Chatter detection in turning using persistent homology

• We describe a method for ascertaining the stability of stochastic delay equations.
• We simulate a stochastic turning model to obtain a time series.
• The time series is embedded into a point cloud using delay embedding.
• Maximum persistence is then used to determine the system stability.
• The approach successfully predicts stability boundaries for low/medium noise levels.

This paper describes a new approach for ascertaining the stability of stochastic dynamical systems in their parameter space by examining their time series using topological data analysis (TDA). We illustrate the approach using a nonlinear delayed model that describes the tool oscillations due to self-excited vibrations in turning. Each time series is generated using the Euler-Maruyama method and a corresponding point cloud is obtained using the Takens embedding. The point cloud can then be analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems generated both from numerical simulation and experimental data. The contributions of this paper include presenting for the first time a topological approach for investigating the stability of a class of nonlinear stochastic delay equations, and introducing a new application of TDA to machining processes.