The lifting bifurcation problem on feed-forward networks

Abstract

We consider feed-forward networks, that is, networks where cells can be divided into layers, such that every edge targeting a layer, excluding the first one, starts in the prior layer. A feed-forward system is a dynamical system that respects the structure of a feed-forward network. The synchrony subspaces for a network, are the subspaces defined by equalities of some cells coordinates, that are flow-invariant by all the network systems. The restriction of each network system to each synchrony subspace is a system associated with a smaller network, which may be, or not, a feed-forward network. The original network is then said to be a lift of the smaller network. We show that a feed-forward lift of a feed-forward network is given by the composition of two types of lifts: lifts that create new layers and lifts inside a layer. Furthermore, we address the lifting bifurcation problem on feed-forward systems. More precisely, the comparison of the possible codimension-one local steady-state bifurcations of a feed-forward system and those of the corresponding lifts is considered. We show that for most of the feed-forward lifts, the increase of the center subspace is a sufficient condition for the existence of additional bifurcating branches of solutions, which are not lifted from the restricted system. However, when the bifurcation condition is associated with the internal dynamics and the lifts occurs inside an intermediate layer, we prove that the existence of bifurcating branches of solutions that are not lifted from the restricted system does depend generically on the particular feed-forward system. branch, Definition 5.1. Exploiting the techniques presented in [11], we give a characterization of the steady-state bifurcation branches associated with the internal dynamics on a feed-forward network in terms of their square-root-orders and slopes. See Proposition 5.3.Last, we study the lifting bifurcation problem on feed-forward networks for the two basic types of lifts on feed-forward networks. The restriction of a lift system to a synchrony subspace is a feed-forward system. Thus any bifurcation branch occurring for a feed-forward system corresponds to a bifurcation branch occurring for the lift system. The lifting bifurcation problem asks if there are more bifurcation branches occurring for the lift system. This problem was first raised in [2] where the authors proved that there are networks which have more bifurcations branches on some lift systems than the ones lifted from the original network. This problem was also studied in [9,10]. A well-know result gives a necessary condition for the lifting bifurcation problem: There can be more bifurcation branches on a lifted network than the ones lifted only if the center subspace of the coupled cell systems associated to the original network and the lift network have different dimensions. See Corollary 6.1.Frequently, this condition is also sufficient for the lifting bifurcation problem and we prove it in the following cases. For lifts that create new layers and inside a...

Amplified steady state bifurcations in feedforward networks

“…Remark 3.8. The branching solutions in Theorem 3.7 are the same as the ones described in Proposition 5.1 in [29] for layered feedforward networks (compare to Remark 2. 19) and in Proposition 5.7 of [2] investigating feedforward structure of transitive components.…”

“…Remark 3.29. The branching solutions in Theorems 3.25, 3.26 and 3.28 contain those that are described in Section 6 in [29] for layered feedforward networks as a special case. Therein, generically all non-maximal cells are critical if the maximal cells are non-critical.…”

“…Each arrow color corresponds to one input map σ : C → C. Note that we have not drawn an arrow for σ = Id corresponding to the internal dynamics which we implicitly assume to be there. This network is clearly a feedforward network as it does not contain any loops besides self-loops (note that it is not a layered feedforward network as in [29]). Its only maximal cell is cell 5.…”

Classification of Networks with Asymmetric Inputs

“…This class of networks have been applied in different fields and theoretical studies of these kind of networks have been addressed. See for example [5,28,6] and references therein to specific applications. Feed-forward systems can exibit dynamical features that are not common in systems without feed-forward structure.…”

“…One example is the occurrence of generic Hopf bifurcation in one-parameter families of coupled cell systems, from an equilibrium to periodic solutions and where there is growth of the amplitude of cells (as a function of the bifurcation parameter) faster than would be expected in systems that do not have the feed-forward structure [24]. See also [20,28] where a similar phenomenon is proved in the steady-state bifurcation case for feed-forward systems and for more recent work [27, and [23].…”