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

Grant number: 21/08895-3
Support type:Scholarships in Brazil - Master
Effective date (Start): December 01, 2021
Effective date (End): July 31, 2023
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal researcher:Francisco Braun
Grantee:Rodrigo Thomaz da Silva
Home Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
Associated research grant:19/07316-0 - Singularity theory and its applications to differential geometry, differential equations and computer vision, AP.TEM

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.

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