The Turing bifurcation on networks: Collective patterns and differentiated nodes

Abstract:

We study the emergence of patterns in a diffusively coupled network system that undergoes a Turing bifurcation. Our main interest is in large and irregular networks. We distingush between two different types of patterns, which appear for different parameter regimes: Collective patterns, which resemble the shape of the linear network modes, and patterns with a small number of differentiated nodes. We show that large network systems display a huge number of coexisting stable stationary states that can be explained in terms of spatial chaos. Our results are based on a bifurcation analysis for the mean-field approximation and on numerical path-following methods for the full system.