Delayed feedback control of fractional-order chaotic systems

TL;DR

This paper explores the use of time-delayed feedback control to stabilize unstable states and periodic orbits in chaotic fractional-order systems, analyzing stability conditions and enhancing control via sinusoidal delay modulation.

Contribution

It introduces a stability analysis framework for fractional-order systems and demonstrates improved control strategies with sinusoidal delay modulation.

Findings

Stability regions depend on feedback gain and delay parameters.

Control fails for equilibria with an odd number of positive eigenvalues.

Sinusoidal delay modulation enlarges stability regions.

Abstract

We study the possibility to stabilize unstable steady states and unstable periodic orbits in chaotic fractional-order dynamical systems by the time-delayed feedback method. By performing a linear stability analysis, we establish the parameter ranges for successful stabilization of unstable equilibria in the plane parametrizad by the feedback gain and the time delay. An insight into the control mechanism is gained by analyzing the characteristic equation of the controlled system, showing that the control scheme fails to control unstable equilibria having an odd number of positive real eigenvalues. We demonstrate that the method can also stabilize unstable periodic orbits for a suitable choice of the feedback gain, providing that the time delay is chosen to coincide with the period of the target orbit. In addition, it is shown numerically that delayed feedback control with a sinusoidally modulated time delay significantly enlarges the stability region of the steady states in comparison to the classical time-delayed feedback scheme with a constant delay.