Kuramoto Model — Order Parameter & Mean Field

N coupled oscillators synchronize when coupling K exceeds K_c = 2σ/π·g(0), where g is the frequency distribution.

N: 80 K: 2.00 σ: 1.00

r = 0.000 | ψ = 0.0°

Kuramoto (1975): dθᵢ/dt = ωᵢ + K/N Σⱼ sin(θⱼ−θᵢ). Order parameter r·e^iψ = (1/N)Σe^iθⱼ. Critical coupling K_c = 2/[π·g(0)] where g is the Lorentzian width σ. Above K_c, r jumps continuously (second-order transition).