Kuramoto Order Parameter

N coupled oscillators synchronize above a critical coupling K_c. The order parameter r = |⟨e^iθ⟩| jumps from 0 to 1.

Coupling K 1.00

N oscillators 40

Spread σ 1.00

dθᵢ/dt = ωᵢ + (K/N)Σⱼ sin(θⱼ − θᵢ)
Natural frequencies ωᵢ ~ Gaussian(0,σ²). Critical coupling: K_c = 2/(πg(0)) where g(0)=1/(σ√2π) is the frequency distribution peak. Red dashed line marks K_c.