In addition to the analytical work, I also ran computer simulations of systems of oscillators.  I wrote a program that simulated the oscillators for a fixed amount of time, and then recorded each oscillator's phase and frequency after they had had a chance to interact.

The three figures below show the recorded frequency of each of 1000 oscillators in a simulation of the Kuramoto model, with varying values for the coupling constant K.  The oscillators are ordered from lowest to highest natural frequency, with natural frequencies selected according to a Lorentzian distribution.  The left plot is for K < KC, the middle plot is for K ~ KC, and the right plot is for K > KC.  Partial frequency synchronization is visible at and above KC, as the oscillators with natural frequencies close to zero synchronize to have a frequency of exactly zero.




The next figures show the corresponding plots for each oscillator's phase after the oscillators have had time to interact.  Partial phase synchronization is visible at and above KC, as the oscillators with natural frequencies close to zero group together and go through their cycle with close to the same phase as their neighbors.



I also ran simulations including random noise, and compared the results with what was obtained analytically.  Below is a plot showing the degree of phase synchronization |r| versus the coupling K.  Notice that the amount of synchronization is zero for low coupling, and then rises when K reaches KC and asymptotically approaches perfect synchronization (|r|=1).  (beta)^2 sets the strength of the noise; notice that higher noise corresponds to a higher amount of coupling required for synchronization, as expected. These results are taken from the simulation of N = 5000 oscillators with natural frequencies distributed according to a Lorentzian distribution.  The simulation was run on a network at the Ohio Supercomputer Center.  The expected values for KC from the analytical work are shown as three vertical lines at 1.5, 2.0, and 2.5.