Grant number: 13/08364-1 Support type: Scholarships in Brazil - Scientific Initiation Effective date (Start): June 01, 2013 Effective date (End): December 31, 2013 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology Principal researcher: Pedro Toniol Cardin Grantee: Melka Carolina Faria Catelan Home 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)