UNSPECIFIED (2006) Symmetry and synchrony in coupled cell networks 1: Fixed-point spaces. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 16 (3). pp. 559-577. ISSN 0218-1274

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Abstract

Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Gamma of permutations of the nodes ('' cells ''). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals - ones that are not fixed-point spaces - can occur for such networks. We also prove the '' folk theorem '' that in any Gamma-equivariant dynamical system on R-k the only flow-invariant subspaces are the fixed-point spaces of subgroups of Gamma.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science
Journal or Publication Title:	INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
Publisher:	WORLD SCIENTIFIC PUBL CO PTE LTD
ISSN:	0218-1274
Date:	March 2006
Volume:	16
Number:	3
Number of Pages:	19
Page Range:	pp. 559-577
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/33505

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