Symmetry groupoids and admissible vector fields for coupled cell networks
Dias, Ana Paula S. and Stewart, Ian, 1945-. (2004) Symmetry groupoids and admissible vector fields for coupled cell networks. Journal of the London Mathematical Society, Vol.69 (No.3). pp. 707-736. ISSN 0024-6107
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Official URL: http://dx.doi.org/10.1112/S0024610704005241
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 group-equivariant 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|
|Official Date:||June 2004|
|Page Range:||pp. 707-736|
|Access rights to Published version:||Open Access|
1. T. Bröcker and L. Lander, Differentiable germs and catastrophes (Cambridge University Press, Cambridge, 1975).
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