Let $\xi$ be an analytic vector field at $0\in \mathbb{R}^3$ with a Hopf singularity, i.e. with eigenvalues $\pm i, c$ with $c\in \mathbb R$. We describe the germ of the subanalytic set $\mathcal{C}(\xi)$ defined by the union of the cycles on small neighborhoods of the singularity. We prove $\mathcal{C}(\xi)$ is the union of a finite number of surfaces or the complement of a curve of singularities of $\xi$. We also prove that the set $\mathcal{C}(\xi)$ is formally determined by any given formal normal form of $\xi$.

This talk is based on a joint work with Nuria Corral and Fernando Sanz Sánchez.