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Keyword: ‘kuramoto’

Harmony on a Donut

April 29, 2017 Leave a comment

“Harmony” was my sloppy look at the Kuramoto model. Looking at 2 oscillators:
\dot{\theta_1}=\omega_1+ k\sin (\theta_2-\theta_1)\\\\ \dot{\theta_2}=\omega_2+ k\sin (\theta_1-\theta_2)

The following starts “knotted” 2:3 with k=0, i.e. no coupling and then increasing the coupling…another visualization of harmony.

msesynchr.gif

Visualizing as motion on a circle:

 
msekuramoto

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Harmony

November 5, 2012 Leave a comment

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…

Categories: Mathematica, Mathematics