Bifurcations on Fully Inhomogeneous Networks

Abstract

Center manifold reduction is a standard technique in bifurcation theory, reducing the essential features of local bifurcations to equations in a small number of variables corresponding to critical eigenvalues. This method can be applied to admissible differential equations for a network, but it bears no obvious relation to the network structure. A fully inhomogeneous network is one in which all nodes and couplings can be different. For this class of networks, there are general circumstances in which the center manifold reduced equations inherit a network structure of their own. This structure arises by decomposing the network into path components, which connect to each other in a feedforward manner. Critical eigenvalues can then be associated with specific components, and the network structure on the center manifold depends on how these critical components connect within the network. This observation is used to analyze codimension-1 and codimension-2 local bifurcations. For codimension-1, only one critical component is involved, and generic local bifurcations are saddle-node and standard Hopf. For codimension-2, we focus on the case when one component is downstream from the other in the feedforward structure. This gives rise to four cases: steady or Hopf upstream combined with steady or Hopf downstream. Here the generic bifurcations, within the realm of network-admissible equations, differ significantly from generic codimension-2 bifurcations in a general dynamical system. In each case, we derive singularity-theoretic normal forms and unfoldings, present bifurcation diagrams, and tabulate the bifurcating states and their stabilities.