M.A.D. Aguiar, A.P.S. Dias, M. Golubitsky and M.C.A. Leite
Bifurcations from regular quotient networks: A first insight
Physica D.
238 (2) (2009) 137-155.
We consider regular (identical-edge identical-node) networks whose
cells can be grouped into classes by an equivalence relation. The
identification of cells in the same class determines a new network -
the quotient network. In terms of the dynamics this corresponds to
restricting the coupled cell systems associated with a network to
flow-invariant subspaces given by equality of certain cell
coordinates. Assuming a bifurcation occurs for a coupled cell system
restricted to the quotient network, we ask how that bifurcation lifts
to the overall space. Surprisingly, for
certain networks, new branches of solutions occur besides the ones
that occur in the quotient network. To investigate this phenomenon
we develop a systematic method that enumerates all networks with
a given quotient. We also prove necessary conditions for the
existence of solutions branches not predicted by the quotient.
We then apply our method to two particular
quotient networks; namely, two- and three-cell bidirectional rings.
We show there are no additional bifurcating solution branches when
the quotient network is a two-cell bidirectional ring. However,
two of the 12 five-cell networks that have the three-cell
bidirectional ring as a quotient network exhibit bifurcating
solutions that do not occur in the quotient itself. Thus, network
architecture sometimes forces the existence of bifurcating
branches in addition to the ones determined by the quotient.