Surfaces with central configuration and Dulac's problem for a three dimensional isolated Hopf singularity

Abstract

Let E be a real analytic vector field with an elementary isolated singularity at 0ER3 and eigenvalues ±bi,c with b,cER and b( no=) 0. We prove that all cycles of o 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.