A General Method for Computer-Assisted Proofs of Periodic Solutions in Delay Differential Problems

Jan Bouwe van den Berg, Chris Groothedde, Jean Philippe Lessard*

*Corresponding author for this work

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Abstract

In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.

Original languageEnglish
Pages (from-to)853-896
Number of pages44
JournalJournal of Dynamics and Differential Equations
Volume34
Issue number2
Early online date2 Nov 2020
DOIs
Publication statusPublished - Jun 2022

Bibliographical note

Funding Information:
Funding was provided by NWO-VICI (Grant No. 639033109) and Natural Sciences and Engineering Research Council of Canada.

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Computer–assisted proofs
  • Contraction mapping
  • Delay differential equations
  • Fourier series
  • Mackey–Glass equation
  • Periodic solutions

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