Abstract: © 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.
    Thematic Areas: Mathematics Matemática / probabilidade e estatística Interdisciplinar Engenharias iii Ciências agrárias i Ciência da computação Astronomia / física Applied mathematics Analysis
    licence for use: https://creativecommons.org/licenses/by/3.0/es/
    Author's mail: jordi.villadelprat@urv.cat
    Author identifier: 0000-0002-1168-9750
    Record's date: 2023-02-19
    Papper version: info:eu-repo/semantics/acceptedVersion
    Link to the original source: https://www.sciencedirect.com/science/article/abs/pii/S0022039620306021
    Papper original source: Journal Of Differential Equations. 275 684-732
    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
    Licence document URL: https://repositori.urv.cat/ca/proteccio-de-dades/
    Article's DOI: 10.1016/j.jde.2020.11.020
    Entity: Universitat Rovira i Virgili
    Journal publication year: 2021
    Publication Type: Journal Publications
  • Keywords:

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