Phase oscillators

Patterns and Simulations

Print
This is a snapshot in time of the relative phases in an array of 400x400 nearest neighbor phase oscillators in a regime where synchrony is asymptotically stable. Initial conditions are a "spiral wave" which accounts for the symmetry of the pattern. The equations are of the form:
$$\frac{d x_j}{dt} = 1 + \sum_k H(x_k-x_j)$$
where the sum is over the 4 nearest neighbors. Boundary conditions are "natural" - the oscillators at the corneres or edges couple to 2 or 3 others.
$$H(u) = a_1 \sin(u) + a_2 \sin(2u) + a_3 \sin(3u) + b_1 (1-\cos(u))+b_2(1-\cos(2u))+b_3 (1-\cos(3u))$$
The plot shows $$y_j = x_j -x_0$$ (subtracting upper left phase)