Coupled Networks, Patterns and Complexity - Abstract

Kouvaris, Nikos

Studying pattern formation in networks of diffusively coupled bistable elements

In this talk we focus on traveling and stationary patterns in bistable systems organized on random networks and on hierarchical trees. Numerical simulations reveal that traveling fronts exist in such network-organized systems. They represent waves of transition from the one stable state into the other, spreading over the entire network. The fronts can furthermore be pinned, thus forming stationary patterns. While pinning of fronts has been already considered for chains of diffusively coupled bistable elements, the network architecture brings about significant differences. Particularly, an important role is played by the number of links (i.e. the degree) of a node. For regular trees with a fixed branching factor, the pinning conditions can be analytically determined. It can also be shown that the transition from traveling to standing fronts corresponds to a saddle-node bifurcation. For large Erdös-Rényi and scale-free networks, stationary patterns are approximately described by a mean-field theory.