Characterization of centers by its complex separatrices

Abstract

In this work we deal with analytic families of real planar vector fields ${\mathcal{X}}_{\hat{I}}»$ having a monodromic singularity at the origin for any $\hat{I}»\in \hat{I}›\subset {\mathbb{R}}^p$ and depending analytically on the parameters $\hat{I}»$. There naturally appears the so-called center-focus problem which consists in describing the partition of $\hat{I}›$ induced by the centers and the foci at the origin. We give a characterization of the centers (degenerated or not) in terms of a specific integral of the cofactor associated to a real invariant analytic curve passing through the singularity, which always exists. Several consequences and applications are also stated.