Nonlinear Network Dynamics with Consensus–Dissensus Bifurcation

Abstract

We study a nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form. The system has several interesting interpretations: as an interconnection of several pitchfork systems, a gradient dynamical system and the dominating behaviour of a general class of nonlinear dynamical systems. The equilibrium behaviour of the system exhibits a global bifurcation with respect to the system parameter, with a transition from a single constant stationary state to a large range of possible stationary states. Our main result classifies the stability of (a subset of) these stationary states in terms of the effective resistances of the underlying graph; this classification clearly discerns the influence of the specific topology in which the local pitchfork systems are interconnected. We further describe exact solutions for graphs with external equitable partitions and characterize the basins of attraction on tree graphs. Our technical analysis is supplemented by a study of the system on a number of prototypical networks: tree graphs, complete graphs and barbell graphs. We describe a number of qualitative properties of the dynamics on these networks, with promising modelling consequences.