Regions of stability for a linear differential equation with two rationally dependent delays

Joseph M. Mahaffy; Timothy C. Busken

doi:10.3934/dcds.2015.35.4955

Article Contents

2015, Volume 35, Issue 10: 4955-4986. Doi: 10.3934/dcds.2015.35.4955

This issue Previous Article Wavefronts of a stage structured model with state--dependent delay Next Article Singly periodic free boundary minimal surfaces in a solid cylinder of $\mathbb{R}^3$

Regions of stability for a linear differential equation with two rationally dependent delays

Joseph M. Mahaffy^1, and
Timothy C. Busken^2,

1.
Department of Mathematics and Statistics, Nonlinear Dynamical Systems Group, Computational Sciences Research Center, San Diego State University, San Diego, CA 92182-7720
2.
Department of Mathematics, Grossmont College, El Cajon, CA 92020

Received: June 30, 2013

Revised: December 31, 2014

Published: September 2015

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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 ratio of $1/3$ asymptotically produces a stability region about 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.

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Mathematics Subject Classification: Primary: 37C75, 37G15; Secondary: 39B82.

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References

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