Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: General setting

Given a C ? family of planar vector fields { X ? ˆ } ? W having hyperbolic saddle, we study the Dulac map D ( s ; ) and time T between two transverse sections located in separatrices at arbitrary distance from saddle. We show (Theorems A B, respectively) that, for any 0 L > , functions have an asymptotic expansion = ? with remainder being uniformly -flat respect to parameters. The principal part both expansions is given monomial scale containing deformation logarithm, so-called Roussarie-Ecalle compensator. coefficients these monomials are “universally” defined, meaning that their existence established before fixing flatness unfolded parameter . Moreover preserved after derivation also provide (Theorem C) explicit upper bound number zeros ? bifurcating as This result enables tackle finiteness problems critical periodic orbits along lines those theorems on finite cyclicity around Hilbert's 16th problem. As application prove results (Corollaries E) about polynomial fields.