Characteristic matrix functions for delay differential equations with symmetry

TL;DR

This paper develops a method using characteristic matrix functions to analyze the stability of discrete wave solutions in delay differential equations with symmetry, leveraging equivariant Floquet theory and applications in feedback stabilization.

Contribution

It introduces a novel approach to determine stability of symmetric delay differential equations' solutions via characteristic matrix functions, extending existing theories to equivariant systems.

Findings

Stability of discrete wave solutions can be determined using characteristic matrix functions.

The approach applies to delay differential equations with symmetry and rationally related periods.

Applications include delayed feedback stabilization of periodic orbits.

Abstract

A characteristic matrix function captures the spectral information of a bounded linear operator in a matrix-valued function. In this article, we consider a delay differential equation with one discrete time delay and assume this equation is equivariant with respect to a compact symmetry group. Under this assumption, the delay differential equation can have discrete wave solutions, i.e. periodic solutions that have a discrete group of spatio-temporal symmetries. We show that if a discrete wave solution has a period that is rationally related to the time delay, then we can determine its stability using a characteristic matrix function. The proof relies on equivariant Floquet theory and results by Kaashoek and Verduyn Lunel on characteristic matrix functions for classes of compact operators. We discuss applications of our result in the context of delayed feedback stabilization of periodic orbits.