Topological Signal Processing (TSP) is the study of time series data through the lens of Topological Data Analysis (TDA)-a process of analyzing data through its shape. This work focuses on developing novel TSP tools for the analysis of dynamical systems. A dynamical system is a term used to broadly refer to a system whose state changes in time. These systems are formally assumed to be a continuum of states whose values are real numbers. However, real-life measurements of these systems only provide finite information from which the underlying dynamics must be gleaned. This necessitates making conclusions on the continuous structure of a dynamical system using noisy finite samples or time series. The interest often lies in capturing qualitative changes in the system's behavior known as a bifurcation through changes in the shape of the state space as one or more of the system parameters vary. Current literature on time series analysis aims to study this structure by searching for a lower-dimensional representation; however, the need for user-defined inputs, the sensitivity of these inputs to noise, and the expensive computational effort limit the usability of available knowledge especially for in-situ signal processing. This research aims to use and develop TSP tools to extract useful information about the underlying dynamical system's structure. The first research direction investigates the use of sublevel set persistence-a form of persistent homology from TDA-for signal processing with applications including parameter estimation of a damped oscillator and signal complexity measures to detect bifurcations. The second research direction applies TDA to complex networks to investigate how the topology of such complex networks corresponds to the state space structure. We show how TSP applied to complex networks can be used to detect changes in signal complexity including chaotic compared to periodic dynamics in a noise-contaminated signal. The last research direction focuses on the topological analysis of dynamical networks. A dynamical network is a graph whose vertices and edges have state values driven by a highly interconnected dynamical system. We show how zigzag persistence-a modification of persistent homology-can be used to understand the changing structure of such dynamical networks.
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