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Asymptotic stability and stability switching for a system of delay differential equations

Wataru Saito, Ikki Fukuda

TL;DR

This work addresses the problem of determining when the zero solution of a linear delay differential system with coupled components is asymptotically stable. By performing a detailed spectral analysis of the characteristic equation $G(\lambda)=0$, the authors derive necessary and sufficient conditions that capture stability and stability switching as the delay parameter $\tau$ varies. The main contribution is a complete characterization that splits into cases based on the sign of $bc$ and provides explicit formulas for critical delays and frequencies, enabling precise prediction of stability intervals. The results have potential implications for nonlinear extensions in applications such as population dynamics, neural networks, and traffic flow, where delayed couplings commonly arise.

Abstract

In this paper, we consider the asymptotic stability for a system of linear delay differential equations. By analysing of the characteristic equation in detail, we have established the necessary and sufficient condition for the asymptotic stability for the zero solution of the system including the stability switching which describe the transition between stability and instability.

Asymptotic stability and stability switching for a system of delay differential equations

TL;DR

This work addresses the problem of determining when the zero solution of a linear delay differential system with coupled components is asymptotically stable. By performing a detailed spectral analysis of the characteristic equation , the authors derive necessary and sufficient conditions that capture stability and stability switching as the delay parameter varies. The main contribution is a complete characterization that splits into cases based on the sign of and provides explicit formulas for critical delays and frequencies, enabling precise prediction of stability intervals. The results have potential implications for nonlinear extensions in applications such as population dynamics, neural networks, and traffic flow, where delayed couplings commonly arise.

Abstract

In this paper, we consider the asymptotic stability for a system of linear delay differential equations. By analysing of the characteristic equation in detail, we have established the necessary and sufficient condition for the asymptotic stability for the zero solution of the system including the stability switching which describe the transition between stability and instability.

Paper Structure

This paper contains 3 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Proof of the Main Theorem

Key Result

Theorem 2.1

Suppose that $\alpha^{2}\neq a^{2}$ when $bc<0$. The zero solution of the system $(1.1)$ is asymptotically stable if and only if any one of the following conditions holds: $\rm{(i)}$$bc\ge 0$ and and where $\beta:=\sqrt{bc}$. $\rm{(ii)}$$bc<0$ and $\rm{(ii-1)}$$\alpha^{2}>a^{2}$ and $\rm{(ii-2)}$$\alpha^{2}<a^{2}$ and $a+\alpha>0$, where $d:=\sqrt{-bc}$, $D:=\alpha^{2}-a^{2}$, $E:=-\frac{a}{\al

Theorems & Definitions (17)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • ...and 7 more