Chaos: an introduction to the dynamical systems theory

Abstract

In this project we will study the basic concepts of discrete dynamical systems, which will serve as a tool for establishing the knowledge and also as the basis for the study of important results and techniques from the dynamical system area, especially for nonlinear systems, which are not seen during the graduation. Some topics that will be covered are the concepts of periodic orbits, bifurcation diagram of the logistic family, conjugation, Schwarziana derivative, sensitive dependence with respect to initial conditions, transitivity, chaos, hyperbolicity, symbolic dynamics, structural stability, circle applications, Sarkoviskii Theorem, etc.. We will see, for example, that simple systems defined in real dimension 1 presents the main phenomena of the theory, allowing the student addresses relevant issues of the theory in order to understand various problems that today are at the boundary of knowledge of the area. Thus we aim at enabling the student can improve and deepen their mathematical knowledge, particularly in the area of dynamical systems. Furthermore, the development of this project will provide to the student an introduction to the mathematical research, to dealing with scientific literature, and their preparation for more advanced searches.