Simulation
Synchrony
Kuramoto coupled oscillators · N = 60 · drag K to watch the transition
Sixty oscillators, each with its own natural frequency drawn from a bell curve. When uncoupled, they drift — each spinning at its own rate, phases spreading across the circle. When coupled strongly enough, they snap into collective motion.
The coupling strength K is the only dial. Below a critical value Kc, coupling is too weak to overcome the spread of natural frequencies — the system stays incoherent. Above it, a fraction of oscillators lock together, pulling more with them. The transition is sharp.
The order parameter r measures collective coherence: r = 0 means phases scattered uniformly, r = 1 means all phases identical. Watch the dot at the center of the phase circle — that's the mean-field vector. Its length is r.
Order parameter r
0.00
Mean phase Ψ
—
State
incoherent
Kc (estimated)
—
Phase circle · each dot is one oscillator
Order parameter r over time
Yoshiki Kuramoto, 1975. The model is dθi/dt = ωi + (K/N) Σj sin(θj − θi).
Each oscillator's phase θi is pulled toward the phases of all others, weighted by K/N.
The critical coupling is Kc = 2/πg(0) where g(ω) is the frequency distribution at its peak.
For a Lorentzian distribution with spread σ, Kc = 2σ.
Kuramoto's insight: rather than solving the full N-body problem, track only the mean-field vector
reiΨ = (1/N) Σj eiθj. The system becomes self-consistent.
The Millennium Bridge (London, 2000) opened June 10, closed June 12.
Walkers on the bridge began to sway slightly due to minor lateral oscillations.
To maintain balance, they unconsciously synchronized their footfalls with the bridge frequency —
which amplified the oscillation, which caused more synchrony. K exceeded Kc in real time.
The fix was structural: tuned mass dampers installed in 2001 absorbed energy and suppressed the feedback.
No behavioral intervention to the walkers was attempted or needed.