Kuramoto Model — Order Parameter

dθᵢ/dt = ωᵢ + (K/N)Σ sin(θⱼ−θᵢ) — synchronization transition at K_c = 2/πg(0)

The Kuramoto model describes N phase oscillators with random natural frequencies ωᵢ ~ N(0,σ²) coupled by strength K. The order parameter r = |N⁻¹Σe^{iθⱼ}| ∈ [0,1] measures synchrony. A phase transition occurs at K_c = 2σ/π: for K>K_c, r grows as √(1−K_c/K).