## Harmony on a Donut

“Harmony” was my sloppy look at the Kuramoto model. Looking at 2 oscillators:

The following starts “knotted” 2:3 with , i.e. no coupling and then increasing the coupling…another visualization of harmony.

Visualizing as motion on a circle:

## Harmony

This post is motivated by the Joy Of X and a video lecture given by Professor Strogatz in which he introduces the Kuramoto model for synchronization of random harmonic oscillators. This continues a theme of contemplations or pointing to resources such as the wonderful story on a Mobius strip.

I made the following animated gif hastily in *Mathematica* and, perhaps, if time permits I will contemplate a better CDF (including rotating frame to better illustrate the frequency locking as well as the “rogue” oscillators”). Ten harmonic oscillators with frequencies derived from a normal distribution (1,1)[ scale/units not represented in gif). Varying coupling strength is illustrated.

There a number of poor aspects to the animation. Unfortunately, to make the file size smaller I compromise the number of cycles rather than image size. Consequently, the jerky effect (given the frequency locking develops rather than a constant angular frequency). This lack of cycles prevents appreciation of the development of frequency locking within a given coupling strength.

Oh well, a public trial and error iteration…