CLARKSON UNIVERSITY Comparing Dynamical Systems by Mostly Conjugacy

A primary concern of this thesis is to develop principles and methods to compare dynamical systems when they are not necessarily conjugate (topologically the same). The first main body of this thesis provides an understanding of “mostly conjugacy (mostly homeomorphism)” between “dynamically close” systems, which enables us to measure and interpret the distance from being conjugate. We also generalize this idea from comparing deterministic systems to stochastic systems. As a second theme, we extend and interpret the concepts of “mostly conjugacy” in symbolic dynamics, where we resort to a variant of the classic Monge-Kantorovich optimization problem to both built a useful change of variables and measure quality of the comparison through the underlying cost called the Wasserstein distance. Later, we build up a bundle structure, visualize as a bundle plot, to show the evolution of symbolic space as we vary a system’s parameter. The main object is a specific structure “joint”, which happens shortly after bifurcation, implies qualitative changes of system where the kneading points become periodic. Finally, we apply the above techniques to study time series analysis and modeling on heart rate data, which we have shown to be a one-dimensional nonlinear map that has a stochastic parameter with persistence causing the heart rate and rhythm system to wander about a bifurcation point.