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The lattice of balanced equivalence relations of a coupled cell network

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Stewart, Ian, 1945-. (2007) The lattice of balanced equivalence relations of a coupled cell network. Cambridge Philosophical Society. Mathematical Proceedings, Vol.143 (No.1). pp. 165-183. ISSN 0305-0041

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Official URL: http://dx.doi.org/10.1017/S0305004107000345

Abstract

A coupled cell system is a collection of dynamical systems, or ‘cells’, that are coupled together. The associated coupled cell network is a labelled directed graph that indicates how the cells are coupled, and which cells are equivalent. Golubitsky, Stewart, Pivato and Török have presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of the concept of a ‘balanced equivalence relation’, which depends solely on the network architecture. In their approach the network is assumed to be finite. We prove that the set of all balanced equivalence relations on a network forms a lattice, in the sense of a partially ordered set in which any two elements have a meet and a join. The partial order is defined by refinement. Some aspects of the theory make use of infinite networks, so we work in the category of networks of ‘finite type’, a class that includes all locally finite networks. This context requires some modifications to the standard framework. As partial compensation, the lattice of balanced equivalence relations can then be proved complete. However, the intersection of two balanced equivalence relations need not be balanced, as we show by a simple example, so this lattice is not a sublattice of the lattice of all equivalence relations with its usual operations of meet and join. We discuss the structure of this lattice and computational issues associated with it. In particular, we describe how to determine whether the lattice contains more than the equality relation. As an example, we derive the form of the lattice for a linear chain of identical cells with feedback.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Dynamics, Equivalence relations (Set theory), Lattice theory
Journal or Publication Title:	Cambridge Philosophical Society. Mathematical Proceedings
Publisher:	Cambridge University Press
ISSN:	0305-0041
Date:	24 July 2007
Volume:	Vol.143
Number:	No.1
Page Range:	pp. 165-183
Identification Number:	10.1017/S0305004107000345
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC)
References:	[1] J.ALDIS. A polynomial time algorithm to determine maximal balanced equivalence relations. Internat J. Bifur Chaos Appl. Sci. Engrg., to appear. [2] F. ANTONELI AND I. STEWART. Symmetry and synchrony in coupled cell networks 2: group networks. Internat. J. Bifur Chaos Appl. Sci. Engrg., to appear. [3] B. A. DAVEY AND H. A. PRIESTLEY. Introduction to Lattices and Order (Cambridge University Press, 1990). Lattice of balanced equivalence relations 183 [4] W. FULTON AND J. HARRIS. Representation Theory. Graduate Texts in Math. 129 (Springer, 1991). [5] M. GOLUBITSKY AND I. STEWART. Nonlinear dynamics of networks: the groupoid formalism. Bull. Amer. Math. Soc. 43(2006), 305–364. [6] M.GOLUBITSKY, I. STEWART AND A. T ¨OR ¨OK. Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dynam. Sys. 4(1) (2005), 78–100. [7] R.L.GRAHAM, D. E. KNUTH AND O. PATASHNIK. Concrete Mathematics (Addison-Wesley, 1994). [8] P. J.HIGGINS. Notes on Categories and Groupoids. Van Nostrand Reinhold Mathematical Studies 32 (Van Nostrand Reinhold, 1971). [9] S. C. MANRUBIA, A. S. MIKHAILOV AND D. H. ZANETTE. Emergence of Dynamical Order (World Scientific, 2004). [10] E.MOSEKILDE, Y. MAISTRENKO AND D. POSTONOV. Chaotic Synchronization (World Scientific, 2002). [11] I. STEWART, M. GOLUBITSKY AND M. PIVATO. Patterns of synchrony in coupled cell networks. SIAM J. Appl. Dynam. Sys 2 (2003), 609–646. [DOI: 10.1137/S1111111103419896]. [12] R. J.WILSON. Introduction to Graph Theory (3rd ed.) (Longman, 1985). [13] C.W.WU. Synchronization in Coupled Chaotic Circuits and Systems (World Scientific, 2002). [14] C.W.WU. Synchronization in networks of nonlinear dynamical systems coupled via a directed graph. Nonlinearity 18 (2005), 1057–1064.
URI:	http://wrap.warwick.ac.uk/id/eprint/651