Kuramoto-Sakaguchi Model

dθᵢ/dt = ωᵢ + (K/N)Σⱼ sin(θⱼ − θᵢ − α) · frustration phase α

r = —
50
2.5
0°
1.0
The Kuramoto-Sakaguchi model extends Kuramoto synchronization with a frustration phase α. When α ≠ 0, the coupling term sin(θⱼ − θᵢ − α) shifts the preferred phase difference. Critical coupling: K_c = 2σ (Lorentzian distribution). For α ≥ π/2, synchronization is impossible — oscillators form traveling waves or chimera states. The order parameter r = |N⁻¹ Σ eⁱᶿ| measures coherence (r=1: full sync, r≈0: incoherence).