Phase circle, frequency histogram, r(t) and the r(K) transition against Kuramoto theory.
Sweeps K from 0.4 K_c to 2.2 K_c with N = 300 and compares the time-averaged r with the self-consistent theory.
Not run yet.
The science
Model. N oscillators with natural frequencies ωi drawn from a density g(ω) obey the Kuramoto equation dθi/dt = ωi + (K/N) Σj sin(θj − θi). The complex order parameter r eiψ = (1/N) Σ eiθj measures coherence: r ≈ 1/√N when phases are scattered, r → 1 when all fireflies flash together. The circle shows phases in the frame rotating at the mean frequency; the arrow is r eiψ.
Critical coupling. Kuramoto (1975) showed that for a unimodal, symmetric g the incoherent state loses stability at Kc = 2 / (π g(0)): Gaussian (σ) gives Kc = 2σ√(2/π) ≈ 1.596 σ, Lorentzian (half-width γ) gives Kc = 2γ, uniform (half-width Δ) gives Kc = 4Δ/π. Above Kc the stationary order parameter is the root of the self-consistency condition 1 = K ∫−π/2π/2 cos²θ g(K r sinθ) dθ; for a Lorentzian it is exactly r = √(1 − Kc/K). The dashed curve in the lower-right graph is this theory (computed numerically by the same code as the 3D page).
What the 2D view adds. The phase circle shows the locked cluster (oscillators with |ωi| < K r) and the drifting tail, the histogram shows g(ω) with the locking window ±K r shaded, and r(t) shows the growth rate of coherence after a quench.
Honest simplifications. Mean-field (all-to-all) coupling only, sinusoidal phase response, frequencies are deterministic quantiles of g (removes sampling noise), N = 120 for the live view and 300 for the sweep; finite-N fluctuations of r are of order 1/√N, and near Kc relaxation is slow (critical slowing down), so the measured points just above Kc sit a little below theory. Real fireflies (e.g. Pteroptyx malaccae) are pulse-coupled; the 3D page also offers the Mirollo–Strogatz pulse model.