Autor segons l'article: Marín D; Villadelprat J
Departament: Enginyeria Informàtica i Matemàtiques
Autor/s de la URV: Villadelprat Yagüe, Jordi
Paraules clau: Uniform flatness Hilberts 16th problem Dulac time Dulac map Criticality Asymptotic expansion uniform flatness families dulac time cyclicity criticality critical periods asymptotic expansion
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: Mathematics Matemática / probabilidade e estatística Interdisciplinar Engenharias iii Ciências agrárias i Ciência da computação Astronomia / física Applied mathematics Analysis
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
URL Document de llicència: http://repositori.urv.cat/ca/proteccio-de-dades/
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
DOI de l'article: 10.1016/j.jde.2020.11.020
Entitat: Universitat Rovira i Virgili
Any de publicació de la revista: 2021
Tipus de publicació: Journal Publications
Repositori URV