Delayed feedback control near Hopf bifurcation

Fatihcan M. Atay

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Submission date: 28. Mar. 2007 (revised version: October 2008)
Pages: 12
published in: Discrete and continuous dynamical systems / S, 1 (2008) 2, p. 197-205 
DOI number (of the published article): 10.3934/dcdss.2008.1.197
Bibtex
MSC-Numbers: 34K35, 93C23, 93D15, 34K20
Keywords and phrases: stability, feedback, hopf bifurcation, distributed delays, {H}opf bifurcation
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Abstract:
The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that when the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. Thus, increasing the variance of the delay distribution near a discrete delay reduces the effect of delays in the feedback.