Topological Machine Learning Framework for Phase Portrait Classification of Nonlinear Dynamical Systems

Source abstract

Nonlinear dynamical systems exhibit complex behaviors such as periodicity and chaos, which are traditionally analyzed using time-series data. However, these approaches often fail to capture the intrinsic geometric structure of the system dynamics represented in the phase space. In this study, we address this limitation by proposing a topological machine learning framework that leverages phase portrait images to classify dynamical regimes. The primary objective of this study is to investigate whether the topological features extracted from phase portraits can effectively distinguish between periodic and chaotic behaviors across different nonlinear systems. To achieve this, we employed the Topological Data Analysis (TDA) technique of cubical homology, which enables the extraction of topological descriptors, such as persistence diagrams and Betti curves. We used these features to train multiple machine learning (ML) classifiers, including XGBoost, Support Vector Machine (SVM), K-Nearest Neighbors (KNN), Gaussian Naïve Bayes (GNB), and Random Forest (RF). The experimental results across benchmark systems, including the Chua, Lorenz, Mathieu–Duffing, and erbium-doped fiber laser models, demonstrate that the proposed approach achieves high classification accuracy, with performance improving from approximately 93% under H0 features to 99–100% under H1 and combined feature representations. These findings highlight that topological features, particularly H1, effectively capture the underlying geometric structure of dynamical systems. Overall, the proposed framework provides a robust, interpretable, and generalizable approach for phase portrait classification, with potential applications in nonlinear system analysis, pattern recognition, and early detection of chaotic transitions.