Optimal and resonant time-delayed feedback control of unstable steady states: self-adaptive tuning of coupling phase

Abstract

We suggest an optimized scheme for the time-delayed feedback control of an unstable fixed point. For the normal form of an unstable focus, we derive the optimal feedback phase that yields the strongest stability and the shortest transient times for control. For this optimal value, the domain of control becomes largest in the plane of feedback strength and time delay. We present an adaptive algorithm for an automatic adjustment of the feedback phase for the optimal control without knowledge for the intrinsic parameters of the system. We extend the optimized control scheme to a network of supercritical Hopf normal forms coupled with delay. Analysis of eigenvalues using a master stability function approach shows that amplitude death is considerably facilitated if the coupling phase is equal to the optimized coupling phase.