Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles : Coefficient properties
Marín Pérez, David (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Villadelprat Yagüe, Jordi (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2024
Abstract:	We consider a C∞ family of planar vector fields {Xµˆ}µˆ∈Wˆ having a hyperbolic saddle and we study the Dulac map D(s; ˆµ) and the Dulac time T(s; ˆµ) from a transverse section at the stable separatrix to a transverse section at the unstable separatrix, both at arbitrary distance from the saddle. Since the hyperbolicity ratio λ of the saddle plays an important role, we treat it as an independent parameter, so that µˆ = (λ, µ) ∈ Wˆ = (0, +∞) × W, where W is an open subset of R N . For each µˆ0 ∈ Wˆ and L > 0, the functions D(s; ˆµ) and T(s; ˆµ) have an asymptotic expansion at s = 0 and 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 Ecalle-Roussarie compensator. In this paper we are interested in the coefficients of these monomials, which are functions depending on µˆ that can be shown to be C ∞ in their respective domains and “universally” defined, meaning that their existence is stablished before fixing the flatness L and the unfolded parameter µˆ0. Each coefficient has its own domain and it is of the form ((0, +∞) \ D) × W, where D a discrete set of rational numbers at which a resonance of the hyperbolicity ratio λ occurs. In our main result, Theorem A, we provide explicit expressions for some of these coefficients and to this end a fundamental tool is the employment of a sort of incomplete Mellin transform. With regard to these coefficients we also prove that they have poles of order at most two at D × W and we give the corresponding residue, that plays an important role when compensators appear in the principal part. Furthermore we prove a result, Corollary B, showing that in the analytic setting each coefficient given in Theorem A is meromorphic on (0, +∞) × W and has only poles, of order at most two, along D × W.
Grants:	Agencia Estatal de Investigación PID2021-125625NB-I00 Agencia Estatal de Investigación PID2020-118281GB-C33 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-01015 Agencia Estatal de Investigación CEX2020-001084-M
Note:	Altres ajuts: acords transformatius de la UAB
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Dulac map ; Dulac time ; Asymptotic expansion ; Incomplete Mellin transform
Published in:	Journal of differential equations, Vol. 404 (September 2024) , p. 43-107, ISSN 1090-2732

DOI: 10.1016/j.jde.2024.05.037

65 p, 853.6 KB

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