When machine learning meets fractional-order chaotic signals: detecting dynamical variations

Abstract

The challenge of classifying multivariate time series generated by discrete and continuous dynamical systems according to their chaotic or non-chaotic behavior has been studied extensively in the literature. The examination of noise or the variation of variables that affect a dynamic system's chaoticity will not be beneficial in analyzing structures employing random number generators (RNG) that are already assured to be chaotic. However, detecting the structural changes and their time intervals in deterministic systems with proven chaoticity can contribute to the literature in encryption applications. Machine Learning algorithms provide flexible possibilities to analyze and predict manipulations that may occur in the dynamics of chaotic and complex systems. This study proposes a deep Long-Short-Term-Memory (LSTM) network with a classification process to predict dynamical changes in a fractional-order chaotic (FOC) system. First, the appropriate system parameters are calculated to satisfy the chaotic behavior in the fractional-order Chen system. The predictive-corrective Adams-Bashforth-Moulton algorithm is used to simulate the FOC Chen system in the time domain. The Lyapunov exponents of the system were obtained according to the Wolf method. Next, three different scenarios have been designed to test and demonstrate the effectiveness of the proposed method. Synthetic FOC signals obtained after sub-sampling and statistical feature extraction processes fed the input of the deep bidirectional LSTM (BiLSTM) network to perform the training and testing process. The classification performance for "q" and "c" classes reaches 100% with the proposed model. The overall average testing accuracy, sensitivity, specificity, precision, F1 score and MCC are 98%, 98%, 99.3%, 98.1%, 98%, and 97.3%, respectively. Our results demonstrate the utility of using a deep BiLSTM network for detecting dynamical variations in nonlinear FOC systems.