An analytical limitation for time-delayed feedback control in autonomous systems

TL;DR

This paper establishes an analytical limitation on the effectiveness of time-delayed feedback control for stabilizing periodic orbits in autonomous systems, extending the understanding of stability boundaries beyond the traditional odd number limitation.

Contribution

It introduces a new analytical limitation based on Floquet multipliers, applicable to autonomous systems, and confirms its accuracy through a two-dimensional example.

Findings

The limitation depends on the number of real Floquet multipliers greater than one.

It generalizes the odd number limitation for autonomous systems.

The limitation accurately predicts stability boundaries in specific examples.

Abstract

We prove an analytical limitation on the use of time-delayed feedback control for the stabilization of periodic orbits in autonomous systems. This limitation depends on the number of real Floquet multipliers larger than unity, and is therefore similar to the well-known odd number limitation of time-delayed feedback control. Recently, a two-dimensional example has been found, which explicitly demonstrates that the unmodified odd number limitation does not apply in the case of autonomous systems. We show that our limitation correctly predicts the stability boundaries in this case.