Leveraging topological noise for dynamic state change detection using persistent homology

Characterization and classification of dynamical states of a system using persistent homology (PH) is proving to be rather fruitful in recent years. The marked success of the approach lies in mapping the topological features extracted by PH with the corresponding dynamical states of the system. However, there are two major drawbacks that cripple its usage widely. First, for effective identification of significant features extracted by the PH procedure, extensive human intervention and validation are inevitable. Second, identification and benchmarking of the periodic orbits via significant PH barcodes and Betti numbers stop short when the system transitions into a chaotic regime. In this paper, we address both the shortcomings of the PH procedure using a two-pronged approach. First, to minimize human intervention and validation, we employ a machine learning (ML) based binary classifier. We train the ML algorithm to demarcate true features from noise in the barcode data at a single instance of the dynamical parameter and allow the trained ML model to characterize the data at other parameter values. Second, we have proposed novel metrics derived from the insignificant feature count (short-lived features) that are normally discarded as noise. In this paper, we demonstrate that our metrics can clearly classify periodic and chaotic states as well as identify the parameters of dynamical transition sufficiently accurately, and compare their performance with a conventional method. We have assessed the performance of our metrics rigorously using three standard evaluators and benchmarked them against the maximum Lyapunov exponent, obtaining strong positive correlation coefficients ranging from 0.75 to 0.97. Furthermore, we assert that when the available data are sparse, as is typical in real-world systems, our proposed metrics can yield robust classification performance.