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

Jan Bouwe van den Berg; Chris Groothedde; Jean Philippe Lessard

doi:10.1007/s10884-020-09908-6

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

Mathematics

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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 language	English
Pages (from-to)	853-896
Number of pages	44
Journal	Journal of Dynamics and Differential Equations
Volume	34
Issue number	2
Early online date	2 Nov 2020
DOIs	https://doi.org/10.1007/s10884-020-09908-6
Publication status	Published - 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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10.1007/s10884-020-09908-6

A General Method for ComputerAssisted Proofs of Periodic Solutions in Delay Differential ProblemsFinal published version, 892 KBLicence: Article 25fa Dutch Copyright Act

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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.",

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

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