Structural parameter tuning of the first-order derivative of an adaptive neuro-fuzzy system for chaotic function modeling

Estimation of the Lyapunov Exponents of a chaotic dynamical system is highly dependent on the tools one may use. Function approximators such as artificial neural networks and polynomial based systems posses an approximation error which effectively impacts LEs precision. Thus, we look for an efficient tool in the approximation of partial derivatives of a function and the function is to be critical of importance. In this paper, the Adaptive Neuro-Fuzzy Inference System and its first order derivative are used to model the nonlinear behavior of the chaotic functions. The structures of the Neuro-Fuzzy system and its first-order derivative are constructed according to the training samples. Premise parameters and consequent parameters which have been initialized during a learning procedure are tuned by a deterministic chaotic-based genetic algorithm based on the Logistic map. Four non-linear chaotic dynamics are modeled with the proposed structure trained by the proposed chaotic genetic algorithm and the approximation capability of the proposed structure is studied. It has been shown that the chaotic behavior of the genetic algorithm enhances its capability in parameter tuning and structure optimization of a nonlinear network. Furthermore, the proposed method comprising the adaptive neuro-fuzzy network with its first order derivative is validated in modeling the chaotic behavior of four chaotic functions. Modeling error, optimization parameters and computation time are analyzed to verify the statements and presentations.