Math Colloquium: Bifurcations and Patterns in the Spatially Extended Kuramoto Model
Wednesday, November 4, 2020
3:00 PM-4:00 PM
Georgi Medvedev, PhD, Drexel University
Abstract: The Kuramoto model (KM) stands for a collection of phase oscillators rotating with random frequencies and interacting with each other through nonlinear coupling. In a spatially extended model, one also includes a graph describing the connectivity of the network. In the thermodynamic limit, the KM is approximated by the Vlasov equation, a hyperbolic PDE describing the evolution of the probability distribution of the oscillators in the phase space.
We will review the linear stability analysis of mixing, a steady state solution of the Vlasov equation and will relate the bifurcations of mixing to spatiotemporal patterns observed in the KM right after mixing loses stability. These include stationary and travelling clusters, twisted states, chimera states, and their combinations. In contrast to reaction-diffusion systems, where patterns are expressed by smooth functions, they are described by tempered distributions for the model at hand.
In the second part of the talk, we will discuss stability of clusters in the KM with inertia. This modification of the KM is used for modeling power grids. This talk is based on the joint work with Hayato Chiba and Matthew Mizuhara.