Surfaces with Central Configuration and Dulac’s Problem for a Three Dimensional Isolated Hopf Singularity

Abstract

Let \(\xi \) be a real analytic vector field with an elementary isolated singularity at \(0\in \mathbb {R}^3\) and eigenvalues \(\pm bi,c\) with \(b,c\in \mathbb {R}\) and \(b\ne 0\). We prove that all cycles of \(\xi \) in a sufficiently small neighborhood of 0, if they exist, are contained in the union of finitely many subanalytic invariant surfaces, each one entirely composed of a continuum of cycles. In particular, we solve Dulac’s problem for such vector fields, i.e., finiteness of limit cycles.