David Hien: Topological Signatures for Identifying and Classifying Oscillations

Abstract.

Nonlinear dynamical systems often exhibit rich and complicated recurrent dynamics. Understanding these dynamics is challenging, especially in higher dimensions where visualization is limited. Additionally, in many applications, time series data is all that is available. This motivates our TDA-based approach to study such systems. More precisely, we introduce the cycling signature which is constructed by taking persistent homology of time series segments in a suitable ambient space. Oscillations in a time series can then be identified by analyzing the cycling signatures of its segments. We demonstrate this through several examples. In particular, we identify and analyze 6 oscillations in a 4d system of ordinary differential equations. The talk will start with the necessary background in dynamics, no previous exposure to dynamical systems is necessary to follow the talk.