# Math Insight

### Applet: The Kuramoto order parameters

Given a collection of phase oscillators, the Kuramoto order parameters $r$ and $\psi$ capture the degree of their synchrony and the average phase, respectively. The phases $\theta_j$ of $N$ oscillators are shown as green points around the unit circle in the complex plane. You can change $N$ by dragging the red point on the slider and change the phases by dragging the green points. The location of the point corresponding to $\theta_j$ is $e^{i\theta_j}$ where $i=\sqrt{-1}$, so $\theta_j$ measures the counterclockwise angle from the positive real axis (i.e., what you'd think of as the positive $x$-axis). The vector $z$ is the average of the $N$ points viewed as vectors in the complex plane: $$z = \frac{1}{N}\sum_{j=1}^N e^{i\theta_j}.$$ The order parameter $r$ is the magnitude of $z$, $r=|z|$. It achieves its maximum $r=1$ when all the phases are identical. It achieves its minimum $r=0$ when the phases are balanced around the circle, such as evenly spread or in clusters that balance each other out. The parameter $r$ is a synchrony measure of the population of oscillators often called the phase coherence. The average phase $\psi$ is the argument of $z$, i.e., counterclockwise angle between $z$ and the positive real axis. One can write $z=re^{i\psi}$.

Applet file: kuramoto_order_parameters.ggb