Symmetry and synchrony in coupled cell networks 1: Fixed-point spaces
UNSPECIFIED. (2006) Symmetry and synchrony in coupled cell networks 1: Fixed-point spaces. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 16 (3). pp. 559-577. ISSN 0218-1274Full text not available from this repository.
Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Gamma of permutations of the nodes ('' cells ''). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals - ones that are not fixed-point spaces - can occur for such networks. We also prove the '' folk theorem '' that in any Gamma-equivariant dynamical system on R-k the only flow-invariant subspaces are the fixed-point spaces of subgroups of Gamma.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics
|Journal or Publication Title:||INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS|
|Publisher:||WORLD SCIENTIFIC PUBL CO PTE LTD|
|Official Date:||March 2006|
|Number of Pages:||19|
|Page Range:||pp. 559-577|
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