Forced Symmetry-Breaking Bifurcations in Networks with Disordered Parameters

Abstract

Emergent behavior in complex networks can be predicted and analyzed via the mechanism of spontaneous symmetry-breaking bifurcation, in which solutions of related bifurcation problems lose symmetry as some parameters are varied, even though the equations that such solutions satisfy retain the full symmetry of the system. A less common mechanism is that of forced symmetry-breaking, in which either a bifurcation problem has symmetry on both the state variables and the parameters, or one where the equations have less symmetry when a certain parameter is varied. In this manuscript, it is shown that in certain networks with parameter mismatches the governing equations remain unchanged when the group of symmetries acts on both the state variables and the parameter space. Based on this observation we study the existence and stability of collective patterns in symmetric networks with parameters mismatches from the point of view of forced symmetry-breaking bifurcations. Treating the parameters as state variables, we perform center manifold reductions, which allow us to understand how the disorder in parameters affects the bifurcation points as well as the stability properties of the ensuing patterns. Theoretical results are validated with numerical simulations.