Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay

We propose a simple adaptive delayed feedback control algorithm for stabilization of unstable periodic orbits with unknown periods. The state dependent time delay is varied continuously towards the period of controlled orbit according to a gradient-descent method realized through three simple ordinary differential equations. We demonstrate the efficiency of the algorithm with the Rössler and Mackey–Glass chaotic systems. The stability of the controlled orbits is proven by computation of the Lyapunov exponents of linearized equations.

► A simple adaptive modification of the delayed feedback control algorithm is proposed.
► It enables the control of unstable periodic orbits with unknown periods.
► The delay time is varied continuously according to a gradient descend method.
► The algorithm is embodied by three simple ordinary differential equations.
► The validity of the algorithm is proven by computation of the Lyapunov exponents.