Oscillator death in coupled functional differential equations near Hopf bifurcation

Fatihcan M. Atay

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Submission date: 07. Apr. 2004 (revised version: April 2004)
published in: Journal of differential equations, 221 (2006) 1, p. 190-209 
DOI number (of the published article): 10.1016/j.jde.2005.01.007
Bibtex
MSC-Numbers: 34C15, 34K20
Keywords and phrases: oscillator death, stability, delay, graph laplacian

Abstract:
Coupled systems of functional differential equations near a supercritical Hopf bifurcation are considered, and the stability of the equilibrium solution is analyzed. Necessary and sufficient conditions are derived for the asymptotic stability of the equilibrium under general coupling conditions. The stabilizing effects of some common coupling types are investigated and the role of the frequency differences and connection delays in inducing stability is displayed. The effect of the connection topology on stability is characterized by the spectral properties of the graph Laplacian. Bounds are given for the parametric stability regions for arbitrary connection topologies.