# Math Insight

### Applet: Desynchronizing 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 $t=0$, all oscillators start at exactly the same phase. However, as time evolves, the oscillators tend to spread out across the whole circle of phases, i.e., the system tends toward asynchrony. You can control how they desynchronize by changing the parameter $\sigma$, which controls the spread of oscillator speeds. When $\sigma=0$, the oscillators will stay synchronized forever.

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: desynchronizing_oscillators.ggb