Validated error bounds for pseudospectral approximation of delay differential equations: unstable manifolds

2024/05/13 by Shane Kepley, Kepley, Shane, Babette A. J. de Wolff +1
Mathematics · Computer Science · #Numerical methods for differential equations #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods

paper · pdf · doi:10.48550/arxiv.2405.07727

Abstract

Pseudospectral approximation provides a means to approximate the dynamics of delay differential equations (DDE) by ordinary differential equations (ODE). This article develops a computer-aided algorithm to determine the distance between the unstable manifold of a DDE and the unstable manifold of the approximating pseudospectral ODE. The algorithm is based upon the parametrization method. While a-priori the parametrization method for a vector-valued ODE involves computing a sequence of vector-valued Taylor coefficients, we show that for the pseudospectral ODE, due to its specific structure, the problem reduces to finding a sequence of scalars, which significantly simplifies the problem.

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