On the Existence and Stability of Rotating Wave Solutions to Lattice Dynamical Systems

Abstract

Rotating wave solutions to evolution equations have been shown to govern many important biological and chemical processes. Much of the rigorous mathematical investigations of rotating waves rely on the model exhibiting a continuous Euclidean symmetry, which is only present in an idealized situation. Here we investigate the existence of rotationally propagating solutions in a discrete spatial setting, in which typical symmetry methods cannot be applied, thus presenting an unique perspective on rotating waves. Our goal in this thesis is to demonstrate the existence and potential stability of rotating wave solutions to a spatially discretized infinite systems of coupled differential equations. This goal is achieved by considering so-called Lambda-Omega systems, which have frequently been used to model typical oscillatory dynamics. Our work is broken into three major components: 1. An infinite system of coupled phase equations is investigated and we demonstrate that under some mild assumptions the system exhibits a phase-locked rotating wave solution. The phase system is derived from a limiting case of the original Lambda-Omega system, and therefore solutions of the phase equation will be useful in finding rotating wave solutions to the full Lambda-Omega system. 2. We examine the stability of the rotating wave solution found in the coupled phase equations. This is achieved by providing a link with an underlying graph-theoretic geometry endowed by the spatially discretized system. We use results from random walks on infinite graphs to provide a general stability theorem for coupled phase equations. 3. We use the rotating wave solution of the phase equations to extend to a rotating wave solution of the full Lambda-Omega system. This result is achieved using a non-standard Implicit Function Theorem, since we show that typical implicit function arguments cannot be applied to our present situation.