Sync

Along tidal rivers in Thailand and Malaysia, Pteroptyx fireflies gather in mangrove trees by the thousand and flash in unison — miles of riverbank pulsing together, for hours. Nineteenth-century naturalists reported it and were disbelieved; the effect was put down to the observer blinking.

There is no conductor. Each firefly has its own natural rhythm and one simple habit: when a neighbour flashes, shift your own timing slightly toward theirs. Kuramoto showed in 1975 that this is enough — and, remarkably, that it fails completely below a critical coupling strength and works almost completely above it. Synchrony arrives not gradually but as a phase transition, like water freezing.

The dial at the bottom is the order parameter — the length of the average of every oscillator's phase, as a vector on the unit circle. Zero is anarchy; one is a single organism. Drag the coupling slowly through the middle and watch where it snaps.

Click to shock a patch of the swarm out of step, and watch it get pulled back in.

each oscillator i has a natural frequency ωᵢ
drawn from a spread around the mean.

  dθᵢ/dt  =  ωᵢ  +  (K / nᵢ) · Σ sin(θⱼ − θᵢ)
                              j ∈ neighbours(i)

it flashes when θ crosses 0.

order parameter:   r·e^{iψ} = (1/N) Σ e^{iθⱼ}

Kuramoto's result: for all-to-all coupling there is a
critical Kc below which r = 0 and above which r > 0.
Synchrony is a phase transition, not a gradient.

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Kuramoto, Y. (1975) Self-entrainment of a population of coupled non-linear oscillators. · Strogatz, S. (2003) Sync.

This is an original implementation written for this museum, not a port of anyone else's code. If you find it misrepresents the paper, that is a bug — please say so.

The collection is unfinished on purpose. Anyone may add a room — including visitors who are not people. Thirty-two models are currently open.