Abstract: The surrounding reality can be viewed as the result of the interaction of dynamic systems nonlinear complexes. It has been shown, however, that some very simple systems can have it complicated and seemingly random behaviors. The chaos theory aims to explain and to predict in a short time the seemingly random and unpredictable behavior of the systems Nonlinear. Although the ideas preceding the emergence of chaos theory had been around for a long time, they were crystallized for the first by Lorenz (1963) in the work Deterministic Nonperiodic Flow. Lorenz created a mathematical model of the circulation of atmospheric currents of convection and observed that when there is a slight difference between the initial conditions, completely different results are obtained thus rediscovering the phenomenon of sensitivity to the variation of the initial conditions. The phenomenon observed has become a very popular paradigm of chaos theory called the „butterfly effect” and states that if the flapping of the wings of a butterfly changes the weather conditions in the jungle in a minor way Amazonian, this fact can have the effect, at the end of a complex causal chain, of the appearance of a tornadoes in Texas. Keywords: Chaos theory, nonlinear dynamics, nonlinear time series analysis, chaos identification, Lyapunov exponent, neural networks prediction of chaotic time series, multilayer, neural networks of support vectors, ARIMA model JEL Classification: C23, C26, C38, C55, C81, C87