Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: general setting
The paper contains the investigation of unfoldings for a \(\mathcal{C}^{\infty}\) family of planar vector fields \(X_{\hat{\mu}}\), \({\hat{\mu}\in\hat{W}}\) having a hyperbolic saddle. The study of so-called Dulac map of the saddle has closed connection with Hilbert's 16th problem. This work is the continuation of a previous paper of the authors [J. Differ. Equations 269, No. 10, 8425--8467 (2020; Zbl 1447.34041)] where a local version of the similar results is proved. The authors study the Dulac map and the Dulac time between two transverse sections located in the separatrices at arbitrary distance from the saddle. They show that the corresponding functions have an asymptotic expansion with a 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 Roussarie-Ecalle compensator. The coefficients of these monomials are \(\mathcal{C}^{\infty}\) functions ``universally'' defined, meaning that their existence is established before fixing the flatness \(L\) of the remainder and the unfolded parameter \(\hat{\mu}_0\). The derived results enable to tackle finiteness problems for the number of critical periodic orbits along the lines of the proved theorems on finite cyclicity around Hilbert's 16th problem. As an application it is proved two finiteness results about the number of critical periodic orbits of polynomial vector fields.
scientific article; zbMATH DE number 7291354
scientific article