Characterization of dynamical systems with scanty data using Persistent Homology and Machine Learning

Determination of the nature of the dynamical state of a system as a function of its parameters is an important problem in the study of dynamical systems. This problem becomes harder in experimental systems where the obtained data is inadequate (low-res) or has missing values. Recent developments in the field of topological data analysis have given a powerful methodology, viz. persistent homology, that is particularly suited for the study of dynamical systems. Earlier studies have mapped the dynamical features with the topological features of some systems. However, these mappings between the dynamical features and the topological features are notional and inadequate for accurate classification on two counts. First, the methodologies employed by the earlier studies heavily relied on human validation and intervention. Second, this mapping done on the chaotic dynamical regime makes little sense because essentially the topological summaries in this regime are too noisy to extract meaningful features from it. In this paper, we employ Machine Learning (ML) assisted methodology to minimize the human intervention and validation of extracting the topological summaries from the dynamical states of systems. Further, we employ a metric that counts in the noisy topological summaries, which are normally discarded, to characterize the state of the dynamical system as periodic or chaotic. This is surprisingly different from the conventional methodologies wherein only the persisting (long-lived) topological features are taken into consideration while the noisy (short-lived) topological features are neglected. We have demonstrated our ML-assisted method on well-known systems such as the Lorentz, Duffing, and Jerk systems. And we expect that our methodology will be of utility in characterizing other dynamical systems including experimental systems that are constrained with limited data.