Introduction to geometric theory of ordinary differential equations

• Grant number: 13/08364-1
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: June 01, 2013
• End date: December 31, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Pedro Toniol Cardin
• Grantee: Melka Carolina Faria Catelan
• Host Institution: Faculdade de Engenharia (FEIS). Universidade Estadual Paulista (UNESP). Campus de Ilha Solteira. Ilha Solteira , SP, Brazil

Abstract

The main goal of this project is to study the concepts and the fundamental results of the Qualitative Theory of Ordinary Differential Equations. We will study the phase portrait of vector fields on $\R^n$, analyzing the local behavior of the trajectories near to regular points, singular points (hyperbolic and semi-hyperbolic), and periodic orbits. We will study the notions of $\alpha$-and $\omega$-limit sets of an orbit, the Poincaré-Bendixson Theorem which characterizes such limit sets, and we will use the Lyapunov functions in order to study the stability and asymptotic stability. Finally, we will make a qualitative study of the Van der Pol equation. (AU)