What is the Kuramoto model?

Introduced by Yoshiki Kuramoto in 1975, it describes N oscillators each with a natural frequency ωᵢ drawn from a distribution g(ω), connected by a mean-field coupling of strength K. The equation dθᵢ/dt = ωᵢ + (K/N)·Σⱼ sin(θⱼ−θᵢ) captures how phase differences drive oscillators toward a common rhythm.

How does the order parameter r measure synchrony?

The complex order parameter r·e^{iψ} = (1/N)·Σⱼ e^{iθⱼ} treats each oscillator as a unit vector on the circle. Its magnitude r ranges from 0 (incoherent: phases uniformly spread) to 1 (fully synchronised: all phases equal). The critical coupling Kc = 2/[π·g(0)] for a Lorentzian distribution.

Why is synchronisation a phase transition?

Below Kc the incoherent state r=0 is stable. Above Kc a bifurcation occurs and r grows as r ≈ √(8(K−Kc)/Kc·π·g(0)). This continuous (second-order) transition has mean-field critical exponent β=1/2, analagous to a ferromagnetic transition with K playing the role of inverse temperature.

What real phenomena does the Kuramoto model describe?

The model describes firefly flash synchrony, cardiac pacemaker cells, circadian rhythms in the suprachiasmatic nucleus, neural oscillations and gamma-band synchrony, power-grid frequency locking, Josephson junction arrays, and chemical oscillators (Belousov-Zhabotinsky reaction).

Home▸Chaos & Dynamics▸Kuramoto Coupled Oscillators: Synchronisation Emergence

🧪 Kuramoto Coupled Oscillators: Synchronisation Emergence

Simulate the Kuramoto model: N coupled oscillators synchronise spontaneously above a critical coupling K. Watch the order parameter r rise from chaos to coherence. Interactive with 6 presets.

Chaos & Dynamics2DModerate60 FPS

⚙ Under the hood

This simulation demonstrates the emergence of synchronized behavior in a system of coupled oscillators modeled by the Kuramoto model. It illustrates how interactions between oscillating systems can lead to collective patterns.

KuramotoCoupled OscillatorsSynchronization

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install