Qualitative study of linear and nonlinear ordinary differential equations

Abstract

The objective of this project is to study the concepts and basic results of the qualitative theory of Ordinary Differential Equations in RÆn, as well as some examples of applications of this theory. We will deal first of linear differential equations and after the nonlinear differential equations. For linear equations, we will study topics such as the classification of planar systems, the flow of a linear equation and linear attractors. For nonlinear equations, we will study the local behavior of trajectories near regular points, singular points (hyperbolic and semi-hyperbolic) and periodic orbits. We will study the concepts of alpha- and omega-limit sets of a orbit, the Poincaré-Bendixson theorem characterizing such limit sets. The stability of equilibrium points, the asymptotic stability and the Lyapunov stability will also be dealt.