STUDY #07 · 2026 · IN OBSERVATION

Kuramoto Model

A model-driven visual study of spontaneous synchronization.

MOVING IMAGE — PHASE DISC K breathing across Kc · r 0 → 0.85

WHAT IS THIS

Synchronization is what a crowd of oscillators does when each one, ticking at its own natural rate, nudges the others just a little. Fireflies flashing in unison, pacemaker cells, an audience clapping into rhythm — all the same story. Below a critical coupling the phases stay scattered; above it the population locks into a common beat. Nothing tells them to; the order emerges on its own.

The Kuramoto model strips this to its minimum: each oscillator is just a phase on a circle, pulled toward the average of all the others. This study runs a mean-field population of these oscillators in real time on the GPU, sweeping the coupling across the synchronization transition.

the coherent arm — a locked cluster out of the drifting halo K 3.0 · lorentzian · phase · r 0.58

Motif coupled phase oscillators / order parameter / phase transition

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer — a mean-field population of coupled phase oscillators, shown as a phase circle and as a rotor field of trails. The visual output was selected through curated parameter exploration.

Observation Above a critical coupling the population condenses into a single rotating arm — synchrony reading as a spinning rainbow collapsing to one hue — while the extreme-frequency oscillators never join and drift as a halo at the rim.

Reference Yoshiki Kuramoto, "Self-entrainment of a population of coupled non-linear oscillators," International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, vol.39, 420-422 (1975).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

A NEW CHAPTER

From a field in space to a population in phase.

Studies #01–06 — space Study #07 — population

What is drawn a field over the plane — concentration, height, intensity a population of oscillators, sorted onto a disc by frequency

Colour a concentration, a temperature, a path difference phase itself — the angle each oscillator has reached

The order patterns fixed by the geometry of diffusion synchrony — emerging in time, past a critical coupling

PARAMETERS EXPLORED

param meaning effect on the image

K the coupling strength between oscillators the transition itself: below Kc = 2γ a sheared rainbow (incoherent), above it a coherent arm (synchronised)

γ / width the spread of natural frequencies sets the critical coupling Kc = 2 / (π g(0)) — a wider spread is harder to synchronise

g(ω) the shape of the frequency distribution lorentzian is the classic closed-form case; gaussian locks tighter; bimodal splits the crowd into two groups

colour phase → hue, or ω → temperature phase condenses to one hue at synchrony; ω shows the split between the locked band and the drifting rim

decay / exposure the trail exposure the length and depth of the rotor trails — how engraved the rings read

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

Each oscillator is a single phase on a circle, pulled toward the average of all the others. The whole crowd collapses to one mean field.

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

N oscillators, each with its own natural frequency ω_i, coupled all-to-all at strength K.

r\,e^{\,i\psi} = \frac{1}{N}\sum_{j} e^{\,i\theta_j}

The complex order parameter — the centroid of the crowd on the circle. r ∈ [0,1] is the degree of synchrony; ψ the mean phase.

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

The mean-field form: every oscillator now follows only the average (r, ψ) — the O(N) reduction the GPU runs.

K_c = \frac{2}{\pi\,g(0)}\,,\qquad r = \sqrt{1 - \frac{K_c}{K}}\ \ (K > K_c)

The critical coupling from the frequency distribution g(ω); for a Lorentzian spread the synchrony grows along this closed-form branch.

Inspired by the Kuramoto model of coupled phase oscillators — a visual interpretation, not an exact reproduction.

SELECTED STILLS — 5

the coherent arm — above Kc K 3.0 · lorentzian · phase · aurora · r 0.58

onset — the arm just emerging K 2.5 · near-critical · phase · r 0.44

differential shear — below Kc K 1.0 · incoherent · phase · r 0.005

two groups — a bimodal distribution K 3.0 · bimodal · ω-colour · ember

firefly — the real-world anchor K 3.0 · width 0.8 · ω-colour · firefly

PROCESS — PARAMETER SWEEPS

The whole transition in one frame: as the coupling K climbs past the critical value, the sheared rainbow disc condenses into a single coherent arm and the order parameter r rises from nothing toward one.

A coupling sweep across the transition K 0.6→6 · lorentzian · Kc 2.0 · r 0.01→0.84

COLOUR = PHASE

Here colour is not a material property but the phase itself. A phase is an angle that wraps around, and the colour wheel is the canonical way to draw a cyclic quantity — so mapping phase to hue is an honest encoding, not decoration. Synchrony then reads directly: a spinning rainbow condensing into one hue.

The radius orders oscillators by their natural frequency, so the coherent core and the drifting rim are the phase-locked and drifting populations of the model. The firefly palette nods to the most literal real-world case — synchronous fireflies, whose "phase" is simply when each one flashes.

These are artistic mappings of dynamical quantities (phase, frequency), not measurements — unlike #01, where colour was a real catalyst oxidation state.

firefly — natural frequency as temperature, the synchronous flash as the bright arc K 3.0 · width 0.8 · ω-colour · firefly

Palette aurora / ember / firefly — hue = phase (or temperature = natural frequency) · radius sorts oscillators by frequency.

REFERENCES

Yoshiki Kuramoto, "Self-entrainment of a population of coupled non-linear oscillators," International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, vol.39, 420-422 (1975).
Steven H. Strogatz, "From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators," Physica D, vol.143, 1-20 (2000).