In a dynamical system, the time series and phase space play vital roles, and we applied topological data analysis to these characteristics. More precisely, we consider the well-known Rössler-like attractor to analyze time-series and phase-space images. We studied persistent homology representations directly from the time series of the system to obtain point cloud data. In our approach, we converted the time series to a point cloud and computed homology using the Rips complex. This enabled us to measure the topological features of the system behavior. We also applied cubical homology to phase-space images for the first time, a novel contribution that represents an image-based approach to analyze phase portraits. This article provides a review of the topological data analysis of time series using examples with the Python function. Finally, we computed topological machine learning features, such as persistent landscapes, persistence images, and Betti curves. These features enable the automated analysis and classification of dynamical behaviors and, hence, connect topological data analysis with machine learning. This study is new in that it presents a comprehensive topological data analysis pipeline tailored to dynamical systems. The goal is to make these approaches accessible and usable for nonlinear dynamics to analyze their temporal series and phase portraits.

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.

The dynamics and synchronization properties of a network of three erbium-doped fiber lasers coupled via higher-order interactions (HOI) are investigated using simplicial complexes. Three coupling configurations are analyzed: first-order only, second-order only, and their combination. Several synchronization metrics, including the average synchronization error, similarity index, and correlation coefficient, are employed to characterize the effects of HOI. The results demonstrate that incorporating second-order coupling substantially enriches the system’s dynamical behavior, revealing a greater diversity of collective states than first-order coupling alone. While complete synchronization is not achieved within the bounded coupling strengths considered, both phase synchronization and anticipation synchronization emerge within distinct parameter intervals.

The dynamics and synchronization properties of a network of three erbium-doped fiber lasers coupled via higher-order interactions (HOI) are investigated using simplicial complexes. Three coupling configurations are analyzed: first-order only, second-order only, and their combination. Several synchronization metrics, including the average synchronization error, similarity index, and correlation coefficient, are employed to characterize the effects of HOI. The results demonstrate that incorporating second-order coupling substantially enriches the system’s dynamical behavior, revealing a greater diversity of collective states than first-order coupling alone. While complete synchronization is not achieved within the bounded coupling strengths considered, both phase synchronization and anticipation synchronization emerge within distinct parameter intervals.