Time-Delayed Feedback Control Design Beyond the Odd Number Limitation

TL;DR

This paper introduces a novel algorithm for designing time-delayed feedback control to stabilize certain unstable periodic orbits previously considered uncontrollable due to the odd number limitation, validated on Lorenz and Chua systems.

Contribution

The paper presents a new control design algorithm that overcomes the odd number limitation by leveraging Floquet multiplier relationships, supported by theoretical correction and practical demonstrations.

Findings

Successfully stabilizes orbits with odd numbers of positive Floquet exponents.

Refutes the odd number theorem with a counterexample and provides a corrected version.

Demonstrates effectiveness on Lorenz and Chua systems.

Abstract

We present an algorithm for a time-delayed feedback control design to stabilize periodic orbits with an odd number of positive Floquet exponents in autonomous systems. Due to the so-called odd number theorem such orbits have been considered as uncontrollable by time-delayed feedback methods. However, this theorem has been refuted by a counterexample and recently a corrected version of the theorem has been proved. In our algorithm, the control matrix is designed using a relationship between Floquet multipliers of the systems controlled by time-delayed and proportional feedback. The efficacy of the algorithm is demonstrated with the Lorenz and Chua systems.