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 $H_0$ features to 99–100% under $H_1$ and combined feature representations. These findings highlight that topological features, particularly $H_1$, 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.

A string under the influence of an alternating electric current and a non-uniform magnetic field is described by a complicated nonlinear mathematical model that, in a certain approximation, yields a driven Mathieu-Duffing (MD) oscillator. To apply the Lyapunov exponent (LE) stability analysis to a driven MD oscillator, we need to transform it into an autonomous form. For that purpose, we transform the nonautonomous (NA) driven MD oscillator into six-dimensional (6D) and three-dimensional (3D) autonomous forms. A comparative numerical analysis is carried out by computing global bifurcation diagrams (BDs) to examine the equivalence of the dynamics between the NA and autonomous forms of the system. Also, for the particular initial conditions (ICs), BDs and corresponding LE spectra were calculated to characterize the system's stability. The results show that the 3D autonomous form of the MD oscillator replicates the dynamics of the original NA system. Meanwhile, the 6D autonomous system, preserving the overall pitchfork-like bifurcation structure, exhibits only partial equivalence. Also, a dominant frequency (DF) analysis using the Fast Fourier Transform (FFT) is performed on the time series of the driven MD oscillator over a wide range of the main control parameter. The obtained spectral evolution demonstrates harmonic, subharmonic, and super-harmonic oscillatory behaviors in agreement with the corresponding bifurcation structures.

In this work, we study the global dynamics of the Mathieu-duffing (MD) driven oscillator, which naturally comes from the model of a stretched string under the influence of alternating electric current in a non-uniform magnetic field. This system exhibits rich nonlinear dynamics that occur from the combined influence of thermal, electromagnetic, and elastic effects. We extracted the bifurcation diagrams (BD), time series, and corresponding phase portraits by numerical calculation of inhomogeneous and adjoint homogeneous equations. We uncover the emergence of periodic, bistable, and multistable regimes, revealing the intricate transition between different states of the system. This study provides a unified framework for investigating the bifurcation structure and temporal dynamics of electrically driven elastic structures in magnetic fields.