Bifurcations from regular quotient networks: A first insight

Abstract

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 * CMUP is supported by FCT through the programmes POCTI and POSI, with Portuguese and European Community structural funds. 1 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.

Amplified Hopf Bifurcations in Feed-Forward Networks

“…Synchronous solutions may moreover undergo bifurcations with quite unusual features. Such synchrony breaking bifurcations have for example been studied in [1], [3], [6], [7], [11] and [22].…”

On bifurcations in lifts of regular uniform coupled cell networks

“…To be precise, in [6], a one-parameter steady-state or Hopf bifurcation problem occurring in a coupled cell system consistent with the structure of a given regular network is considered; also, considering a lift of this network, it asks how does the bifurcation lift to the overall space? The study suggests some other important questions, such as: is it always possible to find, for a given regular network, lifts that just exhibit the bifurcating branches determined by the quotient?…”

“…Bifurcation theory has been applied to the theory of coupled cell networks in various ways (e.g. [6][7][8][9]). …”

“…This issue was raised by Aguiar et al [6,10], and examples are given where, assuming a bifurcation occurs for a coupled cell system restricted to a fixed (quotient) network, new branches of solutions occur in some of the lifts. More precisely: assume that Q is a quotient of G determined by a synchrony subspace and impose a degeneracy condition on F, implying that the centre subspace of J| is non-trivial.…”

“…Hence, if a lift of a regular network is spectrum-preserving, then generically no new branches of solutions can occur for the lift equations, assuming that a synchrony-breaking bifurcation occurs for the quotient equations. In [6], the following question is addressed: assuming that a codimension-1 synchrony-breaking bifurcation occurs in a coupled cell system consistent with the structure of a given regular network, and considering a lift of this network, how does the bifurcation lift to the overall space? It is proved in particular that every lift of the 2-ring is spectrum-preserving.…”