College of Liberal Arts & Sciences

# Math Biology Seminar

**Abstract: **

In two dimensional spatial domain, oscillatory and excitable media are able to produce spiral waves and other types of rotating behavior. In typical experiments phase gradients are computed from filtered local field potentials, thus, a natural mathematical framework is coupled phase equations. In this talk, I will start with a system of nonlocally coupled phase equations and discuss the existence and stability of rotating waves on annulus. I will show that as the inner radius shrinks, rigid rotating waves lose existence through a saddle-node bifurcation and this results in the birth of so-called chimeras. I will also introduce a system of locally coupled phase oscillators on an $N\times N$ grid and show that when the coupling includes non-odd components, rotating waves emerge. I will show that as $N\rightarrow \infty$ that the dynamics can be understood by a Bessel equation on an annulus with inner radius proportional to $1/N$.