Patterns from bifurcations: A symmetry analysis of networks with delayed coupling

Fatihcan M. Atay and Haibo Ruan

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Submission date: 25. Nov. 2013
Pages: 28
published in: Nonlinearity, 28 (2015) 3, p. 795-824 
DOI number (of the published article): 10.1088/0951-7715/28/3/795
Bibtex
with the following different title: Symmetry analysis of coupled scalar systems under time delay
MSC-Numbers: 15A18, 34C14, 34C23
PACS-Numbers: 02.30.Oz, 31.30.-j
Keywords and phrases: symmetry, equivariant degree, bifurcation theory, delay, dynamical patterns
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Abstract:
We study systems of coupled units in a general network configuration with a coupling delay. We show that the destabilizing bifurcations from an equilibrium are governed by the extreme eigenvalues of the coupling matrix of the network. Based on the equivariant degree method and its computational packages, we perform a symmetry classification of destabilizing bifurcations in bidirectional rings of coupled units, for bifurcating solutions either of steady-states or of oscillating states. We also introduce the concept of secondary dominating orbit types to capture bifurcating solutions of submaximal nature.