I also studied how random noise affects synchronization. Noise in the system makes it harder for the oscillators to synchronize. Noise is usually present in any physical system — as one example, the behavior of an array of Josephson junctions could be greatly affected by thermal noise — so it is important to study how random noise can change the synchronization behavior seen in the Kuramoto model.
With some powerful math techniques, it turns out that we can still solve for KC in the presence of noise! The technique used is a stability analysis, where we study the stability of the incoherent, unsynchronized state to small perturbations. If the incoherent state is stable, then we must be below KC, because the system returns to its completely unsynchronized state. If a perturbation makes the incoherent state unstable, then we must be above KC. By finding the transition from stability to instability, we can calculate the value of KC.
For example, for a Lorentzian distribution of half-width gamma and noise of strength (beta)^2, we get:
Again, other distributions have more complex formulas for the critical coupling, but we can always calculate KC for the noisy system, a rather remarkable fact.
Next section: Numerical Results