Symmetry groupoids and admissible vector fields for coupled cell networks

• Type: Article in International Scientific Journal
• Date: 2004

Authors

dias, aps (Author) FCUP; stewart, i (Author) Other

Journal

Title: Journal of the London Mathematical Society

Vol. 69

Pages: 707-736

ISSN: 0024-6107

Publisher: Wiley-Blackwell

Indexing

ISI Web of Knowledge - 14 Citations

Scopus - 16 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-000-A7F

DOI: 10.1112/s0024610704005241

Abstract (EN): 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.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 30