Stability of infinite systems of coupled oscillators via random walks on weighted graphs

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 work. 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 differ through time-independent phase-lags. The main stability result of this work comes from adapting a series of investigations into random walks on infinite weighted graphs. We provide an interesting link between the seemingly unrelated areas of coupled oscillators and random walks to obtain algebraic decay rates of small perturbations off the phase-locked solutions under some minor technical assumptions. We also provide some interesting and motivating examples that demonstrate the stability of phase-locked solutions, particularly that of a rotating wave solution arising in a well studied paradigm in the theory of coupled oscillators.