Equivariant Versions of Odd Number Theorem

TL;DR

This paper extends the Odd Number Theorem to equivariant systems with symmetries, analyzing stabilization of periodic solutions via Pyragas control and proposing variants for systems with finite symmetry groups.

Contribution

It introduces new versions of the Odd Number Theorem applicable to symmetric systems and modifies control methods for stabilizing specific solutions within symmetric orbits.

Findings

Stabilization of entire orbits in rotationally symmetric systems.

Control variants for systems with finite symmetry groups.

Illustrations with coupled oscillators and lasers.

Abstract

We consider the problem of stabilization of unstable periodic solutions to autonomous systems by the non-invasive delayed feedback control known as Pyragas control method. The Odd Number Theorem imposes an important restriction upon the choice of the gain matrix by stating a necessary condition for stabilization. In this paper, the Odd Number Theorem is extended to equivariant systems. We assume that both the uncontrolled and controlled systems respect a group of symmetries. Two types of results are discussed. First, we consider rotationally symmetric systems for which the control stabilizes the whole orbit of relative periodic solutions that form an invariant two-dimensional torus in the phase space. Second, we consider a modification of the Pyragas control method that has been recently proposed for systems with a finite symmetry group. This control acts non-invasively on one selected periodic solution from the orbit and targets to stabilize this particular solution. Variants of the Odd Number Limitation Theorem are proposed for both above types of systems. The results are illustrated with examples that have been previously studied in the literature on Pyragas control including a system of two symmetrically coupled Stewart-Landau oscillators and a system of two coupled lasers.