Regions of Stability for a Linear Differential Equation with Two Rationally Dependent Delays

TL;DR

This paper analyzes the stability regions of a linear differential equation with two delays, revealing that rationally dependent delays significantly enlarge the stability region and providing geometric methods to determine stability boundaries.

Contribution

It introduces a geometric approach to analyze stability regions for equations with two delays, especially when delays are rationally dependent, and characterizes asymptotic stability shapes for specific delay ratios.

Findings

Rationally dependent delays increase stability region size.

Delay ratio 1/n leads to predictable asymptotic stability shapes.

Delay ratio 1/3 yields a 44.3% larger stability region asymptotically.

Abstract

Stability analysis is performed for a linear differential equation with two delays. Geometric arguments show that when the two delays are rationally dependent, then the region of stability increases. When the ratio has the form 1/n, this study finds the asymptotic shape and size of the stability region. For example, a delay ration of 1/3 asymptotically produces a stability region 44.3% larger than any nearby delay ratios, showing extreme sensitivity in the delays. The study provides a systematic and geometric approach to finding the eigenvalues on the boundary of stability for this delay differential equation. A nonlinear model with two delays illustrates how our methods can be applied.