Direct lifts of coupled cell networks

• Type: Article in International Scientific Journal
• Date: 2018

Authors

dias, aps (Author) FCUP; moreira, cs (Author) FCUP

Journal

Title: Nonlinearity

Vol. 31

Pages: 1299-1312

ISSN: 0951-7715

Publisher: Institute of Physics Publishing

Indexing

ISI Web of Knowledge - 3 Citations

Scopus - 3 Citations

Other information

Authenticus ID: P-00N-NND

DOI: 10.1088/1361-6544/aa9e5b

Abstract (EN): In networks of dynamical systems, there are spaces defined in terms of equalities of cell coordinates which are flow-invariant under any dynamical system that has a form consistent with the given underlying network structure-the network synchrony subspaces. Given a network and one of its synchrony subspaces, any system with a form consistent with the network, restricted to the synchrony subspace, defines a new system which is consistent with a smaller network, called the quotient network of the original network by the synchrony subspace. Moreover, any system associated with the quotient can be interpreted as the restriction to the synchrony subspace of a system associated with the original network. We call the larger network a lift of the smaller network, and a lift can be interpreted as a result of the cellular splitting of the smaller network. In this paper, we address the question of the uniqueness in this lifting process in terms of the networks' topologies. A lift G of a given network Q is said to be direct when there are no intermediate lifts of Q between them. We provide necessary and sufficient conditions for a lift of a general network to be direct. Our results characterize direct lifts using the subnetworks of all splitting cells of Q and of all split cells of G. We show that G is a direct lift of Q if and only if either the split subnetwork is a direct lift or consists of two copies of the splitting subnetwork. These results are then applied to the class of regular uniform networks and to the special classes of ring networks and acyclic networks. We also illustrate that one of the applications of our results is to the lifting bifurcation problem.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 14