This demonstration shows artistic patterns that may arise in a two-dimensional Kuramoto model of coupled oscillators with variable phase shift. The oscillators are coupled to their nearest neighbors within an adjustable radius on a grid with periodic boundary conditions. The image shown displays the phase of each oscillator after evolving the system starting with random phases.
Contributed by: Oliver K. Ernst
Snapshot 1&2: Two examples of spiral patterns produced at phase shifts .
Snapshot 3: Patterns that resemble plane waves at phase shift .
The Kuramoto model describes a set of oscillators which are coupled sinuosoidally according to their phase differences.
In this demonstration, a 100x100 grid of oscillators is initialized with random phases . Each oscillator is coupled to it's nearest neighbors within a variable radius . The time evolution of the phases is governed by the differential equation
where the phase shift is , and the sum goes over all oscillators at positions for which . The differential equation is solved for sufficient timesteps such that interesting patterns are observed, and the final phases of each oscillator displayed above.
For further details, see published work  on which this demonstration is based. For general information on the Kuramoto model, see the Wikipedia article of the same name.
 P.-J. Kim, et al. "Pattern Formation in a Two-Dimensional Array of Oscillators with Phase-Shifted Coupling," Phys. Rev. E., 70(6): 065201, 2004.