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

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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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Funding

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

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Authors and Affiliations

  1. Department of Mathematics, VU Amsterdam, De Boelelaan 1081, 1081 HV, Amsterdam, The Netherlands

    Jan Bouwe van den Berg & Chris Groothedde

  2. Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada

    Jean-Philippe Lessard

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  1. Jan Bouwe van den Berg
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Correspondence to Jean-Philippe Lessard.

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van den Berg, J.B., Groothedde, C. & Lessard, JP. A General Method for Computer-Assisted Proofs of Periodic Solutions in Delay Differential Problems. J Dyn Diff Equat 34, 853–896 (2022). https://doi.org/10.1007/s10884-020-09908-6

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