Métodos numéricos para previsão em sistemas dinâmicos

Abstract

Many natural and artificial behaviors can be reproduced by dynamical systems described by differential equations, whose solutions are time series. Therefore, an important problem is how to determine which mathematical model is able of describing the behavior of a given dynamical system, as well as how to determine what are the values of the parameters governing this model. Solving this problem enables one to obtain information about the evolution of the system and to determine its possible future states. This information is especially important when dealing with systems of difficult forecast, as is the case of complex or nonlinear systems, which are extremely sensitive to their initial conditions. To try to reproduce the dynamics of systems whose equations are not known, we will study the Lorenz system. Our goal is to show that knowledge of time series data from dynamical systems can be used to recover the dynamics originating the observed behavior. Then, using this recovered dynamics, we will address the problem of predicting the future state on time series. We have developed a technique of artificial intelligence using polynomial functions to calculate the evolution velocities of the state variables of the system, aiming to recover its underlying dynamics, and then forecast its future evolution. The algorithm is capable of adapting to the system, with flexible parameters extracted from the previously observed data, even when the time series is contaminated by observation noise. We hope that our method will be useful beyond basic science in many applications, such as making predictions of systems that may be physical, biological, financial, geographic, meteorological, etc.