Characterization of centers by their complex separatrices

Abstract

In this work, we address analytic families of real planar vector fields $X_{\lambda }$ having a monodromic singularity at the origin for all parameters $\lambda \in \Lambda$, where Λ is an open subset of the real finite-dimensional Euclidean space. We assume that $X_{\lambda }$ depends analytically on λ. This naturally leads to the so-called center-focus problem, which consists of describing the partition of Λ induced by the centers and the foci at the origin. We provide a characterization of the centers (whether degenerate or not) in terms of a specific integral of the cofactor associated with a real invariant analytic curve passing through the singularity, which always exists. Several consequences and applications are also discussed.