Watch oscillators synchronise: the Kuramoto model
Run the mean-field Kuramoto model in the browser, watch the order parameter climb past the critical coupling, and read the evidence bundle the run produces.
The model
N oscillators each run at their own natural frequency ω_i (drawn from a standard normal distribution) and pull on each other through their phase differences: dθ_i/dt = ω_i + (K/N)·Σ_j sin(θ_j − θ_i). The runner integrates the algebraically identical mean-field form dθ_i/dt = ω_i + K·r·sin(ψ − θ_i), where r·e^{iψ} is the population's complex mean phase, using explicit Euler steps of dt = 0.05.
The order parameter
The single number to watch is r = |(1/N)·Σ_j e^{iθ_j}|, between 0 (incoherent phases) and 1 (perfect synchrony). Kuramoto's classical result: below a critical coupling K_c the population stays incoherent; above it, a synchronised cluster forms and r settles at a finite value. For standard-normal frequencies the infinite-N theory gives K_c = 2/(π·g(0)) = 2·√(2/π) ≈ 1.596 — at finite N you will see the transition smeared and r fluctuating, which is itself part of the lesson.
The claim boundary
This is a bounded educational model: explicit Euler (not an adaptive integrator), a fixed-seed PRNG so every run with the same parameters replays identically, and finite N far from the thermodynamic limit. The evidence bundle the run emits says exactly that — a numerical-model claim admitted only inside its stated ranges, never a measurement.
Run the simulation
Start below the critical coupling (K ≈ 1) and watch r stay low and noisy. Then push K past ≈ 1.6 and run again: the trajectory sparkline shows the order parameter climbing and settling. More oscillators make the transition cleaner; more steps give the settled value more time to stabilise.
Inspect the evidence
The bundle records the settled and final order parameter with the exact parameters that produced them, and admits the claim only as a bounded model. The same phase-dynamics discipline runs at production scale in the SCPN Phase Orchestrator — a domain-agnostic coherence control compiler built on Kuramoto/UPDE dynamics (see /studio/scpn-phase-orchestrator/ and its publication with a resolvable DOI on /publications/).
Part of these topics.
The research topics this lesson sits inside — the science, the studios, and the literature.