Exploring Equilibria in Stochastic Delay Differential Equations Using Persistent Homology

This paper explores the possibility of using techniques from topological data analysis for studying datasets generated from dynamical systems described by stochastic delay equations. The dataset is generated using Euler-Maryuama simulation for two first order systems with stochastic parameters drawn from a normal distribution. The first system contains additive noise whereas the second one contains parametric or multiplicative noise. Using Taken’s embedding, the dataset is converted into a point cloud in a high-dimensional space. Persistent homology is then employed to analyze the structure of the point cloud in order to study equilibria and periodic solutions of the underlying system. Our results show that the persistent homology successfully differentiates between different types of equilibria. Therefore, we believe this approach will prove useful for automatic data analysis of vibration measurements. For example, our approach can be used in machining processes for chatter detection and prevention.