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Pattern Formation in the Kuramoto Model

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

SNAPSHOTS

DETAILS

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 [1] on which this demonstration is based. For general information on the Kuramoto model, see the Wikipedia article of the same name.

[1] 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.