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Pàgina inicial > Articles > Articles publicats > Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles : |
Data: | 2021 |
Resum: | 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. |
Ajuts: | 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 |
Drets: | 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. |
Llengua: | Anglès |
Document: | Article ; recerca ; Versió acceptada per publicar |
Matèria: | Dulac map ; Dulac time ; Asymptotic expansion ; Uniform flatness ; Criticality |
Publicat a: | Journal of differential equations, Vol. 275 (February 2021) , p. 684-732, ISSN 1090-2732 |
Postprint 38 p, 790.4 KB |