Journal
DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL
Volume 38, Issue 1, Pages 30-51Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/14689367.2022.2132136
Keywords
Characteristic matrices; spatio-temporal symmetries; equivariant Pyragas control
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This article investigates the stability of delay differential equations with discrete wave solutions. The stability can be determined using a characteristic matrix function. The applications of this result in delayed feedback stabilization of periodic orbits are discussed.
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.
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