Delayed Feedback Control near Hopf Bifurcation

TL;DR

This paper analyzes how delayed feedback influences the stability of systems near a Hopf bifurcation, deriving conditions for stability and comparing different delay strategies using averaging theory.

Contribution

It provides necessary and sufficient conditions for stability under delayed feedback near Hopf bifurcation, including comparisons between discrete and distributed delays.

Findings

Delayed feedback can stabilize systems where undelayed feedback cannot.

Discrete delays are most stabilizing among delays with the same mean.

Global stability results hold for symmetrically distributed delays.

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.