State-dependent act-and-wait time-delayed feedback control algorithm

Abstract

Time-delayed feedback control (TDFC) method is one of the most popular experimental tools used for stabilization of unstable periodic orbits in nonlinear dynamical systems. The theory of the method is difficult since systems controlled by time-delayed feedback evolve in an infinite-dimensional function space. Here we modify the TDFC by appending the feedback loop with a state-dependent switch. The switch closes and opens the feedback loop in the dependence of the phase of the output signal, which is measured by a specially designed phase detector. This modification represents a further development of the act-and-wait concept applied to time-delayed feedback controllers. The concept allows to reduce the phase space dimension of time-delayed feedback systems and to significantly simplify the theory. In previous works, the time-delayed feedback loop was tuned on and off periodically, independently of the state of the system. We discuss the advantages of our algorithm with respect to previously considered modifications. The efficacy of the proposed algorithm is numerically demonstrated with autonomous Rössler and Lorenz systems as well as a nonautonomous Duffing oscillator.