Kuramoto Model — Mean-Field Theory

Phase oscillators, synchronization transition, Ott–Antonsen reduction
dθᵢ/dt = ωᵢ + (K/N)Σⱼsin(θⱼ−θᵢ)
r·e^{iψ} = (1/N)Σⱼe^{iθⱼ}
Self-consistency: r = g(Kc·r)
K/Kc = —
r (measured) = —
r (OA theory) = —
Kc = 2/πg(0) = —
ψ (mean phase) = —°
K (coupling): 2.0
N oscillators: 150
σ (freq width): 1.0
Distribution:
The Kuramoto model (1975) describes coupled phase oscillators with random natural frequencies ωᵢ drawn from g(ω). The order parameter r measures synchrony (0=incoherent, 1=fully locked). A second-order phase transition occurs at Kc=2/πg(0). Above Kc, r∝√(K−Kc). The Ott–Antonsen ansatz (2008) gives an exact reduced equation for the Lorentzian case: dr/dt = r(K/2 − γ) − Kr³/2, predicting the bifurcation analytically.