Nonlinear chaotic Lorenz-Lü-Chen fractional order dynamics: A novel machine learning expedition with deep autoregressive exogenous neural networks

Abstract

This exhaustive study entails fractional processing of the unified chaotic Lorenz-Lü-Chen attractors using machine learning expedition with Levenberg-Marquardt optimized deep nonlinear autoregressive exogenous neural networks (NARX-NNs-LM). The fractional Lorenz-Lü-Chen attractors (FLLCA) system is unified by three Caputo-based fractional differential equations reflecting Lorenz, Lü, Chen attractors exacted by the single control parameter. The Fractional Adams-Bashforth-Moulton predictor-corrector method is efficaciously employed for the FLLCA models for different variation of fractional orders to generate synthetic datasets for temporal anticipation and processing. Acquired datasets of FLLCA systems were arbitrarily split into a training, validation and test sets for the execution of nonlinear autoregressive exogenous neural networks optimized sequentially using the Levenberg-Marquardt algorithm. This refined NARX-NNs-LM strategy is validated across the reference numerical solutions via scrutiny on mean square error (MSE) convergence graphs, error histograms, regression indices, error autocorrelations, error input autocorrelations and time series response on exhaustive experimentation study on FLLCA systems. The predictive strength of the NARX-NNs-LM strategy is analyzed by means of step-ahead and multistep ahead predictors. Diminutive error metrics on sundry FLLCA scenarios reflect the expert utilization of NARX-NNs-LM for the precise examination, anticipation and forecasting of nonlinear chaotic fractional attractors.