Antoneli, Fernando and Stewart, Ian, 1945-. (2008) Symmetry and synchrony in coupled cell networks 3 : exotic patterns. International Journal of Bifurcation and Chaos, Vol.18 (No.2). pp. 363-373. ISSN 0218-1274

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Abstract

This paper continues the study of patterns of synchrony ( equivalently, balanced colorings or flowinvariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Our aim is to provide a group-theoretic explanation of the "exotic" balanced coloring previously discussed in Part 2. Here we show that the pattern can be obtained as a projection into two dimensions of a fixed-point pattern in a three-dimensional lattice. We prove a general theorem giving sufficient conditions for such a construction to lead to a balanced coloring, for an arbitrary direct product of group networks.

Item Type:	Journal Article
Subjects:	Q Science > Q Science (General) Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	System theory, Group theory, Symmetry groups, Symmetry (Physics), Chaotic behavior in systems, Pattern formation (Physical sciences)
Journal or Publication Title:	International Journal of Bifurcation and Chaos
Publisher:	World Scientific Publishing Co. Pte. Ltd.
ISSN:	0218-1274
Date:	February 2008
Volume:	Vol.18
Number:	No.2
Number of Pages:	11
Page Range:	pp. 363-373
Identification Number:	DOI:10.1142/S0218127408020331
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC), Universidade do Porto. Centro de Matemática (CMUP), Fundação para a Ciência e a Tecnologia (FCT)
Grant number:	DMS-0244529 (NSF)
References:	Antoneli, F., Dias, A. P. S., Golubitsky, M. & Wang, Y. [2005] “Patterns of synchrony in lattice dynamical systems,” Nonlinearity 18, 2193–2209. Antoneli, F. & Stewart, I. [2006] “Symmetry and synchrony in coupled cell networks 1: Fixed-point spaces,” Int. J. Bifurcation and Chaos 16, 559–577. Antoneli, F. & Stewart, I. [2007] “Symmetry and synchrony in coupled cell networks 2: Group networks,” Int. J. Bifurcation and Chaos 17, 935–951. Baake, M., Kramer, P., Schlottmann, M. & Zeidler, D. [1990] “Planar patterns with fivefold symmetry as sections of periodic structures in 4-space,” in Quasicrystals, Networks, and Molecules of Fivefold Symmetry, ed. Hargittai, I. (VCH, New York), pp. 127–157. Davey, B. A. & Priestley, H. A. [1990] Introduction to Lattices and Order (Cambridge University Press, Cambridge). Golubitsky, M. & Stewart, I. [2002] The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics, Vol. 200 (Birkh¨auser, Basel). Golubitsky, M., Stewart, I. & T¨or¨ok, A. [2005] “Patterns of synchrony in coupled cell networks with multiple arrows,” SIAM J. Appl. Dyn. Syst. 4, 78–100. Hahn, T. [1992] International Tables for Crystallography: Volume A, Space-Group Symmetry (Kluwer, Dordrecht). Hall, Jr. M. [1959] The Theory of Groups (Macmillan, NY). Neumann, P. M., Stoy, G. A. & Thompson, E. C. [1994] Groups and Geometry (Oxford University Press, Oxford). Stewart, I., Golubitsky, M. & Pivato, M. [2003] “Patterns of synchrony in coupled cell networks,” SIAM J. Appl. Dyn. Syst. 2, 609–646. [DOI: 10.1137/ S1111111103419896] Stewart, I. [2007] “The lattice of balanced equivalence relations on a coupled cell network,” Math. Proc. Camb. Phil. Soc. 143, 165–183. Tutte, W. T. [1984] Graph Theory, Encyclopaedia of Mathematics and Its Applications, Vol. 21, ed. Rota, G.-C. (Addison-Wesley, Menlo Park). Wang, Y. & Golubitsky,M. [2005] “Two-color patterns of synchrony in lattice dynamical systems,” Nonlinearity 18, 631–657. Wilson, R. J. [1985] Introduction to Graph Theory, 3rd edition (Longman, Harlow).
URI:	http://wrap.warwick.ac.uk/id/eprint/29760

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