Stabilization of unstable steady states by variable delay feedback control

Abstract

We report on a dramatic improvement of the performance of the classical time-delayed autosynchronization method (TDAS) to control unstable steady states, by applying a time-varying delay in the TDAS control scheme in a form of a deterministic or stochastic delay-modulation in a fixed interval around a nominal value $T_0$. The successfulness of this variable delay feedback control (VDFC) is illustrated by a numerical control simulation of the Lorenz and R\"{o}ssler systems using three different types of time-delay modulations: a sawtooth wave, a sine wave, and a uniform random distribution. We perform a comparative analysis between the VDFC method and the standard TDAS method for a sawtooth-wave modulation by analytically determining the domains of control for the generic case of an unstable fixed point of focus type.