Figure 2: Characterizing various coherent states in the phase diagram.

R [long time average of R(t)] vs. K for the uniform frequency distribution , . Branches 1 − −5 are the incoherent state, the (unstable) traveling wave state predicted by the mean-field theory, the standing wave state, the unstable and the stable two-cluster synchronous states, respectively. The blue and red lines denote the forward and the backward transitions, respectively. In both directions, K is changed adiabatically in simulations. There is a hysteresis region of the standing wave solution within K = 1.725 − 1.8. In the simulations oscillators number N = 50,000, and a fourth-order Runge-Kutta integration method with time step 0.01 is used.