Antoneli, Fernando and Stewart, Ian, 1945-. (2007) Symmetry and synchrony in coupled cell networks 2: group networks. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Vol.17 (No.3). pp. 935-951. ISSN 0218-1274

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Abstract

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flow-invariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Let Gamma be a permutation group acting on the set of cells. We de. ne the group network G(Gamma), whose architecture is entirely determined by the group orbits of Gamma. We prove that if Gamma has the "balanced extension property" then every balanced coloring of G(Gamma) is a fixed-point coloring relative to the automorphism group of the group network. This theorem applies in particular when Gamma is cyclic or dihedral, acting on cells as the symmetries of a regular polygon, and in these cases the automorphism group is Gamma itself. In general, however, the automorphism group may be larger than Gamma. Several examples of this phenomenon are discussed, including the finite simple group of order 168 in its permutation representation of degree 7. More dramatically, for some choices of Gamma there exist balanced colorings of G(Gamma) that are not fixed-point colorings. For example, there exists an exotic balanced 2-coloring when Gamma is the symmetry group of the two-dimensional square lattice. This coloring is doubly periodic, and its reduction modulo 8 leads to a finite group with similar properties. Although these patterns do not arise from fixed-point spaces, we provide a group- theoretic explanation of their balance property in terms of a sublattice of index two.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science
Divisions:	Faculty of Science > Mathematics
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:	March 2007
Volume:	Vol.17
Number:	No.3
Number of Pages:	17
Page Range:	pp. 935-951
Identification Number:	10.1142/S0218127407017641
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
URI:	http://wrap.warwick.ac.uk/id/eprint/31123

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