Kuramoto Model
N phase oscillators · Global coupling · Synchronization transition at K_c
Kuramoto model: dθᵢ/dt = ωᵢ + (K/N)Σⱼ sin(θⱼ − θᵢ), where ωᵢ ~ g(ω) (Lorentzian or Gaussian frequency distribution).
Order parameter: r(t)e^{iψ} = (1/N)Σⱼ e^{iθⱼ}. r=0: incoherent; r=1: fully synchronized.
Critical coupling: K_c = 2/πg(0) (Kuramoto 1984). For Lorentzian g(ω) = γ/π(ω²+γ²): K_c = 2γ.
Bifurcation: For K > K_c, r grows as r ~ √((K−K_c)/K_c) — a supercritical pitchfork bifurcation on the mean-field level. Finite-N fluctuations round the transition.