Discrete Chaotic Dynamics and Fractals

Abstract

Dynamical states are present in our everyday lives without us even being aware of their existence. Any model, from the simplest, such as a leaf swinging in the wind, to the most complicated, such as the motion of the planets in our solar system, can be seen as a dynamical system. These dynamical states can be presented in continuous form or also in discrete form and both have important applications in many areas of science.Much mathematical research has been done with the focus of understanding qualitatively the behavior of these dynamical states and their properties. And one important behavior that can be verified in some of these systems is, among others, chaotic dynamics. Thus, this dynamics opens the door to the study of more complex geometries beyond Euclidean geometry, such as, for example, Fractal Geometry.In this work, the student will study the chaotic behavior present in a family of real functions with real values, then, he will see some equivalences in this definition of Chaos, ending the studies with an introduction to Fractal Geometry and, with this, leaving the field open for future studies.