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

Abstract

We consider a C∞ family of planar vector fields {Xμˆ}μˆ∈Wˆ having a hyperbolic saddle and we study the Dulac map D(s;μˆ) and the Dulac time T(s;μˆ) 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 μˆ=(λ,μ)∈Wˆ=(0,+∞)×W, where W is an open subset of RN. For each μˆ0∈Wˆ and L>0, the functions D(s;μˆ) and T(s;μˆ) have an asymptotic expansion at s=0 and 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 Ecalle-Roussarie compensator. In this paper we are interested in the coefficients of these monomials, which are functions depending on μˆ 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 μˆ0. 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.