Kuramoto Model

N coupled phase oscillators with natural frequencies ωᵢ and global coupling K/N. Watch the order parameter r rise as K increases and oscillators synchronize.

Top notes — order parameter r and critical coupling.

Side notes — frequency distribution and partial locking.

N: K: σ (freq spread): dt: Speed:

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Side notes — r = |(1/N) ∑ e^{iθᵢ}|; dot and arrow show coherence.

Bottom notes — mean‑field interpretation; thresholds depend on g(ω).