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Symmetry groupoids and admissible vector fields for coupled cell networks
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Dias, Ana Paula S. and Stewart, Ian. (2004) Symmetry groupoids and admissible vector fields for coupled cell networks. Journal of the London Mathematical Society, Vol.69 (No.3). pp. 707736. ISSN 00246107

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Official URL: http://dx.doi.org/10.1112/S0024610704005241
Abstract
The space of admissible vector fields, consistent with the structure of a network of coupled dynamical systems, can be specified in terms of the network's symmetry groupoid. The symmetry groupoid also determines the robust patterns of synchrony in the network – those that arise because of the network topology. In particular, synchronous cells can be identified in a canonical manner to yield a quotient network. Admissible vector fields on the original network induce admissible vector fields on the quotient, and any dynamical state of such an induced vector field can be lifted to the original network, yielding an analogous state in which certain sets of cells are synchronized. In the paper, necessary and sufficient conditions are specified for all admissible vector fields on the quotient to lift in this manner. These conditions are combinatorial in nature, and the proof uses invariant theory for the symmetric group. Also the symmetry groupoid of a quotient is related to that of the original network, and it is shown that there is a close analogy with the usual normalizer symmetry that arises in groupequivariant dynamics.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Ergodic theory, Differential equations, Vector field, Groupoids, Vector valued groupoids 
Journal or Publication Title:  Journal of the London Mathematical Society 
Publisher:  Cambridge University Press 
ISSN:  00246107 
Official Date:  June 2004 
Volume:  Vol.69 
Number:  No.3 
Page Range:  pp. 707736 
Identification Number:  10.1112/S0024610704005241 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  1. T. Bröcker and L. Lander, Differentiable germs and catastrophes (Cambridge University Press, Cambridge, 1975). 
URI:  http://wrap.warwick.ac.uk/id/eprint/756 
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