Non-linear network dynamics with consensus-dissensus bifurcation

We study a non-linear 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 non-linear 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 modeling consequences.