P. Gandhi, M. Golubitsky, C. Postlethwaite, I. Stewart and Y. Wang
Bifurcations on Fully Inhomogeneous Networks
reprint.
Submitted.
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 one and two local bifurcations.
For codimension-1 only one critical component is involved,
and generic local bifurcations are saddle-node and standard Hopf. For
codimension two, 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.