Kuramoto Mean-Field Synchronization

N coupled phase oscillators: dθᵢ/dt = ωᵢ + (K/N)Σ sin(θⱼ−θᵢ). The order parameter r = |⟨e^{iθ}⟩| measures global synchrony. Above the critical coupling K_c = 2g(0)⁻¹, collective synchronization emerges.

Order parameter r = 0.000  |  K_c ≈ 0.000