This explorable illustrates pattern formation, interesting and beautiful properties of oscillators that are spatially arranged on a lattice and interact with their neighbors. Oscillators and their interaction are described by the famous Kuramoto Model for phase coupled oscillators. The model is amazing, because on one hand it is conceptually quite simple, on the other it holds a number of unexpected dynamical secrets that you can discover here.

Press Play and keep on reading….

This is how it works:

Each pixel in the display panel 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, the state of each oscillator can be represented by a color in a cyclic color spectrum.

Initially, the phases advance with a constant angular speed $\omega_ n = \omega$, identical for each oscillator. Also, 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 described by the differential equation

[ \dot\theta_ n=\omega_ n ]

When you press play, the simulation will initially start without interactions and you can observe that each pixel changes its state continuously through the color range. Relative to one another, the oscillators do 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 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 intuitive way of modeling this is the Kuramoto model defined by

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

The parameter $K>0$ is the coupling constant that quantifies how strongly neighboring oscillators try to align their phase.

You can now increase the coupling strength using the slider and observe the resulting pattern. You should see that quickly oscillators align their phase with their neighborhood and a smooth, dynamic pattern emerges.

Yet,….

…. 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 because they are surrounded by all other phases around them. In the overall pattern these are phase singularities.

Initially there are many pinwheels. But, as time goes on, you will see that their number gradually declines.

What is happening?

As the pattern equilibrates to a dynamic, globally oscillatory, wavy state, very slowly the remaining pinwheels start moving around. You can trace them by toggling the “Show pinwheels” option. You can see that pinwheels of opposite rotational direction attract each other, and, once they collide, they annihilate. You have to be patient to observe these events.

Eventually, only one or zero phase singularities will survive.

Oscillator Heterogeneity

All of this depends on the fact that all oscillators have the same natural frequency. You can change this situation by increasing the oscillator heterogeneity setting (slider). When you do this each oscillator will have a slightly different natural frequency from the rest.

Pick a coupling strength that isn’t to high but sufficient to generate synchrony and a smooth, dynamic pattern. Then increase the heterogeneity. You should witness that at some point, synchrony will break down.