Sketch · 2026-08-21
Kuramoto oscillators, drifting into sync.
A hundred and twenty-eight oscillators sit on a unit circle. Each one turns at its own natural frequency, drawn once from a Lorentzian centred at zero. Coupling them nudges each phase toward the mean of the rest. Below a critical K they scatter and drift; push K past it and a cluster forms, then reels the stragglers in.
Critical coupling for γ = 0.3 is Kc = 2γ = 0.60
The rule
Each oscillator i has a phase θᵢ and a natural frequency ωᵢ. Coupling to the others sums pairwise sines of phase differences:
dθᵢ/dt = ωᵢ + (K / N) · Σⱼ sin(θⱼ − θᵢ)Kuramoto's trick is to fold that sum into a single complex number. Define the order parameter as the mean of the unit vectors:
r · e^(i·ψ) = (1 / N) · Σⱼ e^(i·θⱼ)Then every oscillator sees the same driver, K·r·sin(ψ − θᵢ). r is the amber vector inside the ring: length zero when the phases are spread evenly, length one when they all point the same way.
Why Kc = 2γ
An oscillator locks when its detuning |ωᵢ − Ω| is small enough for the driver K·r to hold it in place, roughly |ωᵢ − Ω| < K·r. With a Lorentzian density of half-width γ, a self-consistency equation for r closes in the infinite-N limit and gives the exact threshold Kc = 2γ. Below it the only fixed point is r = 0. Above it, r grows continuously like √(K − Kc); the drifters and the lockers coexist, and the locked fraction widens as K climbs.
What to try
Start at K = 0 and the dots slide around the ring at their own rates; r hovers near 0.09, the finite-N floor. Nudge K to 0.4 and the arrow starts to breathe. Cross 0.6 and a cluster nucleates somewhere on the ring, then holds shape as it rotates. Push to 2.5 and almost every dot follows the arrow; only the fastest outliers still slip past. Redraw the frequencies to shuffle who locks first.
Sources
- Winfree, A. T. (1967). Biological rhythms and the behavior of populations of coupled oscillators. Journal of Theoretical Biology, 16(1), 15–42. The problem Kuramoto answered.
- Kuramoto, Y. (1975). Self-entrainment of a population of coupled nonlinear oscillators. In H. Araki (Ed.), International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics 39, 420–422. The original solution.
- Strogatz, S. H. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143(1–4), 1–20. A modern review.
- Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. & Spigler, R. (2005). The Kuramoto model: a simple paradigm for synchronization phenomena. Reviews of Modern Physics, 77(1), 137–185. The full derivation of Kc = 2γ, and much else.