Qualitative methods for ordinary differential equations applied to the study of Lienard systems

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.