Nonlinear Network Dynamics with Consensus–Dissensus Bifurcation

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