Kamei, Hiroko. (2009) Construction of lattices of balanced equivalence relations for regular homogenous networks using lattice generators and lattice indices. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Vol.19 (No.11). pp. 3691-3705. ISSN 0218-1274

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Abstract

Regular homogeneous networks are a class of coupled cell network, which comprises one type of cell (node) with one type of coupling (arrow), and each cell has the same number of input arrows (called the valency of the network). In coupled cell networks, robust synchrony (a flow-invariant polydiagonal) corresponds to a special kind of partition of cells, called a balanced equivalence relation. Balanced equivalence relations are determined solely by the network structure. It is well known that the set of balanced equivalence relations on a given finite network forms a complete lattice. In this paper, we consider regular homogeneous networks in which the internal dynamics of each cell is one-dimensional, and whose associated adjacency matrices have simple eigenvalues (real or complex). We construct explicit forms of lattices of balanced equivalence relations for such networks by introducing key building blocks, called lattice generators, along with integer numbers called lattice indices. The properties of lattice indices allow construction of all possible lattice structures for balanced equivalence relations of regular homogeneous networks of any number of cells with any valency. As an illustration, we show all 14 possible lattice structures of balanced equivalence relations for four-cell regular homogeneous networks.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science
Divisions:	Faculty of Science > Molecular Organisation and Assembly in Cells (MOAC)
Journal or Publication Title:	International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Publisher:	World Scientific Publishing Co. Pte. Ltd.
ISSN:	0218-1274
Date:	November 2009
Volume:	Vol.19
Number:	No.11
Number of Pages:	15
Page Range:	pp. 3691-3705
Identification Number:	10.1142/S0218127409025067
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	EP/D043859 (EPSRC)
URI:	http://wrap.warwick.ac.uk/id/eprint/6347

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