Sistemas dinâmicos e a teoria do caos: uma abordagem a partir do mapa logístico e do sistema de Lorenz

Abstract

This work deals with Chaos Theory, an important topic within Mathematics, more precisely, in the theory of dynamic systems. Chaos is observed in many phenomena in nature, from the evolution of species and population growth in biology, to the movement of celestial bodies throughout the universe. Dynamic systems that exhibit chaotic behaviors can be analyzed and characterized by means of theorems and methods of mathematical analysis, both analytically and numerically, depending on their nature. One of the objectives of the study of Chaos Theory is to act in the control of such systems, through predictions about their future behavior based on the measurement of the parameters of a state space. In this sense, in this work an initial theoretical approach and chaos characterization was made through the analysis of two dynamic systems: the Logistic Map and the Lorenz System. Lyapunov exponents were used to analyze the chaotic behavior on the Logistic Map and a Lyapunov function to show that the Lorenz system is dissipative. From there, it was observed how such systems evolve into chaos due to their parameters.