Networks with asymmetric inputs: lattice of synchrony subspaces

Abstract

We consider coupled cell networks with asymmetric inputs and study their lattice of synchrony subspaces. For the particular case of 1-input regular coupled cell networks we describe the join-irreducible synchrony subspaces for their lattice of synchrony subspaces, first in terms of the eigenvectors and generalized eigenvectors that generate them, and then by giving a characterization of the possible patterns of the associated balanced colourings. The set of the join-irreducible synchrony subspaces is join-dense for the lattice, that is, the lattice can be obtained by sums of those join-irreducible elements (M. Aguiar, P. Ashwin, A. Dias, and M. Field. Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation, J. Nonlinear Sci. 21 (2) (2011) 271-323), and we conclude about the possible patterns of balanced colourings associated to the synchrony subspaces in the lattice. We also consider the disjoint union of two regular coupled cell networks with the same cell-type and the same edge-type. We show how to obtain the lattice of synchrony subspaces for the network union from the lattice of synchrony subspaces for the component networks. The lattice of synchrony subspaces for a homogeneous coupled cell network is given by the intersection of the lattice of synchrony subspaces for its identical-edge subnetworks per each edge-type (M. A. D. Aguiar and A. P. S. Dias. The lattice of synchrony subspaces of a coupled cell network: Characterization and computation algorithm, Journal of Nonlinear Science, 24 (6) (2014), 949-996). This, together with the results in this paper, on the lattice of synchrony subspaces for 1-input regular networks and on the lattice of synchrony * Partial support by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020 subspaces for the disjoint union of networks, define a procedure to obtain the lattice of synchrony subspaces for homogeneous coupled cell networks with asymmetric inputs. 2010 Mathematics Subject Classification: 34C15 37C10 06B23 15A18 Coupled cell systems have been for long a focus of interest to the scientific community, including biologists, physicists and mathematicians, since these systems are used as models in a wide range of real-world applications. See, for example, Albert and Barabási [9], Newman [32], Boccaletti et al. [13], Arenas et al. [10], and references therein. The structure of a coupled cell system can be abstracted by a coupled cell network -each cell represents an individual dynamical system and the connections represent the mutual interactions between those individual dynamics. See, for example, the formalism of Golubitsky and Stewart [41], [27], [24] which is more algebraic and the formalism of Field [16] which is more combinatorial.Coupled cell networks can be represented by directed graphs where the vertices are the cells and the edges represent the connections between them. Edges of the same type indicate the same k...

Towards a classification of networks with asymmetric inputs

“…For example, it is proved by Aguiar et al [4] that these networks can support robust heteroclinic cycles, even in low dimension. The synchrony lattice of networks with asymmetric inputs is studied by Aguiar [3]. Bifurcation problems have been considered by Rink and Sanders [24,25], Nijholt et al [19][20][21] and Aguiar et al [9].…”

“…Concerning (a), the list of the six ODE-distinct minimal three-cell networks with one input was obtained by Leite and Golubitsky [17]. Here, we provide the complete list of the 48 ODEdistinct minimal networks with three cells and two inputs (theorem 5.2 and tables [3][4][5][6]. In particular, this list contains the ten ODE-classes of strongly connected networks, with three cells, two asymmetric inputs and one or two two-dimensional synchrony subspaces, considered in Aguiar et al [4] and the seven networks with a monoid symmetry with three elements given by Rink and Sanders [24].…”