Comportamiento caótico en sistemas dinámicos discretos unidimensionales

Abstract

This monograph focuses on the analysis of one-dimensional discrete dynamical systems, exploring fundamental concepts such as stability, bifurcation and chaos. These systems exhibit a variety of behaviors under given initial conditions: they may stabilize in orbits, fail to converge, or display unpredictable and random behavior, typical characteristics of a chaotic system. The fundamental purpose of this work is to show in detail the behavior of orbits defined by the family of functions fμ(x) = μx(1 − x), whose dynamics varies with the parameter μ. The bifurcation phenomenon and its relation to changes in the behavior of the systems are analyzed. Finally, two definitions of chaos are provided: in the Li-Yorke sense and in the Devaney sense. The validity of the theoretical result is supported by a set of simulations.