Synchronization of Kuramoto oscillators in dense networks

Abstract

Abstract We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let G = ( V , E ) be a connected graph and ( a i j ) i , j = 1 n denotes its adjacency matrix. Let the function f : T n → R be given by f ( θ 1 , … , θ n ) = ∑ i , j = 1 n a i j cos ( θ i − θ j ) . This function has a global maximum when θ i = θ for all 1 ⩽ i ⩽ n . It is known that if every vertex is connected to at least μ ( n − 1) other vertices for μ sufficiently large, then every local maximum is global. Taylor proved this for μ ⩾ 0.9395 and Ling, Xu & Bandeira improved this to μ ⩾ 0.7929. We give a slight improvement to μ ⩾ 0.7889. Townsend, Stillman & Strogatz suggested that the critical value might be μ c = 0.75.