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