# Math Insight

### Applet: Synchronizing oscillators

Illustration of $N=100$ phase oscillators whose state can be represented by a point around the unit circle. The state of oscillator $j$ at time $t$ is represented by the state variable $\theta_j(t)$, or phase, which we can map to the unit circle by plotting a green point in the complex plane at the location $e^{i\theta_j(t)}$ where $i=\sqrt{-1}$. For each oscillator, $\theta_j(t)$ increases with time, so each point moves counterclockwise around the circle. At time $t=0$, the oscillators are randomly spread over state space. However, as time evolves, the oscillators tend to group together to have similar phase, i.e., they tend to synchronize. You can control how well they synchronize by changing the parameter $\sigma$, which controls the final spread of points. When $\sigma=0$, the oscillators would eventually completely synchronize so that all $\theta_j(t)$ become identical.

The degree of synchrony is captured by the order parameter $$z=\frac{1}{N}\sum_{i=1}^N e^{i\theta_j(t)},$$ plotted as a blue vector and by its magnitude $r=|z|$. When $r=1$, the oscillators are completely synchronized, and $r=0$ indicates that the oscillators are asynchronous, or at least balanced around the circle.

Applet file: synchronizing_oscillators.ggb