Global geometry of the stable regions for two delay differential equations
type: Item
instance of: scholarly article
title: Global geometry of the stable regions for two delay differential equations
zbMATH Open document ID: 0787.34062
author: Jack K. Hale
author: Wenzhang Huang
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-05-15T00:00:00Z
review text: The stable region for the two delay differential equation \[ \dot x(t) +ax(t)+bx(t-r)+cx(t-\sigma)=0,\quad r \geq 0,\;\sigma \geq 0, t \geq 0 \] is investigated. The previously obtained numerical results show that the boundary of a stable region may be a very complicated curve which has infinitely many kinks, therefore some interesting questions arise: What is the asymptotical behavior of the boundary of a stable region? Can it become chaotic? In the paper a complete geometrical description of the stable region in the \(r-\sigma\) plane is given. In particular it is proved that: (i) If a stable region is unstable, then its boundary will approach a straight line parallel to the \(r\)-axis or \(\sigma\)-axis as \(r+\sigma \to \infty\). (ii) If a half line in the first quadrant of the \(r-\sigma\) plane contains an unstable point, then the intersection of this line and the boundary of the stable region contains at most finitely many points and eventually the half line leaves the stable region.
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 484022
zbMATH Keywords: global geometry for delay differential equations
zbMATH Keywords: stable region
zbMATH Keywords: two delay differential equation
zbMATH Keywords: asymptotical behavior
zbMATH Keywords: chaotic
Wikidata QID: Q115395440
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1312
OpenAlex ID: W1992572795
DOI: 10.1006/JMAA.1993.1312
description: scientific article; zbMATH DE number 484022