Infinitesimal Poincaré–Bendixson Problem in Dimension 3

Abstract

We describe the sets of accumulation of secants for orbits of real analytic vector fields in dimension 3 having the origin as only ω-limit point. It is a kind of infinitesimal Poincaré–Bendixson problem in dimension 3. These sets have structure of a cyclic graph when the singularities are isolated under one blow-up. In the case of hyperbolic reduction of singularities with conditions of Morse–Smale-type, we prove that the accumulation set is at most a single poly-cycle isomorphic to S1.