Order parameter r(K): phase transition from incoherence to synchrony
r = 0.000 | ψ = 0.000
Coupling K1.50
Oscillators N80
Freq spread σ1.00
Noise D0.00
The Kuramoto model: dθᵢ/dt = ωᵢ + (K/N)Σⱼsin(θⱼ-θᵢ). Frequencies ωᵢ drawn from Lorentzian g(ω)=γ/[π((ω-ω₀)²+γ²)]. Order parameter r·e^(iψ) = (1/N)Σe^(iθᵢ). Critical coupling Kc = 2/πg(0) = 2σ for Gaussian, 2γ for Lorentzian. Above Kc, r grows as r ~ √(K-Kc) (mean-field). Exact solution by Kuramoto (1984) via self-consistency equation.