Kuramoto Synchronization

N=50 oscillators with random natural frequencies ω_i. θ̇_i = ω_i + (K/N)Σ sin(θ_j - θ_i). Order parameter r = |⟨e^{iθ}⟩| jumps from ~0 to ~1 at K_c = 2/(π·g(0)) ≈ 1.6 (Lorentzian g).

Coupling

1.00
K_c ≈ 1.6 for Lorentzian dist.

Order Parameter

r = |⟨e^{iθ}⟩|—
Mean phase ψ—
Time0

r(t) history

Each dot: oscillator on unit circle
Arrow: order parameter vector
Colors: phase angle (hue)
Below K_c: incoherent (r→0)
Above K_c: partial sync (r>0)
Far above K_c: full sync (r→1)