Estimating the Lyapunov exponents of chaotic time series: A model based method

In this paper, the problem of Lyapunov Exponents (LEs) computation from chaotic time series based on Jacobian approach by using polynomial modelling is considered. The embedding dimension which is an important reconstruction parameter, is interpreted as the most suitable order of model. Based on a global polynomial model fitting to the given data, a novel criterion for selecting the suitable embedding dimension is presented. By considering this dimension as the model order, by evaluating the prediction error of different models, the best nonlinearity degree of polynomial model is estimated. This selected structure is used in each point of the reconstructed state space to model the system dynamics locally and calculate the Jacobian matrices which are used in QR factorization method in the LEs estimation. This procedure is also applied to multivariate time series to include information from other time series and resolve probable shortcoming of the univariate case. Finally, simulation results are presented for some well-known chaotic systems to show the effectiveness of the proposed methodology.