Kuramoto Model in 3D

N oscillators on a sphere, each with a natural frequency ω_i. Coupling K drives them toward synchrony. Order parameter r = |⟨e^{iθ}⟩| measures synchronization: r≈0 is incoherent, r≈1 is fully synchronized. Phase transition at K_c = 2/πg(0).

Parameters

Oscillators N 60

Coupling K 1.00

Freq spread σ 1.00

Time step dt 0.02

Order r0.00

Mean phase0.00

K_c estimate—

Time0.0

dθ_i/dt = ω_i + (K/N)Σ sin(θ_j - θ_i)

r·e^{iψ} = (1/N)Σ e^{iθ_k}

K_c = 2/[π·g(0)]
For Lorentzian g: K_c = 2γ