The Kuramoto Model
Module 2: Synchronization and Collective Behaviour
The Kuramoto model describes a large population of coupled oscillators with different natural frequencies. Each oscillator \(i\) has a phase \(\theta_i\) and a natural frequency \(\omega_i\) (kuramoto1975self?; Kuramoto 2003; Strogatz 2000). It is the standard example of how collective synchronization emerges from simple interactions among oscillators that are not identical. Physicists, biologists and engineers use it to study circadian rhythms, synchronized neurons and the stability of power grids (Strogatz 2000).
From limit cycles to phases
For any system of weakly coupled, nearly identical limit-cycle oscillators, the long-term dynamics reduce to phase equations of the form
\[ \dot\theta_i = \omega_i + \sum_{j=1}^N \Gamma_{ij} \left( \theta_j - \theta_i \right),\quad i = 1, \dots, N, \tag{1}\]
where \(\Gamma_{ij}\) is the coupling function that describes how oscillator \(j\) influences oscillator \(i\) through their phase difference (Kuramoto 2003; Strogatz 2000). This reduction keeps the timing of each oscillator and discards its amplitude. (winfree1967biological?) first proposed studying populations in this way.
Kuramoto’s choice
Kuramoto took the simplest case: equally weighted, all-to-all, purely sinusoidal coupling,
\[ \Gamma_{ij}(\theta) = \frac{K}{N} \sin \theta. \tag{2}\]
Substituting into Equation 1 gives the Kuramoto model:
\[ \dot\theta_i = \omega_i + \frac{K}{N}\sum_{j=1}^N \sin(\theta_j - \theta_i), \quad i = 1, \dots, N. \tag{3}\]
- \(K \ge 0\) is the coupling strength.
- The factor \(1/N\) keeps the total input to each oscillator finite as \(N\) grows.
- The natural frequencies \(\omega_i\) are drawn from a density \(g(\omega)\) that is unimodal and symmetric about its mean \(\Omega\) (Strogatz 2000).
Two oscillators with \(\theta_j\) slightly ahead of \(\theta_i\) give \(\sin(\theta_j - \theta_i) > 0\), so \(i\) speeds up toward \(j\). The coupling always pulls phases together. What competes against it is the spread of natural frequencies.
Rotating frame
Move to a frame that rotates at the mean frequency, \(\theta_i \to \theta_i + \Omega t\). The coupling depends only on phase differences, so Equation 3 keeps its form and the new frequencies have mean zero. From now on you can assume \(g(\omega) = g(-\omega)\) (Strogatz 2000).
Frequency densities used in this module
| Density | \(g(\omega)\) | \(g(0)\) |
|---|---|---|
| Normal, standard deviation \(\sigma\) | \(\dfrac{1}{\sqrt{2\pi}\,\sigma}\, e^{-\omega^2/(2\sigma^2)}\) | \(\dfrac{1}{\sqrt{2\pi}\,\sigma}\) |
| Lorentzian (Cauchy), scale \(\gamma\) | \(\dfrac{\gamma}{\pi(\gamma^2 + \omega^2)}\) | \(\dfrac{1}{\pi\gamma}\) |
The Lorentzian case gives closed-form results (Critical Coupling). The normal case is the default in the lab. The initial phases are drawn uniformly on \([0, 2\pi)\) unless stated otherwise.
Show that Equation 3 with \(N = 2\) reduces to the firefly equation of Phase Locking for the phase difference \(\phi = \theta_2 - \theta_1\). What is the locking condition?
With \(N = 2\), \(\dot\theta_1 = \omega_1 + \tfrac{K}{2}\sin(\theta_2 - \theta_1)\) and \(\dot\theta_2 = \omega_2 + \tfrac{K}{2}\sin(\theta_1 - \theta_2)\). Subtracting,
\[ \dot\phi = (\omega_2 - \omega_1) - K\sin\phi. \]
This is the phase-difference equation of Phase Locking with detuning \(\omega_2 - \omega_1\) and strength \(K\). The two oscillators lock when \(|\omega_2 - \omega_1| \le K\).