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Patterns of synchrony in coupled cell networks with multiple arrows
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Golubitsky, Martin, Stewart, Ian, 1945 and Torok, Andrei. (2005) Patterns of synchrony in coupled cell networks with multiple arrows. SIAM Journal on Applied Dymanical Systems, Vol.4 (No.1). pp. 78100. ISSN 15360040

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Official URL: http://dx.doi.org/10.1137/040612634
Abstract
A coupled cell system is a network of dynamical systems, or “cells,” coupled together. The architecture
of a coupled cell network is a graph that indicates how cells are coupled and which cells are
equivalent. Stewart, Golubitsky, and Pivato presented a framework for coupled cell systems that
permits a classification of robust synchrony in terms of network architecture. They also studied
the existence of other robust dynamical patterns using a concept of quotient network. There are
two difficulties with their approach. First, there are examples of networks with robust patterns of
synchrony that are not included in their class of networks; and second, vector fields on the quotient
do not in general lift to vector fields on the original network, thus complicating genericity arguments.
We enlarge the class of coupled systems under consideration by allowing two cells to be coupled in
more than one way, and we show that this approach resolves both difficulties. The theory that we
develop, the “multiarrow formalism,” parallels that of Stewart, Golubitsky, and Pivato. In addition,
we prove that the pattern of synchrony generated by a hyperbolic equilibrium is rigid (the pattern
does not change under small admissible perturbations) if and only if the pattern corresponds to
a balanced equivalence relation. Finally, we use quotient networks to discuss Hopf bifurcation in
homogeneous cell systems with twocolor balanced equivalence relations.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Coupled systems 
Journal or Publication Title:  SIAM Journal on Applied Dymanical Systems 
Publisher:  Society for Industrial and Applied Mathematics 
ISSN:  15360040 
Official Date:  22 February 2005 
Volume:  Vol.4 
Number:  No.1 
Page Range:  pp. 78100 
Identification Number:  10.1137/040612634 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
Funder:  National Science Foundation (U.S.) (NSF), Norman Hackerman Advanced Research Program (NHARP) 
Grant number:  DMS0244529 (NSF), 00365200322001 (ARP) 
References:  [1] H. Brandt, ¨ Uber eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), pp. 360–366. 
URI:  http://wrap.warwick.ac.uk/id/eprint/183 
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