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The Lifting Bifurcation Problem on Feed-Forward Networks

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<p>We consider feed-forward networks, that is, networks where cells can<br /> be divided into layers, such that every edge targeting a layer, exclud-<br /> ing the first one, starts in the prior layer. A feed-forward system<br /> is a dynamical system that respects the structure of a feed-forward<br /> network. The synchrony subspaces for a network, are the subspaces<br /> defined by equalities of some cells coordinates, that are flow-invariant<br /> by all the network systems. The restriction of each network system<br /> to each synchrony subspace is a system associated with a smaller net-<br /> work, which may be, or not, a feed-forward network. The original<br /> network is then said to be a lift of the smaller network. We show that<br /> a feed-forward lift of a feed-forward network is given by the composi-<br /> tion of two types of lifts: lifts that create new layers and lifts inside<br /> a layer. Furthermore, we address the lifting bifurcation problem on<br /> feed-forward systems. More precisely, the comparison of the possi-<br /> ble codimension-one local steady-state bifurcations of a feed-forward<br /> system and those of the corresponding lifts is considered. We show<br /> that for most of the feed-forward lifts, the increase of the center sub-<br /> space is a sufficient condition for the existence of additional bifurcating<br /> branches of solutions, which are not lifted from the restricted system.<br /> However, when the bifurcation condition is associated with the in-<br /> ternal dynamics and the lifts occurs inside an intermediate layer, we<br /> prove that the existence of bifurcating branches of solutions that are<br /> not lifted from the restricted system does depend generically on the<br /> particular feed-forward system.</p>

Pedro Soares

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Year of publication: 2017

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