Analysis, Dynamics and Applications Seminar

Turing bifurcations are the primary pattern forming mechanism in spatially extended systems. In this talk we will demonstrate how to understand Turing bifurcations on one-dimensional random ring networks, characterized by the fact that the probability of a connection between two nodes depends on the distance between the two nodes. We will discuss an approach that uses the theory of graphons to approximate the graph Laplacian in the limit as the number of nodes tends to infinity by a nonlocal operator - the graphon Laplacian. For large networks we derive estimates that relate the eigenvalues and eigenvectors of the finite graph Laplacian to those of the graphon Laplacian. We then turn to studying pattern formation in the Swift-Hohenberg equation, which is a phenomenological model for systems undergoing a Turing bifurcation. We prove that with high probability the bifurcations that occur in the finite graph are well approximated by those in the graphon limit, allowing one to understand pattern formation in random landscapes simply through the limiting nonlocal graphon Swift-Hohenberg equation.

Place: Hybrid, Math, 402 and Zoom: https://arizona.zoom.us/j/81150211038  Password: “arizona” (all lower case)