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 a finite number of subanalytic invariant surfaces entirely composed by a continuum of cycles. In particular, we solve Dulac's problem, i.e. finiteness of limit cycles, for such vector fields.