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Synchrony-breaking bifurcation at a simple real eigenvalue for regular networks 2 : higher-dimensional cells
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Stewart, Ian (2014) Synchrony-breaking bifurcation at a simple real eigenvalue for regular networks 2 : higher-dimensional cells. SIAM Journal on Applied Dynamical Systems, Volume 13 (Number 1). pp. 129-156. doi:10.1137/130917636 ISSN 1536-0040.
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Official URL: http://dx.doi.org/10.1137/130917636
Abstract
Bifurcation from a trivial branch of equilibria at a simple real eigenvalue can be characterized by a singularity-theoretic normal form $\lambda x \pm x^r$. The case $r=2$ corresponds to transcritical bifurcation and $r=3$ to pitchfork bifurcation. A previous paper [I. Stewart and M. Golubitsky, SIAM J. Appl. Dyn. Syst., 10 (2011), pp. 1404--1442] discussed such bifurcations in the context of regular coupled cell networks. When cells are 1-dimensional, there exist networks in which all such bifurcations are 3-degenerate, that is, have normal forms with $r \geq 4$. We extend the analysis to cells of arbitrary dimension and prove that if the adjacency matrix has triangular eigenstructure, then the same degree of degeneracy occurs, subject to a mild condition on the linearization that ensures a simple eigenvalue. We give an example of a network without triangular eigenstructure where 3-degeneracy is not preserved for cells of dimension $\geq 2$.
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | SIAM Journal on Applied Dynamical Systems | ||||||||
Publisher: | Society for Industrial and Applied Mathematics | ||||||||
ISSN: | 1536-0040 | ||||||||
Official Date: | 11 February 2014 | ||||||||
Dates: |
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Volume: | Volume 13 | ||||||||
Number: | Number 1 | ||||||||
Page Range: | pp. 129-156 | ||||||||
DOI: | 10.1137/130917636 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access |
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