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Symmetry groupoids and patterns of synchrony in coupled cell networks
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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. 609646. ISSN 15360040

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Official URL: http://dx.doi.org/10.1137/S1111111103419896
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:  15360040 
Official Date:  2003 
Volume:  Vol.2 
Number:  No.4 
Page Range:  pp. 609646 
Identification Number:  10.1137/S1111111103419896 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  [1] H. Brandt, ¨ Uber eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), pp. 360–366. 
URI:  http://wrap.warwick.ac.uk/id/eprint/182 
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