F. Cano

• University of Valladolid

The properties of oscillation for trajectories of three-dimensional vector fields are “visible” in the sense that they can be detected by means of plane projections. We also recover a three-dimensional version of the classical dichotomy “nonoscillation” versus “spiralling” for plane vector fields.

Let γ0 be an integral curve of an analytic vector field X in a real three dimensional manifold, suppose that γ0 has a single limit point and that it has all iterated tangents. The integral pencil PI(γ0) is the set of all integral curves of X having the same (oriented) iterated tangents as γ0- We prove that two arbitrary curves in PI(γ0) arc either,...

Let γ0 be a integral curve of an analytic vector field on a manifold of dimension 3. We suppose that γ0 has oriented, iterated tangents. The integral pencil PI(γ0) is the set of integral curves γ which have the same oriented, iterated tangent as γ0. The curves of PI(γ0) are either subanalitically separated or asymptotically linked. In this case PI(...

La dynamique oscillante d'un champ de vecteurs analytique en dimension trois s'organise autour d'un nombre fini d'axes de tourbillonnement lorsqu'elle ne se délocalise pas par des éclatements de point.