Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: general setting: General setting

David Marin Perez, Jordi Villadelprat*

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)684-732
Number of pages49
JournalJournal of Differential Equations
Volume275
DOIs
Publication statusPublished - 2021

Keywords

  • Asymptotic expansion
  • Criticality
  • Dulac map
  • Dulac time
  • Uniform flatness

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