Stability Determination in Turning Using Persistent Homology and Time Series Analysis

This paper describes a new approach for ascertaining the stability of autonomous stochastic delay equations in their parameter space by examining their time series using topological data analysis. We use a nonlinear model that describes the tool oscillations due to self-excited vibrations in turning. The time series is generated using Euler-Maruyama method and then is turned into a point cloud in a high dimensional Euclidean space using the delay embedding. The point cloud can then be analyzed using persistent homology. Specifically, in the deterministic case, the system has a stable fixed point while the loss of stability is associated with Hopf bifurcation whereby a limit cycle branches from the fixed point. Since periodicity in the signal translates into circularity in the point cloud, the persistence diagram associated to the periodic time series will have a high persistence point. This can be used to determine a threshold criteria that can automatically classify the system behavior based on its time series. 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.