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

TL;DR

This paper introduces a computer-aided algorithm that accurately estimates the difference between unstable manifolds of delay differential equations and their pseudospectral ODE approximations, leveraging a simplified scalar-based parametrization method.

Contribution

It develops a novel, simplified scalar parametrization approach for pseudospectral ODEs to compute error bounds for unstable manifolds of DDEs.

Findings

The algorithm effectively computes the distance between manifolds.

Scalar reduction simplifies the parametrization process.

Provides validated error bounds for pseudospectral approximations.

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.