EXPLORABLES

This explorable illustrates some interesting and beautiful properties of oscillators that are spatially arranged on a lattice and interact with their neighbors.

#### This is how it works:

Each pixel in the panel on the right represents an oscillator. The state of each oscillator $n$ is defined by its phase $\theta_n(t)$, the angle of a circle. Because of this, we represent the state of each oscillator by a color in the cyclic rainbow spectrum.

We assume that the phases advance with a constant angular speed $\omega$, identical for each oscillator. Initially, each oscillator's state is randomly chosen in the interval $\theta_n(0)\in[0, 2\pi]$, which is why the initial state of the system looks like a random arrangement of colored pixels.

When the oscillators are not coupled, the dynamics of each oscillator is desribed by the differential equation

[
\dot\theta_n=\omega
]

When you click on the lattice, the simulation will initially start without interactions and you can observe that each pixel changes its state continuously through the rainbow colors (another click on the panel stops the simulation). Relative to one another, the oscillators to not change phase, e.g. the differences $\theta_n-\theta_m$ remain constant.

#### Interactions:

We now define a "force" by which oscillators interact with their 8 nearest neighbors on the lattice (the set of neighbors is denoted by $U_n$). We say that neighboring oscillators try to align their phase, the leading oscillator slowing down a bit, and the lagging oscillator speeding up. The most intutitive way of modelling this is the Kuramoto model defined by

[
\dot\theta _n=\omega + K\,\sum _{m\in U _n} \sin (\theta _m-\theta _n)
]

The parameter $K$ is the coupling constant that quantifies how strongly neighboring oscilators try to synchronize.

You can now increase the coupling constant and observe the resulting pattern. You should see that quickly oscillators align their phase with their neighbors and a smooth pattern emerges. However, there are some points where oscillators are surrounded by the complete spectrum of phases. These locations are known as pinwheels. At their center, oscillators are having trouble aligning their phase, in the overall pattern these are phase singularities.

Initially there are many pinwheels. If there are less than 50, you can make them more visible by changing the Pinwheel Opacity slider (which appears after a time). When you decrease the natural frequency $\omega$ (even to zero) you can observe the fate of the singularities. Singularities of opposing "sign attract and annihilate eventually. You have to be patient to observe these events.