A neuro-fuzzy based model for accurate estimation of the Lyapunov exponents of an unknown dynamical system

A neuro-fuzzy based model is proposed in this paper for estimating the Lyapunov exponents of an unknown dynamical system according solely to a set of observations. Several approaches have been presented in recent years; most of them using the approximation of both the function of the trajectory of observations and the partial derivatives, to yield the Jacobian matrix of the function. The Jacobian matrix is then employed in the Jacobian-based methods that extract the Lyapunov exponents by QR-decomposition. While the accurate estimation of Lyapunov exponents has been sought, most of the related papers mainly focus on the accuracy of the trajectory approximation. In this paper, an Adaptive Neuro-Fuzzy Inference System is presented and stated to be an efficient tool for such a purpose. Structural parameters of the proposed model as the embedding dimension and the delay time are calculated by the Takens theorem and autocorrelation function, respectively. Model validation is performed by cross approximate entropy. Then, the promising performance of the proposed model as an accurate estimation of the Lyapunov exponents and its robustness to the measurement noise are finally evaluated.