Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: Coefficient properties.

We consider a C ∞ family of planar vector fields { X μ ˆ } μ ˆ ∈ W ˆ having a hyperbolic saddle and we study the Dulac map D (s ; μ ˆ) and the Dulac time T (s ; μ ˆ) from a transverse section at the stable separatrix to a transverse section at the unstable separatrix, both at arbitrary distance from the saddle. Since the hyperbolicity ratio λ of the saddle plays an important role, we treat it as an independent parameter, so that μ ˆ = (λ , μ) ∈ W ˆ = (0 , + ∞) × W , where W is an open subset of R N. For each μ ˆ 0 ∈ W ˆ and L > 0 , the functions D (s ; μ ˆ) and T (s ; μ ˆ) have an asymptotic expansion at s = 0 and μ ˆ ≈ μ ˆ 0 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 Ecalle-Roussarie compensator. In this paper we are interested in the coefficients of these monomials, which are functions depending on μ ˆ that can be shown to be C ∞ in their respective domains and "universally" defined, meaning that their existence is stablished before fixing the flatness L and the unfolded parameter μ ˆ 0. Each coefficient has its own domain and it is of the form ((0 , + ∞) ∖ D) × W , where D a discrete set of rational numbers at which a resonance of the hyperbolicity ratio λ occurs. In our main result, Theorem A, we provide explicit expressions for some of these coefficients and to this end a fundamental tool is the employment of a sort of incomplete Mellin transform. With regard to these coefficients we also prove that they have poles of order at most two at D × W and we give the corresponding residue, that plays an important role when compensators appear in the principal part. Furthermore we prove a result, Corollary B, showing that in the analytic setting each coefficient given in Theorem A is meromorphic on (0 , + ∞) × W and has only poles, of order at most two, along D × W. [ABSTRACT FROM AUTHOR]