An introduction to dynamical systems

Abstract

The beginning of studies in dynamical systems starts with the Differential and Integral Calculus discovered by Newton and Leibniz, to solve problems due by physical and geometrical considerations. These methods, in its evolution, gradually leading to the consolidation of Differential Equations as a new branch of mathematics, which in middle of the eighteenth century became one of the most important mathematical disciplines and the most effective method for scientific research. The contributions of distinguished mathematicians like Euler, Lagrange and Laplace significantly expanded the knowledge of differential equations in the Calculus of Variations in Celestial Mechanics and Fluid Dynamics. In some of these systems many complicated behaviors are observed through equations. An algebraic form does not indicate that the dynamic behavior of this system is simple, it can even become "chaotic". In this work attempts to study the basic and initial concepts of the Dynamic Systems Theory, such as Equations Discrete and Orbits, Geometric Representation of orbits, Fixed Point attractors and repulsors, Periodicals points, bifurcations, Symbolic Dynamics, Conjugation Topological and finally the Chaos.