Kuramoto Model — Synchronization & Order Parameter
N coupled phase oscillators with natural frequency spread σ and coupling K
Order r = -
K_c = -
Phase ψ = -
Synchronized: -
The Kuramoto model: dθi/dt = ωi + (K/N)Σ sin(θj−θi).
Natural frequencies ωi are drawn from a Lorentzian g(ω) of width σ.
The complex order parameter r·eiψ = (1/N)Σ eiθ_k measures synchrony:
r=0 is incoherent, r=1 is fully locked. Critical coupling Kc = 2σ/π for Lorentzian,
2/(πg(0)) in general. Above Kc, a synchronized cluster grows as r ~ √(1−Kc/K).
Left: unit circle with oscillator phases (arrows) and mean-field vector. Right: r(t) history.