Topological Feature Extraction of Scanty Time Series Data: A Data-Driven Approach for Dynamic State Change Detection
Complex dynamical systems often undergo transitions from periodic to chaotic behaviour as bifurcation parameters vary, making timely detection of these changes essential. Conventional approaches based on the maximal Lyapunov exponent (MLE) generally require either knowledge of the governing equations or sufficiently long, uniformly sampled time series. Their performance degrades when the available data are scanty or contain missing observations, making reliable phase-space reconstruction difficult. We propose a methodology that combines Topological Data Analysis (TDA), specifically 0-D sublevel persistence, with Machine Learning (ML) classifiers to distinguish periodic and chaotic regimes directly from time series. Sublevel persistence extracts topological features by analysing the evolution of minima and maxima, revealing repeating signatures for periodic dynamics and more scattered patterns for chaotic dynamics. These features are used to train logistic regression, support vector machine, and k-nearest neighbour classifiers. Hyperparameters are validated using K-fold cross-validation, yielding average classification accuracies exceeding 90%. The trained classifiers provide binary predictions, identifying periodic and chaotic behaviour in previously unseen data. The proposed methodology is evaluated on the Duffing, R\"ossler, and Lorenz systems, where the detected transitions closely agree with those identified using the MLE, demonstrating the reliability of the approach. It is further applied to real-world ECG signals to classify normal and abnormal heartbeats, producing encouraging performance across standard statistical metrics. The proposed framework provides an effective alternative for analysing sparse or incomplete time series and is particularly useful in experimental settings where conventional nonlinear time-series methods are limited.