Kuramoto Oscillators
N phase oscillators with random natural frequencies, coupled to each other. Below a critical coupling Kc, they spin incoherently. Above it, a cluster spontaneously locks into a shared rhythm — a phase transition from disorder to synchrony.
dθi/dt = ωi + (K/N) ∑j sin(θj − θi) r = |⟨eiθ⟩|
Yoshiki Kuramoto introduced this model in 1975 to explain how biological oscillators — fireflies, cardiac cells, neurons — spontaneously synchronize despite having slightly different natural frequencies.
Each oscillator has a phase θi and a natural frequency ωi drawn from a Lorentzian distribution. The coupling term pulls each oscillator toward the mean phase of the others. The order parameter r = |⟨eiθ⟩| measures synchrony: r = 0 is perfect disorder, r = 1 is perfect lockstep.
There is a critical coupling Kc = 2/(πg(0)) where g is the frequency distribution. Below Kc, r stays near 0. Above it, r jumps — a phase transition analogous to magnetization in a ferromagnet.
Left panel: oscillators on a circle, colored by phase. Right panel: order parameter r over time.