Stability and Phase-locking in the Kuramoto model

Jared Bronski

Department of Mathematics  

University of Illinois at Urbana-Champaign

Synchronization of coupled oscillators is a very important natural phenomenon: originally discovered by Huygens it occurs in such diverse systems as the flashing of fireflies and the unstable oscillations of the Millenium bridge. The Kuramoto model is a system of phase-coupled ordinary differential equations that demonstrates this phenomenon. We show that the Kuramoto model exhibits a phase transition (in the appropriate scaling) as the strength of the coupling is varied, with the probability of synchronization undergoing a sharp transition from zero to one as the coupling strength increases. If time permits we will discuss a closely related problem of determining the qualitative behavior of a dynamical system on a network from the topology of the underlying network.