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Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles : General setting
Marín Pérez, David (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Villadelprat Yagüe, Jordi (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)

Date: 2021
Abstract: Given a C∞ family of planar vector fields{Xˆ µ}ˆ µ∈ ˆ W having a hyperbolic saddle, we study the Dulac map D(s; ˆ µ) and the Dulac time T(s; ˆ µ) between two transverse sections located in these paratrices at arbitrary distance from the saddle. We show (Theorems A and B, respectively) that, for any ˆ µ0 ∈ ˆ W and L > 0, the functions T(s; ˆ µ) and D(s; ˆ µ) have an asymptotic expansion at s = 0 for ˆ µ ≈ ˆ µ0 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 socalled 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 ˆ µ0. 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; ˆ µ) bifurcating from s = 0 as ˆ µ ≈ ˆ µ0. 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.
Grants: Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617
Ministerio de Economía y Competitividad MDM-2014-0445
Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1725
Ministerio de Ciencia e Innovación PGC2018-095998-B-I00
Ministerio de Ciencia e Innovación MTM2015-66165-P
Ministerio de Ciencia e Innovación MTM2017-86795-C3-2-P
Rights: Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades. Creative Commons
Language: Anglès
Document: Article ; recerca ; Versió acceptada per publicar
Subject: Dulac map ; Dulac time ; Asymptotic expansion ; Uniform flatness ; Criticality
Published in: Journal of differential equations, Vol. 275 (February 2021) , p. 684-732, ISSN 1090-2732

DOI: 10.1016/j.jde.2020.11.020


Postprint
38 p, 790.4 KB

The record appears in these collections:
Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles

 Record created 2021-02-12, last modified 2023-03-05



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