Introduction to smooth and piecewise smooth qualitative theory of differential equations with applications to Mechanics and Electrical Engineering

• Grant number: 15/21705-8
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: December 01, 2015
• End date: May 31, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Tiago de Carvalho
• Grantee: Andre do Amaral Antunes
• Host Institution: Faculdade de Ciências (FC). Universidade Estadual Paulista (UNESP). Campus de Bauru. Bauru , SP, Brazil
• Associated research grant: 13/24541-0 - Ergodic and qualitative theory of dynamical systems, AP.TEM

Abstract

We begin this project with the study of several examples modeled by smooth ordinary differential equations. These examples are obtained from electrical problems, biological, chemical, and especially mechanical (spring-mass system or other). We will make the study of these systems through their phase portraits, which provide us some geometric information about the behavior of such systems. Then we will deal with the formal results regarding the Qualitative Theory of Vector Fields where we will study classical results as The Existence and Uniqueness Theorem, Tubular Flow theorem, the Grobman-Hartman Theorem, and the Poincaré-Bendixson theorem. Finally, we will make an introduction to the study of Piecewise-smooth Vector Fields, which is a very current line of research and may guide the studies of a future master by the candidate.