Kuramoto oscillators and the synchronization transition

What you are seeing: 128 phase oscillators with intrinsic frequencies $\omega_i$ drawn from a Lorentzian distribution. Each oscillator advances at its own pace but is pulled by the global mean direction with strength $K$: $\dot{\theta}_i = \omega_i + (K/N) \sum_j \sin(\theta_j - \theta_i)$. The order parameter $r = |\frac{1}{N} \sum_j e^{i\theta_j}|$ measures how much the oscillators agree on a common direction.

Below the critical coupling $K_c = 2 \gamma$ the oscillators are incoherent, scattered around the unit circle, and $r \approx 0$. Above $K_c$ a fraction of oscillators lock onto a common phase $\psi$ and a coherent peak forms in the angular distribution; the laggard oscillators (those with extreme $\omega_i$) still drift but a macroscopic order emerges, with $r$ rising as $\sqrt{K - K_c}$ above threshold.