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. 78-100. ISSN 1536-0040

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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 two-color 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:	1536-0040
Date:	22 February 2005
Volume:	Vol.4
Number:	No.1
Page Range:	pp. 78-100
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:	DMS-0244529 (NSF), 003652-0032-2001 (ARP)
References:	[1] H. Brandt, ¨ Uber eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), pp. 360–366. [2] R. Brown, From groups to groupoids: A brief survey, Bull. London Math. Soc., 19 (1987), pp. 113–134. [3] A. P. S. Dias and I. Stewart, Symmetry groupoids and admissible vector fields for coupled cell networks, J. London Math. Soc. (2), 69 (2004), pp. 707–736. [4] A. P. S. Dias and I. Stewart, Linear Equivalence and ODE-Equivalence for Coupled Cell Networks, submitted. [5] I. R. Epstein and M. Golubitsky, Symmetric patterns in linear arrays of coupled cells, Chaos, 3 (1993), pp. 1–5. [6] D. Gillis and M. Golubitsky, Patterns in square arrays of coupled cells, J. Math. Anal. Appl., 208 (1997), pp. 487–509. [7] M. Golubitsky, M. Nicol, and I. Stewart, Some curious phenomena in coupled cell networks, J. Nonlinear Sci., 14 (2004), pp. 119–236. [8] P. J. Higgins, Notes on Categories and Groupoids, Van Nostrand Reinhold Mathematical Studies 32, Van Nostrand Reinhold, London, 1971. [9] M. Leite and M. Golubitsky, Synchrony-breaking bifurcations in homogeneous three-cell networks, in preparation. [10] I. Stewart, M. Golubitsky, and M. Pivato, Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dynam. Sys., 2 (2003), pp. 609–646. [11] Y. Wang and M. Golubitsky, Two-color patterns of synchrony in lattice dynamical systems, Nonlinearity, 18 (2005), pp. 631–657. [12] N. J. Wildberger, A new look at multisets, preprint, University of New South Wales, Sydney, 2003
URI:	http://wrap.warwick.ac.uk/id/eprint/183

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