An odd-number limitation of extended time-delayed feedback control in autonomous systems

TL;DR

This paper establishes a necessary condition for stabilizing periodic orbits in autonomous systems using extended time-delayed feedback control, highlighting an odd-number limitation related to Floquet multipliers and the orbit's period.

Contribution

It extends the odd-number limitation concept to autonomous systems, linking Floquet multipliers and orbit period mismatches to control success.

Findings

The odd-number limitation applies to autonomous systems with real Floquet multipliers.

The period mismatch between control delay and orbit influences stabilization.

A necessary condition for stabilization depends on the number of Floquet multipliers greater than one.

Abstract

We propose a necessary condition for the successful stabilisation of a periodic orbit using the extended version of time-delayed feedback control. This condition depends on the number of real Floquet multipliers larger than unity and is therefore related to the well-known odd-number limitation in non-autonomous systems. We show that the period of the orbit which is induced by mismatching the delay-time of the control scheme and the period of the uncontrolled orbit plays an important role in the formulation of the odd-number limitation in the autonomous case.