Synchronization of Kuramoto oscillators in dense networks

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, letG= (V,E) be a connected graph and (a(ij))(i,j=1)(n) f:T-n -> R be given by f(theta(1), ..., ,theta(n))= Sigma(n)(i,j=1) cos(theta(i)-theta(j)). This function has a global maximum when theta(i) = theta for all 1 <= i <= n. It is known that if every vertex is connected to at least mu(n- 1) other vertices for mu sufficiently large, then every local maximum is global. Taylor proved this for mu >= 0.9395 and Ling, Xu & Bandeira improved this to mu > 0.7929. We give a slight improvement to mu >= 0.7889. Townsend, Stillman & Strogatz suggested that the critical value might be mu(c)= 0.75.