Acute Lymphoblastic Leukemia (ALL) is a prevalent form of childhood blood cancer characterized by the proliferation of immature white blood cells that rapidly replace normal cells in the bone marrow. The exponential growth of these leukemic cells can be fatal if not treated promptly. Classifying lymphoblasts and healthy cells poses a significant challenge, even for domain experts, due to their morphological similarities. Automated computer analysis of ALL can provide substantial support in this domain and potentially save numerous lives. In this paper, we propose a novel classification approach that involves analyzing shapes and extracting topological features of ALL cells. We employ persistent homology to capture these topological features. Our technique accurately and efficiently detects and classifies leukemia blast cells, achieving a recall of 98.2% and an F1-score of 94.6%. This approach has the potential to significantly enhance leukemia diagnosis and therapy.

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.

Acute lymphoblastic leukemia (ALL) is a hematological malignancy characterized by the rapid proliferation of immature white blood cells in the bone marrow. Early and accurate diagnosis is essential for improving clinical outcomes; however, distinguishing between lymphocytes and lymphoblasts poses significant challenges owing to their subtle morphological similarities. Traditional manual diagnostic methods, which rely on expert evaluations, are inherently time-consuming and subject to human error. In recent years, machine learning and deep learning approaches have emerged as promising tools for automating and enhancing diagnostic processes. This review systematically examines state-of-the-art traditional and deep learning techniques applied for ALL detection and classification. We provide a comprehensive analysis of various methodologies, including supervised machine learning algorithms and advanced deep learning architectures, with a focus on critical stages such as image preprocessing, feature extraction, and blast cell quantification. Furthermore, we discuss the performance metrics and accuracy benchmarks, highlighting the potential of these techniques to match or exceed human diagnostic capabilities. The review concludes with a discussion of the current challenges, recent developments, and future directions in the application of artificial intelligence for ALL diagnosis, underscoring the need for continued innovation to meet emerging clinical demands.

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.