Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Dimitri Breda, Davide Liessi, Sjoerd M. Verduyn Lunel

TL;DR

This work analyzes the spectra of evolution operators for a class of neutral renewal equations (NREs) with a focus on stability of equilibria and periodic solutions. It develops a theoretical characterization of the monodromy operator spectrum for a scalar linear NRE with one discrete delay and periodic coefficients, showing $\sigma( U_{\mathbb{C}} ) = \overline{f(\mathbb{R})}$ and detailing the point, residual, and empty continuous spectrum components; a non-compactness result is also established. A pseudospectral collocation discretization is described to approximate the evolution operator spectrum, and experiments across constant, piecewise, and periodic delay profiles corroborate the theory by illustrating convergence toward the spectral set $\overline{f(\mathbb{R})}$ (and include $0$ from discretization). The paper further explores linear systems with one delay and scalar two-delay cases, providing explicit eigenvalue formulas and confirming convergence behavior in numerical tests. Overall, the work lays theoretical and numerical groundwork for stability analysis of NREs and motivates future convergence proofs and extensions to multi-delay and system settings.

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.

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

TL;DR

This work analyzes the spectra of evolution operators for a class of neutral renewal equations (NREs) with a focus on stability of equilibria and periodic solutions. It develops a theoretical characterization of the monodromy operator spectrum for a scalar linear NRE with one discrete delay and periodic coefficients, showing and detailing the point, residual, and empty continuous spectrum components; a non-compactness result is also established. A pseudospectral collocation discretization is described to approximate the evolution operator spectrum, and experiments across constant, piecewise, and periodic delay profiles corroborate the theory by illustrating convergence toward the spectral set (and include from discretization). The paper further explores linear systems with one delay and scalar two-delay cases, providing explicit eigenvalue formulas and confirming convergence behavior in numerical tests. Overall, the work lays theoretical and numerical groundwork for stability analysis of NREs and motivates future convergence proofs and extensions to multi-delay and system settings.

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.
Paper Structure (8 sections, 9 theorems, 39 equations, 6 figures)

This paper contains 8 sections, 9 theorems, 39 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Theoretical results
  4. Discretization of the evolution operators
  5. Experiments
  6. Linear systems with one delay
  7. Linear scalar equations with two delays
  8. Finally, a perspective on convergence

Key Result

Lemma 1

$U$ is the multiplication operator by $f$, i.e., $U\phi = f \phi$ for each $\phi \in Y$.

Figures (6)

  • Figure 1: Spectra of $U_{\mathbb{C}}$ with $f = f_i$ for $i \in \{1,2,3,4\}$ (see \ref{['f1', 'f2', 'f3', 'f4']}), computed with $L=1$ and $M=30$.
  • Figure 2: Errors on $\lambda$ in the spectrum of $U_{\mathbb{C}}$ for $f = f_3$ in \ref{['f3']} and $f = f_4$ in \ref{['f4']}, computed with $L=1$. The errors are the absolute errors on the approximated eigenvalue closest to $\lambda$. Reference dashed lines show $M^{-2}$.
  • Figure 3: Spectra of $U_{\mathbb{C}}$ with $f = f_3$ in \ref{['f3']} and $f = f_4$ in \ref{['f4']}, computed with $L=1$ and varying $M$. Recall that the spectra are real.
  • Figure 4: Spectrum of $U_{\mathbb{C}}$ for \ref{['nre-system']} with $A$ defined by \ref{['matrixstar']} computed with $L=1$ and $M = 30$ (left) and errors on $\lambda = -1$ varying $M$ (right). The errors are the absolute errors on the approximated eigenvalue closest to $\lambda$. The reference dashed line shows $M^{-2}$.
  • Figure 5: Spectra of $U_{\mathbb{C}}$ computed with $L=1$ and $M=30$ (a,c), compared with the union of the spectra of $A(t)$ for $t$ varying on a uniform grid of $100$ points in $[0, 1]$ (b,d), for $A(t) = A_1(t)$ (a,b) and $A(t) = A_2(t)$ (c,d).
  • ...and 1 more figures

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • Theorem 6
  • ...and 8 more