Kuramoto Model with Noise

dθᵢ/dt = ωᵢ + (K/N)Σ sin(θⱼ−θᵢ) + ξᵢ(t) · K_c = 2/(πg(0))

K (coupling): 2.00 N oscillators: 80 Noise σ: 0.30 ω spread Δ: 1.00

N oscillators with random natural frequencies ωᵢ ~ Lorentzian g(ω) = (Δ/π)/(ω²+Δ²). Kuramoto (1984): order parameter r = |⟨e^(iθ)⟩|. For g Lorentzian: K_c = 2Δ (half-width). Above K_c, r > 0 — spontaneous synchronization. Noise (Langevin ξ) broadens the transition and reduces coherence.