Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Delayed feedback control near Hopf bifurcation

Pages: 197 - 205, Volume 1, Issue 2, June 2008      doi:10.3934/dcdss.2008.1.197

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Fatihcan M. Atay - Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig 04103, Germany (email)

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 if 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. The result also holds globally if one considers delays that are symmetrically distributed about their mean.

Keywords:  Stability, feedback, Hopf bifurcation, distributed delays.
Mathematics Subject Classification:  Primary: 34K35, 93C23, 93D15, 34K20.

Received: October 2006;      Revised: October 2007;      Available Online: March 2008.