Spectra of evolution operators of a class of neutral renewal equations: theoretical and numerical aspects

TL;DR

This paper investigates the spectra of evolution operators in neutral renewal equations, combining theoretical analysis and numerical experiments to understand stability and spectral properties, with potential for generalization.

Contribution

It provides a full spectral characterization for a simple linear periodic neutral renewal equation and explores numerical discretization methods for broader applications.

Findings

Spectral characterization of the monodromy operator for a basic linear periodic equation.

Numerical experiments confirm theoretical spectral results.

Discussion on extending methods to systems and multiple delays.

Abstract

In this work we begin a theoretical and numerical investigation on the spectra of evolution operators of neutral renewal equations, with the stability of equilibria and periodic orbits in mind. We start from the simplest form of linear periodic equation with one discrete delay and fully characterize the spectrum of its monodromy operator. We perform numerical experiments discretizing the evolution operators via pseudospectral collocation, confirming the theoretical results and giving perspectives on the generalization to systems and to multiple delays. Although we do not attempt to perform a rigorous numerical analysis of the method, we give some considerations on a possible approach to the problem.