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

Identifier:  imarina:9372460

Handle: https://hdl.handle.net/20.500.11797/imarina9372460

Authors:  Marín, D.; Villadelprat, J

Abstract:
We consider a family of planar vector fields having a hyperbolic saddle and we study the Dulac map and the Dulac time 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 , where W is an open subset of . For each and , the functions and have an asymptotic expansion at and 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 that can be shown to be in their respective domains and “universally” defined, meaning that their existence is stablished before fixing the flatness L and the unfolded parameter . Each coefficient has its own domain and it is of the form , 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 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 and has only poles, of order at most two, along .