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Permissible symmetries of coupled cell networks

Published online by Cambridge University Press:  24 October 2008

Peter Ashwin  and
Peter Stork
Peter Ashwin
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL
Peter Stork
Affiliation:
Institut für Angewandte Mathematik, Universität Hamburg, Bundesstrasse 55, D-2000 Hamburg, Germany
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Abstract

We consider coupled sets of identical cells and address the problem of which symmetries are permissible in such networks. For example, n linearly coupled cells with one independent variable in each cell cannot be constructed with the symmetry group An, the alternating group on n symbols. Using a graphical technique, we show that it is possible to construct cell networks with any desired finite group of symmetries. In particular, we show that any subgroup of Sn can be realized as the symmetries of a group of n cells. Special forms of coupling (especially low order polynomial coupling) are shown to restrict the possible symmetries. We give some upper and lower bounds for the degree of polynomial required to realize several classes of subgroups of Sn.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 1 , July 1994 , pp. 27 - 36
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Ashwin, P.. Applications of Dynamical Systems with Symmetry. (PhD thesis, University of Warwick, 1991).Google Scholar
[2]Ashwin, P. and Swift, J. W.. The dynamics of n identical oscillators with symmetric coupling. Journal of Nonlinear Science 2 (1992), 69–108.CrossRefGoogle Scholar
[3]Burnside, W. S. and Panton, A. W.. The Theory of Equations. Vol. II (Dublin University Press, 1904 Edition).Google Scholar
[4]Golubitsky, M., Stewart, I. N. and Schaeffer, D.. Singularities and Groups in Bifurcation Theory Volume 2, volume 69 of App. Math. Sci. (Springer, 1988).CrossRefGoogle Scholar
[5]Jordan, C.. Traité de substitutions (Gauthier–Villars, 1870).Google Scholar
[6]Kirkman, T. P.. Theory of groups and many-valued functions. Memoirs of the Manchester Literary and Philosophical Society, Volume 1 (3rd series) Art. XXII, pp. 274–398, 1862.Google Scholar
[7]Kopell, N., Zhang, W. and Ermentrout, G. B.. Multiple coupling in chains of oscillators. SIAM J. Math. Anal. 21 (1990), 935–953.CrossRefGoogle Scholar
[8]Lang, S.. Algebra, 2nd edition (Addison-Wesley, 1984).Google Scholar
[9]Ledermann, W.. Introduction to Group Theory (Longman, 1973).Google Scholar
[10]Murray, J. D.. Mathematical Biology, volume 13 of Biomathematics (Springer, 1989).CrossRefGoogle Scholar
[11]Springer, T. A.. Invariant Theory (Springer, 1977).CrossRefGoogle Scholar
[12]Stork, P.. Statische Verzweigung in Gradientenfeldern mit Symmetrien vom komplexen oder quaternionischen Typ mit numerischer Behandlung. Ph.D. Thesis, Institut für Angewandte Mathematik, University of Hamburg; Wissenschaftliche Beiträge aus europäischen Hochschulen: Reihe 11, Band 7. (Verlag an der Lottbeck, Hamburg, 1993.)Google Scholar
[13]Wielandt, H. W.. Permutation Groups through Invariant Relations and Invariant Functions (Department of Mathematics, Ohio State University, 1969).Google Scholar