Bistable reaction-diffusion on a network

TL;DR

This paper analyzes bistable reaction-diffusion dynamics on networks, proving the existence of stable states, bifurcations, and depinning phenomena through analytical and numerical methods.

Contribution

It provides a comprehensive analytical and numerical study of bistable reaction-diffusion equations on arbitrary networks, including bifurcation analysis and depinning criteria.

Findings

Stable fixed points exist for small diffusion.

Fold bifurcations lead to depinning transitions.

Continuation methods confirm theoretical results.

Abstract

We study analytically and numerically a bistable reaction-diffusion equation on an arbitrary finite network. We prove that stable fixed points (multi-fronts) exist for any configuration as long as the diffusion is small. We also study fold bifurcations leading to depinning and give a simple depinning criterion. These results are confirmed by using continuation techniques from bifurcation theory and by solving the time dependent problem near the treshold. A qualitative comparison principle is proved and verified for time dependent solutions, and for some related models.