Kuramoto Sync

Explore collective phase coherence

Choose N, coupling K and frequency half-width γ. Run starts the seeded ensemble; Pause, One step and PNG inspect it. Changing K or γ preserves current phases and time; γ rescales the same frequency sample. Changing N, seed or Δt starts a new preview. Seed 0 is a valid distinct ensemble. The number field allows finer K values than the slider.

Read the views

θ̇ᵢ=ωᵢ+(K/N)Σⱼsin(θⱼ−θᵢ), with fixed ωᵢ sampled uniformly from[−γ,γ]. The order parameter Z=(1/N)Σexp(iθᵢ)=r exp(iψ). r is phase coherence, between 0 and 1; high r at an instant is not proof of frequency locking. The mean vector has its actual length r. Positive phase motion is counterclockwise. The history uses actual model times; the histogram counts oscillators in 24 equal angular bins.

Compare coupling values

The scan starts every K from exactly the same N, seed and frequencies. It discards the selected warm-up, then averages r over the selected finite window using time-weighted trapezoidal samples. Min/max bars show variation during that window, not statistical confidence or guaranteed convergence. All requested oscillators are included. Progress reflects actual integration work; Cancel, parameter changes or leaving the scan stop it. Values and CSV expose the computed numbers and reproduction parameters.

Kc=4γ/π is a continuum onset reference for the uniform frequency family, not a classifier for this finite realization. The stationary formula sqrt(1−Kc/K) of a Lorentzian family is not used here. Try K=0 to observe uncoupled motion, then larger coupling; change seed or N to compare finite-population variation.

Numerics and scope

Two simultaneous O(N) Heun stages, chosen Δt and fixed-step playback. Refine Δt and lengthen the scan window to assess numerical/transient effects. Frequencies and time are model units, not measured seconds. Hidden tools pause playback; under load it can slow. The last 1200 history samples are retained. PNG pauses and captures current phases, available history and phase counts; scan PNG includes means/ranges. CSV decimal points and column tokens are locale-neutral. This is a deterministic all-to-all phase model, not measured oscillator data or a general network solver.

Strogatz (2000) · Dörfler & Bullo

Keep and compare experiments

Reloading or returning restores paused phases, actual frequencies, recent coherence history, raw inputs, panels and the last completed scan. Invalid live inputs pause playback and keep the last valid ensemble. Changed scan settings retain the previous scan, labelled Previous scan; CSV and PNG still describe that scan’s original settings. A new scan runs only when requested and can be cancelled; an interrupted scan restores the previous completed result. Imports are bounded and checked, and a failed import keeps the current experiment. Live history retains at most 1200 samples; unfinished scans are not resumed.

Only input values are remembered in this browser. Calculated results and experiment histories are not saved.