In this chapter, we give an introduction to the very important subject of dynamical systems, the study of qualitative behavior and the computation of solutions of nonlinear systems of the form
We present some of the most relevant theoretical results in this area, as well as several examples to illustrate better the idea being introduced. We also take care to introduce some numerical techniques necessary for the computation of special solutions. Some particular mathematical models of dynamical systems are studied in detail, and they give us a chance to highlight and apply the theory and techniques introduced.
In most mathematical theory, dealing with linear problems is by far simpler than studying nonlinear ones. Results on linear dynamical systems are well established, and are essential to understanding and developing the theory for their nonlinear counterpart. Thus, our starting point will be focusing on the simpler theory of linear dynamical systems and the qualitative behavior of their solutions.
The study of nonlinear dynamical systems is usually divided into local and global theory. The main ingredient for local theory will be the linearization of the system around equilibrium points or periodic solutions, and then we will study the resulting linear systems to understand the local behavior of the corresponding nonlinear system around those special ...