Semigroup discretization and spectral approximation for linear nonautonomous delay differential equations

TL;DR

This paper presents a spectral approximation method for linear nonautonomous delay differential equations using semigroup discretization and collocation, enabling accurate stability analysis with proven convergence.

Contribution

It introduces a novel approach combining semigroup reduction and collocation with Chebyshev nodes for spectral approximation of delay differential equations.

Findings

Convergence of spectral approximation with infinite order.

Effective determination of stability of equilibria and limit cycles.

Rigorous a priori error bounds established.

Abstract

This paper deals with the approximation of the spectrum of linear and nonautonomous delay differential equations through the reduction of the relevant evolution semigroup from infinite to finite dimension. The focus is placed on classic collocation, even though the requirements that a numerical scheme has to fulfill in order to allow for a correct approximation of the spectral elements are recalled. This choice, motivated by the analyticity of the underlying eigenfunctions, allows for a convergence of infinite order, as rigorously demonstrated through a priori error bounds when Chebyshev nodes are adopted. Fundamental applications such as determination of asymptotic stability of equilibria (autonomous case) and limit cycles (periodic case) follow at once.