Stabilization of unstable steady states by variable delay feedback control

TL;DR

This paper demonstrates that using a variable, time-varying delay in feedback control significantly enhances the stabilization of unstable steady states in chaotic systems, outperforming traditional fixed-delay methods.

Contribution

It introduces a novel variable delay feedback control (VDFC) method and compares its effectiveness with standard TDAS through numerical simulations and analytical domain analysis.

Findings

VDFC outperforms standard TDAS in stabilizing unstable states.

Different delay modulations (sawtooth, sine, random) are effective.

Analytical control domains are derived for specific cases.

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.