The 17th oscillator — an antagonist that prevents synchronisation

A ring of coupled Kuramoto oscillators wants to synchronise. That is the whole point of the coupling term — and it is also the failure mode, because a system that reaches consensus stops computing. Every input produces the same output.

The 17th oscillator sits at the centre of a 16-node ring and does two things at once.

It repels the ring

Where normal coupling attracts, the centre pushes:

rep = -repel * sin(thetaC - th[i])
dth[i] = om[i] + ext + (K/n)*c + rep

The sign is inverted against the standard term. Every node is driven away from the centre's phase, and the gain is set high enough to fight the whole crowd.

And it flees the ring

The centre is not a fixed obstacle. It runs from the mean phase of the nodes chasing it, while drifting on its own:

meanPhase = atan2(sum sin th, sum cos th)
thetaC += dt * (omegaC - 0.5*repel*sin(meanPhase - thetaC))

omegaC = 0.6 is a constant drift — the "keep flowing" term. The result is a perpetual chase: the ring is pushed away from the centre, the centre runs from the ring's average, and neither reaches equilibrium.

Why this is different from the traveling-wave fix

An earlier approach solved the same problem by replacing static phase initialisation with a traveling wave, so there was no fixed point to collapse to. The 17th oscillator leaves the fixed point in place and makes it unreachable — an active antagonist rather than a change of coordinates.

The practical difference is that repulsion is a live, tunable force. repel is a dial: at zero the ring synchronises normally, and raising it holds the system in sustained motion. The traveling wave is a choice made at initialisation and cannot be turned down later.

See it run

The Wave Observatory has the 17th oscillator on a checkbox. Toggle it off and the ring locks; toggle it on and raise REPEL and the order parameter never settles. The harmonic spectrum, mode-coupling products and the coupling matrix G(p,q) update live alongside it.