Nonlinear ordinary differential equations: a first study

Abstract

The project aims to introduce the student to the Qualitative Theory of Ordinary Differential Equations. In a first phase, aspects of local theory will be addressed through the study of some classic results, such as: existence and uniqueness of solutions theorem, theorems of dependency in relation to initial conditions and parameters, flow defined by a differential equation, stable manifold theorem and the Grobman-Hartman theorem. In the final phase of the project, aspects of the global theory of ODE's will be addressed, such as: limit and attractor sets, alpha- and omega-limit, Poincaré-Bendixon theorem, first return Poincaré's map in planar systems, Lyapunov coefficients of a weak focus with the aid of an algebraic computing system. And to conclude the project we are going to study the number of limit cycles that bifurcate from the linear center by a n-degree perturbation.