Introduction to the study of differential equations: a dynamic approach

Abstract

The study of vector fields in the vicinity of a singularity, or stationary point, is a topic widely studied due to its importance in several branches of mathematics and its applications in other areas of science, such as biology, meteorology and so on. Often, when we intend to equate certain phenomena, we come across equations that involve variations of certain quantities considered essential. The use of differential equations to represent these variations is essentially due to the fact that the studied phenomenon is seen in discrete or continuous time. Because many differential equations are not conveniently soluble by analytical methods, it is important to consider qualitative information obtained from their solutions, without actually solving them. We will see how this can be done. Thus, some methods of solving more important or better known differential equations are presented. We will also focus on linear systems, as this theory will be used in the qualitative study of nonlinear systems.Information about the behavior of a vector field near a singular point can be obtained through some invariants. This project aims to develop basic concepts of Differential Equations as a preparation for further study of vector fields and associated invariants. The most basic invariant of a vector field at a singular point is called the Poincaré-Hopf index. A very important concept, and considered a true link between branches of mathematics such as Algebraic Topology and Differential Topology. Such a connection is possible through the important result known as the Poincaré-Hopf Theorem. (AU)