Stewart, Ian, 1945-, Golubitsky, Martin and Pivato, Marcus. (2003) Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM Journal on Applied Dynamical Systems, Vol.2 (No.4). pp. 609-646. ISSN 1536-0040

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Abstract

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only mechanism that can create such states in a coupled cell system and show that it is not. The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with the corresponding internal dynamics and couplings—are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is “robust” if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of “polydiagonal” subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an equivalence relation on cells is “balanced.” The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the “quotient network.” The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Coupled mode theory, Groupoids, Symmetry (Mathematics)
Journal or Publication Title:	SIAM Journal on Applied Dynamical Systems
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	1536-0040
Date:	2003
Volume:	Vol.2
Number:	No.4
Page Range:	pp. 609-646
Identification Number:	10.1137/S1111111103419896
Status:	Peer Reviewed
Access rights to Published version:	Open Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/182

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