Reducibility of steady-state bifurcations in coupled cell systems

• Consider coupled cell system, coupled ODE system, on regular coupled cell network.
• Also consider codimension-one synchrony-breaking steady-state bifurcation.
• Define the notion “reducibility” for the bifurcation regarding quotient network.
• Give the complete classification of the bifurcation on 1-input regular networks.
• Show that the bifurcation in generic systems with DnDn symmetry are reducible.

A general theory for coupled cell systems was formulated recently by I. Stewart, M. Golubitsky and their collaborators. In their theory, a coupled cell system is a network of interacting dynamical systems whose coupling architecture is expressed by a directed graph called a coupled cell network. An equivalence relation on cells in a regular network (a coupled cell network with identical nodes and identical edges) determines a new network called quotient network by identifying cells in the same equivalence class and determines a quotient system as well. In this paper we develop an idea of reducibility of bifurcations in coupled cell systems associated with regular networks. A bifurcation of equilibria from subspace where states of all cells are equal is called a synchrony-breaking bifurcation. We say that a synchrony-breaking steady-state bifurcation is reducible in a coupled cell system if any bifurcation branch for the system is lifted from those for some quotient system. First, we give the complete classification of codimension-one synchrony-breaking steady-state bifurcations in 1-input regular networks (where each cell receives only one edge). Second, we show that under a mild condition on the multiplicity of critical eigenvalues, codimension-one synchrony-breaking steady-state bifurcations in generic coupled cell systems associated with an n  -cell coupled cell network with DnDn symmetry, a regular network, is reducible for n>2n>2.