Sistemas dinâmicos e caos

Abstract

It was through the historical development of astronomy that the Theory of Dynamical Systems, whose field of study is open and recent, emerged. In this work, we will understand this emergence, learning about important concepts concerning the area. We consider dynamical systems defined by successive applications of a function that maps an interval of real numbers to itself and we study the dynamics of some mathematical models. Among which we can highlight simple examples of financial mathematics, which is a branch very close to our reality, and the study of the tent function. We introduce the notion of equivalence between dynamical systems defined by iteration of functions and, through this notion, we get to know the dynamics of new systems. We also study asymptotic stability of a fixed point or periodic point of a dynamical system. We present the topological definition of chaos and discuss some essential features of this important concept. We analyze again the tent function and present, through the binary expansion of real numbers in the interval [0, 1], a proof that the dynamical system defined by this function and, consequently, any other equivalent to it, is chaotic. Finally, we examine the "logistic population model" discussed by May( [6]), highlighting some of its features.