Characteristic matrix functions for delay differential equations with symmetry

Babette A.J. de Wolff*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

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.

Original languageEnglish
Pages (from-to)30-51
Number of pages22
JournalDynamical Systems
Volume38
Issue number1
Early online date24 Oct 2022
DOIs
Publication statusPublished - 2023

Bibliographical note

Funding Information:
The author was partially supported by the Berlin Mathematical School and was an associated member of SFB 910 ‘Control of self-organizing linear systems’. The author is grateful to Bernold Fiedler and Sjoerd Verduyn Lunel for useful discussions and encouragement; and to Alejandro López Nieto, Bob Rink, Isabelle Schneider and an anonymous referee for comments on earlier versions. The contents of this article are based upon contents of the authors doctoral thesis [1 ], written at the Freie Universität Berlin under the supervision of Bernold Fiedler.

Publisher Copyright:
© 2022 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

Funding

The author was partially supported by the Berlin Mathematical School and was an associated member of SFB 910 ‘Control of self-organizing linear systems’. The author is grateful to Bernold Fiedler and Sjoerd Verduyn Lunel for useful discussions and encouragement; and to Alejandro López Nieto, Bob Rink, Isabelle Schneider and an anonymous referee for comments on earlier versions. The contents of this article are based upon contents of the authors doctoral thesis [1 ], written at the Freie Universität Berlin under the supervision of Bernold Fiedler.

Keywords

  • 34K20
  • 34K35
  • 93C23
  • Characteristic matrices
  • equivariant Pyragas control
  • spatio-temporal symmetries

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