Stability region and critical delay

TL;DR

This paper investigates the stability regions of a simple delay differential equation by analyzing its characteristic roots, combining analytical and graphical methods to enhance understanding of delay effects on stability.

Contribution

It introduces a combined analytical and geometric approach to determine stability regions for a transcendental delay differential equation, extending previous results and aiding analysis of higher-dimensional cases.

Findings

Provided a new proof of existing stability results

Developed a combined method for stability analysis

Enhanced understanding of delay effects on stability

Abstract

The location of roots of the characteristic equation of a linear delay differential equation (DDE) determines the stability of the linear DDE. However, by its transcendency, there is no general criterion on the contained parameters for the stability. Here we mainly concentrate on the study of a simple transcendental equation $(*)$ $z + a - w \mathrm{e}^{-\tau z} = 0$ with coefficients of real $a$ and complex $w$ and a delay parameter $\tau > 0$ to tackle this transcendency brought by delay. The consideration is twofold: (i) to give the stability region in the parameter space for Eq.~$(*)$ by using the critical delay and (ii) to compare this with a graphical method (so-called the method of D-partitions) by combining with the delay sequence obtained by conditions for purely imaginary roots. By (i), we obtain another proof of Hayes' and Sakata's results, which reveals the nature of imaginary $w$ case in Eq.~$(*)$. By (ii), we propose a method combining the analytic one and geometric one. This combination is important because it will be helpful in studying characteristic equations having higher-dimensional parameters.