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

TL;DR

This paper analyzes the asymptotic behavior of the Dulac map and time near hyperbolic saddles in planar vector fields, revealing detailed coefficient properties and their dependence on the hyperbolicity ratio and unfolding parameters.

Contribution

It provides explicit expressions for key coefficients in the asymptotic expansion of the Dulac map and time, including their pole structure and residues, using a novel Mellin transform approach.

Findings

Coefficients are smooth functions with explicit formulas.

Poles of coefficients occur at rational resonance points.

Residues at poles are explicitly computed.

Abstract

We consider a $\mathscr C^\infty$ 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 $\mathscr C^\infty$ 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.