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A General Method for Computer-Assisted Proofs of Periodic Solutions in Delay Differential Problems
Journal of Dynamics and Differential Equations ( IF 1.3 ) Pub Date : 2020-11-02 , DOI: 10.1007/s10884-020-09908-6
Jan Bouwe van den Berg , Chris Groothedde , Jean-Philippe Lessard

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.



中文翻译:

时滞微分问题中周期解的计算机辅助证明的一种通用方法

在本文中,我们为延迟微分方程的周期解开发了一种通用的计算机辅助证明方法。考虑的问题类别包括具有任意数量(正向和反向)延迟的延迟微分方程组。当非线性包括非多项式项时,我们引入辅助变量以首先将问题重写为等效多项式。然后,我们在几何衰减傅里叶系数的空间中应用灵活的定点技术。通过在著名的Mackey-Glass延迟微分方程中证明经典参数值的周期解,我们证明了该方法的有效性。

更新日期:2020-11-03
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