Title: Random Walks, Graph Geometry, and the Stability of Phase-Locked Solutions of Coupled Oscillators
Abstract: Weakly coupled oscillators are used throughout the physical sciences, particularly in mathematical neuroscience to describe the interaction of neurons in the brain. Systems of weakly coupled oscillators have a well-known decomposition to a canonical phase model which forms the basis of our investigation in this talk. Particularly, our interest lies in examining the stability of synchronous (phase-locked) solutions to this phase system: solutions with phases having the same temporal frequency but diff er through time-independent phase-lags. I demonstrate how a series of investigations into random walks on in finite weighted graphs can be used to provide an interesting link with coupled oscillators to obtain algebraic decay rates of small perturbations off of the phase-locked solutions. I will also discuss some ongoing research into rotating wave solutions to lattice dynamical systems which motivated and heavily utilizes these graph-theoretic techniques developed for these systems of coupled oscillators.