Kuramoto Synchronization

N=100 phase oscillators with random natural frequencies ωᵢ ~ N(0,1). Coupling K. Above Kc ≈ 1.596, the order parameter r = |⟨e⟩| jumps from 0 — spontaneous synchronization.

Coupling K 1.50
K = 1.50   Kc ≈ 1.596   Order r = 0.00   Mean phase ψ = 0.00 rad   UNSYNCHRONIZED