Qualitative theory of differential equations in the plane: topological classification results

Abstract

It is known that a continuously differentiable vector field in the real plane is completely determined by its separatrix configuration up to topological equivalence, that is, by its critical points, its separatrices and by one orbit in each canonical region of the vector field, as well as by the configuration of such elements. This result is due to Markus (when there are no limit separatrices), or, more generally (including fields over surfaces and allowing the existence of limit separatrices), to Neumann. A recent article by Espín Buendía and Jiménez López puts this result in check, presenting counterexamples and correcting it. The goal in this Master's project is to make the student aware of all classic bibliography about topological equivalence of planar vector fields, from Kaplan (when there are no singularities), through Markus and Neumann, to recent results in this area, in particular the cited Espín Buendía and Jiménez López's work. The student will then make a dissertation about the problems in the original result, as well as the corrections presented. Depending on the progress, we will study the problem of describing the possible equivalence classes of polynomial fields without singularities of fixed degree, a problem which is essentially open.