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.