Grant number: 14/03849-0 Support type: Scholarships in Brazil - Scientific Initiation Effective date (Start): April 01, 2014 Effective date (End): December 31, 2014 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology Principal researcher: Pedro Toniol Cardin Grantee: Gabriel Antonio da Silva Inacio Home Institution: Faculdade de Engenharia (FEIS). Universidade Estadual Paulista (UNESP). Campus de Ilha Solteira. Ilha Solteira , SP, Brazil Abstract The goals of this project are: First, to study the concepts and the fundamental results of Qualitative Theory of Ordinary Differential Equations. Second, to apply such concepts and results in the study of Lienard systems. For this first we will study phase portraits 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 a orbit, the Poincaré--Bendixson Theorem which characterizes such limit sets, and we will use the Lyapunov functions in order to study the stability and assyntotic stability. Finally, we will study the phase portrait of Lienard systems. We will be particularly interested in issues such as the existence and the number of limit cycles for such systems.