Control of unstable steady states by extended time-delayed feedback

TL;DR

This paper explores the application of extended time-delayed feedback control to stabilize an unstable focus, analyzing effects of latency, filtering, and coupling on the control domain through theoretical and numerical methods.

Contribution

It extends the time-delayed feedback control method to unstable steady states, including effects of latency, filtering, and phase coupling, supported by theoretical analysis and numerical validation.

Findings

Control domain depends on latency and filter parameters.

Rotational coupling phase influences control effectiveness.

Numerical results confirm theoretical predictions.

Abstract

Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar et al. to an unstable focus. This system represents a generic model of an unstable steady state which can be found for instance in a Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.