Infinitesimal Poincar\'e-Bendixson Problem in dimension three

Abstract

We describe the sets of accumulation of secants for orbits of real analytic vector fields in dimension three having the origin as only {\omega}-limit point. It is a kind of infinitesimal Poincar\'e-Bendixson problem in dimension three. These sets have structure of cyclic graph when the singularities are isolated under one blow-up. In the case of hyperbolic reduction of singularities with conditions of type Morse-Smale, we prove that the accumulation set is at most a single polycycle isomorphic to {\mathbb S}^1.