An introduction to the qualitative theory and bifurcations of dynamical systems

Abstract

The initial objective of this research project is to study the concepts and basic results of the Qualitative Theory of Ordinary Differential Equations. We will approach local and global aspects of the theory such as: Existence and Uniqueness Theorem, Tubular Flow Theorem, Stable Manifold Theorem, Grobman-Hartman's Theorem, Poincaré's First Return Map, and Poincaré-Bendixson's Theorem. In a second moment, the study on the bifurcations theory of dynamical systems will be introduced. We will study several types of local bifurcations that can occur in one-parameter families of vector fields. In particular, we study bifurcations at non-hyperbolic singularities including bifurcations of periodic orbits from non-hyperbolic singularities. (AU)