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

Abstract

AbstractLet $$\xi $$ ξ be a real analytic vector field with an elementary isolated singularity at $$0\in \mathbb {R}^3$$ 0 ∈ R 3 and eigenvalues $$\pm bi,c$$ ± b i , c with $$b,c\in \mathbb {R}$$ b , c ∈ R and $$b\ne 0$$ b ≠ 0 . We prove that all cycles of $$\xi $$ ξ 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.