Classification of fractional-order chaotic systems using deep learning methods

Abstract

Recent developments in fractional calculus reveal that fractional operators enable the emergence of new chaotic behaviors that cannot be observed in integer-order systems. When the characteristics of chaotic systems are combined with fractional calculus, the complexity and unpredictability of the system are further enhanced. This study examines two simple yet topologically similar six-term chaotic systems, namely Sprott H and Sprott K, to explore their application in deep learning within the framework of fractional calculus. Through the analysis of time series data, phase portraits, Lyapunov exponents, and bifurcation diagrams, it is demonstrated that the fractional-order Sprott H (FOS-H) and fractional-order Sprott K (FOS-K) chaotic systems exhibit chaotic behavior under specific system parameters and fractional orders. A dataset of 28,800 time series samples is generated and classified using pre-trained deep learning models, including GoogleNet, MobileNet-v2, DarkNet-19, DarkNet-53, and EfficientNet-b0, with transfer learning techniques applied. Using the SGDM optimizer with a learning rate of 0.0003, all models, except DarkNet-19, achieve high classification accuracy. To assess generalization, a 3160-sample test dataset from lower order fractional chaotic systems (0.8 for FOS-H, 0.82 for FOS-K) is introduced, where DarkNet-53 achieves 100% accuracy and GoogleNet reaches 99.77%, outperforming other models. Additionally, a comparison with classical machine learning methods (Support Vector Machines (SVM), Decision Tree, Random Forest) reveals that while SVM achieves 99.77% validation accuracy, its performance declines for lower fractional orders. Test results confirm that GoogleNet offers the best balance between accuracy and resource efficiency, making it suitable for real-time applications. These findings demonstrate that deep learning models, particularly DarkNet-53 and GoogleNet, can effectively classify chaotic time series with varying fractional orders, even when encountering previously unseen system dynamics.