2021/05/20 by David Marín, Marín, David, Jordi Villadelprat +1 · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Meromorphic and Entire Functions #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2105.09785
We consider a \mathscr C^∞ family of planar vector fields \Xμ\μ∈ W having a hyperbolic saddle and we study the Dulac map D(s;μ) and the Dulac time T(s;μ) 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 consider it as an independent parameter, so that μ=(λ,μ)∈ W=(0,+∞)× W, where W is an open subset of \mathbb RN. For each μ0∈ W and L>0, the functions D(s;μ) and T(s;μ) have an asymptotic expansion at s=0 and μ≈μ0 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 that can be shown to be \mathscr C^∞ in their respective domains and "universally" defined, meaning that their existence is stablished before fixing the flatness L and the unfolded parameter μ0. 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 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× W and we give the corresponding residue, that plays an important role when compensators appear in the principal part.