Medida probabilística e densidade de probabilidade em sistemas caóticos

Abstract

The study of dynamical systems that exhibit chaotic behavior provides applications in several areas of knowledge such as predictions in engineering, physics, biology, among others. Therefore, studying such systems is important not only from the intrinsic point of view of the search to understand phenomena. Among the various properties that these systems exhibit, one in particular is the characteristic of preserving some measure under the action of dynamics, such as the density of points along the transformation domain. This work aims to verify the existence of this measure through numerical simulations. We will use the Frobenius Perron Operator to verify that this density is invariant in some maps, which are discrete-time systems and which provide the simplest way to study chaotic behavior, such as the logistic map, Bernoulli shift and the tent map. The methodology used will be the construction of histograms to capture the distribution of points generated by the map and compare with the theoretical results present in the literature.