Kuramoto Synchronization: Order Parameter Onset

N coupled oscillators with random natural frequencies. Above critical coupling K_c = 2σ, a synchronized cluster emerges — the order parameter r jumps from 0 to positive.

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dθᵢ/dt = ωᵢ + (K/N)Σⱼsin(θⱼ−θᵢ)  |  r·e^iψ = (1/N)Σe^iθⱼ  |  K_c = 2σ√(π/2)