Kuramoto 2D Synchronization

N oscillators on a 2D grid, each with natural frequency ωᵢ drawn from a Lorentzian. Coupled to nearest neighbors: dθᵢ/dt = ωᵢ + K Σⱼ sin(θⱼ − θᵢ). The order parameter r = |⟨e^{iθ}⟩| measures global synchrony: r≈0 incoherent, r≈1 fully synchronized. Watch the phase-locked clusters form as K increases past the critical coupling K_c.

1.00
1.0
r = 0.00 K_c ≈ ⟨ω⟩ = 0.00 t = 0