Fuzzifying Chaos in Dynamical Systems

Marcos V. Ferreira, Tatiane Nogueira Rios, Ricardo Araújo Rios

Abstract: Fuzzy Time Series has attracted the attention of researchers interested in modeling systems by considering the relationship among its observations. However, most studies were designed to model such a relationship in the time domain. A notable challenge arises when analyzing chaotic time series, where observation behaviors undergo substantial changes in response to any perturbations affecting the system. In this case, temporal observations must be transformed into a higher dimensional space before starting the modeling process. This work has connected Chaos and Fuzzy areas to address this problem in two phases. Firstly, we use the Dynamical System and Chaotic tools to find time series attractors that rule how observations evolve over time. Subsequently, fuzzy clustering was applied to model sets as Gaussian membership functions. Next, fuzzy relationship and defuzzification steps were employed to support the time series forecasting. The results obtained by our work were compared with state-of-the-art, emphasizing that the proposed approach significantly contributes to model time series with chaotic behavior.