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

Abstract

We consider a ${\mathcal{C}}^{\infty }$ family of planar vector fields $\{X_{\hat{\mu }}{\}}_{\hat{\mu }\in \hat{W}}$ having a hyperbolic saddle and we study the Dulac map $D(s;\hat{\mu })$ and the Dulac time $T(s;\hat{\mu })$ 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 $\lambda$ of the saddle plays an important role, we consider it as an independent parameter, so that $\hat{\mu }=(\lambda ,\mu )\in \hat{W}=(0,+\infty )\times W$, where $W$ is an open subset of ${\mathbb{R}}^N.$ For each ${\hat{\mu }}_0\in \hat{W}$ and $L>0$, the functions $D(s;\hat{\mu })$ and $T(s;\hat{\mu })$ have an asymptotic expansion at $s=0$ and $\hat{\mu }\approx {\hat{\mu }}_0$ 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 $\hat{\mu }$ that can be shown to be ${\mathcal{C}}^{\infty }$ in their respective domains and "universally" defined, meaning that their existence is stablished before fixing the flatness $L$ and the unfolded parameter ${\hat{\mu }}_0.$ Each coefficient has its own domain and it is of the form $((0,+\infty )\setminus D)\times W$, where~$D$ a discrete set of rational numbers at which a resonance of the hyperbolicity ratio $\lambda$ occurs. In our main result we give the explicit expression of 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 $D\times W$ and we give the corresponding residue, that plays an important role when compensators appear in the principal part.