Fundamentals of the Qualitative Theory of Differential Equations

Abstract

The theory of dynamical systems is one of the most important tools in quantitative and qualitative study of mathematical models of applied sciences. Most of these models are formulated using differential equations (continuous dynamical systems). Since the first published work by Poincaré in the late nineteenth century, the qualitative theory of differential equations have experienced a significant expansion. The goal of this project is to introduce the student to the fundamentals of the Qualitative Theory of Ordinary Differential Equations and prepare it for advanced studies and a possible academic career. We begin with the study of systems of linear ordinary differential equations and then approach the classical results of the qualitative theory of ODE; Existence and Uniqueness Theorems, Continuous Dependence of the Solutions with Respect to Initial Conditions and Parameters, Poincaré-Bendixson theorem, etc. This project provides the learning of several important concepts of mathematics and inserts the student in the research in Qualitative Theory of the Differential Equations.