experiment 015 · complex systems · synchronisation

Kuramoto Oscillators

N phase oscillators with random natural frequencies, each nudged toward the instantaneous mean field. Below a critical coupling K_c the population is incoherent — oscillators drift independently and the order parameter hovers near zero. Above K_c, a macroscopic cluster locks into phase, the order parameter jumps to a finite value, and the system undergoes a continuous phase transition whose mean-field theory Kuramoto solved exactly in 1975. The same mathematics governs fireflies, cardiac pacemaker cells, Josephson junction arrays, and power-grid frequency stability.

order parameter r(t) — last 400 steps

The model

Each oscillator i has a phase θᵢ and a natural frequency ωᵢ drawn from a Lorentzian distribution g(ω) = $\gamma$ / [π(ω² + γ²)]. In the absence of coupling each oscillator simply rotates at its own rate. The coupling term introduces a restoring force toward the mean field:

$$ \frac{d\theta_i}{dt} = \omega_i + \frac{K}{N} \sum_j \sin(\theta_j - \theta_i) $$

The sum can be written in terms of the complex order parameter z = r e^iψ = (1/N) Σⱼ e^iθⱼ, where r ∈ [0,1] measures global coherence and ψ is the mean phase:

$$ \frac{d\theta_i}{dt} = \omega_i + K r \sin(\psi - \theta_i) $$

Each oscillator sees only the mean field — an exact mean-field theory that becomes asymptotically correct as N → ∞.

The phase transition

Kuramoto's self-consistency analysis (1975) shows that for the Lorentzian distribution, the incoherent state (r = 0) becomes unstable when

$$ K_c = 2\gamma $$

Above K_c, the order parameter grows continuously from zero — a second-order phase transition. Oscillators whose natural frequencies fall within a band of width ~K around the mean lock into the rotating cluster; those outside remain drifting but are biased by the cluster's field, producing partial entrainment. The steady-state order parameter satisfies r = 1 − K_c/K near the transition.

What to observe

Start Incoherent: dots orbit the circle independently, the cyan phasor tip wanders near the origin, and r ≈ 0. Drag K up past K_c = 2γ and watch the transition: a tight cluster crystallises, the phasor swings out, and r rises toward 1. Hit Locked to see near-total synchrony at K = 3. Colours encode natural frequency — warm reds are fast oscillators, cool violets are slow. Near-frequency oscillators lock first; extreme tails are last to join.

Increase $\gamma$ to widen the frequency distribution and push K_c higher. Reduce N to see finite-size fluctuations blur the transition.

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