Kuramoto Synchronization

N coupled oscillators spontaneously synchronize when coupling K exceeds the critical threshold Kc = 2/(πg(0))

Phase distribution on unit circle
Order parameter r(t) over time
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r(t) order
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Kc critical
Incoherent
state
The Kuramoto model (1984): dθᵢ/dt = ωᵢ + (K/N)Σⱼ sin(θⱼ−θᵢ). Natural frequencies ωᵢ are drawn from a Lorentzian with half-width γ. The order parameter r = |N⁻¹Σ eⁱθ| measures coherence: r≈0 is incoherent, r≈1 is fully synchronized. The phase transition at K_c = 2γ is second-order — r grows as √(K−K_c) above threshold.