Rotating waves: from local discrete systems to nonlocal continuous media

Bard Ermentrout
University of Pittsburgh
In two or more spatial dimensions, oscillatory and excitable media are able to produce spiral and other types of rotating behavior. We start with a system of locally coupled phase oscillators on an NxN grid and show that when the coupling includes non-odd components, spiral waves emerge. We show that as $N\to\infty$, that the dynamics can be understood by a Burgers type equation on an annulus with inner radius proportional to 1/N. We then turn to nonlocal coupling on an annulus and show that rotating waves solve a certain one-dimensional integral equation. We investigate the stability of the waves and connect an instability as the inner radius of the annulus shrinks to the formation of so-called chimeras.