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About this book

This book presents the authors' recent work on the numerical methods for the stability analysis of linear autonomous and periodic delay differential equations, which consist in applying pseudospectral techniques to discretize either the solution operator or the infinitesimal generator and in using the eigenvalues of the resulting matrices to approximate the exact spectra. The purpose of the book is to provide a complete and self-contained treatment, which includes the basic underlying mathematics and numerics, examples from population dynamics and engineering applications, and Matlab programs implementing the proposed numerical methods. A number of proofs is given to furnish a solid foundation, but the emphasis is on the (unifying) idea of the pseudospectral technique for the stability analysis of DDEs. It is aimed at advanced students and researchers in applied mathematics, in dynamical systems and in various fields of science and engineering, concerned with delay systems. A relevant feature of the book is that it also provides the Matlab codes to encourage the readers to experience the practical aspects. They could use the codes to test the theory and to analyze the performances of the methods on the given examples. Moreover, they could easily modify them to tackle the numerical stability analysis of their own delay models.

Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract
During the last decades, the interest for systems of differential equations depending on the past history has been increasing.
Dimitri Breda, Stefano Maset, Rossana Vermiglio

Theory

Frontmatter

Chapter 2. Notation and Basics

Abstract
The aim of this chapter is to introduce basic notation and definitions, together with solvability theorems for Cauchy problems for DDEs and a remark on continuous dependence on the data.
Dimitri Breda, Stefano Maset, Rossana Vermiglio

Chapter 3. Stability of Linear Autonomous Equations

Abstract
We focus our attention on linear autonomous DDEs and on the analysis of the stability properties of the zero solution.
Dimitri Breda, Stefano Maset, Rossana Vermiglio

Chapter 4. Stability of Linear Periodic Equations

Abstract
The central subject of this chapter is the stability analysis of the zero solution of linear periodic DDEs.
Dimitri Breda, Stefano Maset, Rossana Vermiglio

Numerical Analysis

Frontmatter

Chapter 5. The Infinitesimal Generator Approach

Abstract
The IG approach consists in approximating the space \(X\) with a finite dimensional linear space \(X_{M}\), called the discretization of \(X\) of index \(M\), and the infinitesimal generator \(\fancyscript{A}\) with a finite dimensional linear operator \(\fancyscript{A}_{M}:X_{M}\rightarrow X_{M}\), called the discretization of \(\fancyscript{A}\) of index \(M\)
Dimitri Breda, Stefano Maset, Rossana Vermiglio

Chapter 6. The Solution Operator Approach

Abstract
In the SO approach, the eigenvalues of an evolution operator.
Dimitri Breda, Stefano Maset, Rossana Vermiglio

Implementation and Applications

Frontmatter

Chapter 7. MATLAB Implementation

Abstract
The book is provided with the following three MATLAB codes:
  • myDDE.m;
  • eigAM.m;
  • eigTMN.m;
freely available, [48].
Dimitri Breda, Stefano Maset, Rossana Vermiglio

Chapter 8. Applications

Abstract
All the following tests and applications refer to the notation and structure of model (7.​2), i.e., from the user’s point of view as explained in Sect. 7.​1.
Dimitri Breda, Stefano Maset, Rossana Vermiglio

Backmatter

Additional information