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Symmetry and synchrony in coupled cell networks 1 : Fixed-point spaces
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UNSPECIFIED (2006) Symmetry and synchrony in coupled cell networks 1 : Fixed-point spaces. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Volume 16 (Number 3). pp. 559-577. ISSN 0218-1274.
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Abstract
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 | ||||
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Subjects: | Q Science > QA Mathematics Q Science |
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Journal or Publication Title: | International Journal of Bifurcation and Chaos in Applied Sciences and Engineering | ||||
Publisher: | World Scientific Publishing Co. Pte. Ltd. | ||||
ISSN: | 0218-1274 | ||||
Official Date: | March 2006 | ||||
Dates: |
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Volume: | Volume 16 | ||||
Number: | Number 3 | ||||
Number of Pages: | 19 | ||||
Page Range: | pp. 559-577 | ||||
Publication Status: | Published |
Data sourced from Thomson Reuters' Web of Knowledge
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