Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations
type: Item
instance of: scholarly article
title: Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations
zbMATH Open document ID: 0929.65054
DOI: 10.1023/A:1018986817622
author: Koen Engelborghs
author: Dirk Roose
published in: Advances in Computational Mathematics
publication date: 2000-01-11T00:00:00Z
review text: This paper is concerned with the stability of steady state solutions (fixed points) of systems of time dependent delay differential equations \[ \dot x= f(x(t), x(t-\tau_1),\dots, x(t-\tau_m)), \] with \(x\in\mathbb{R}^n\), and one or several fixed delays \(\tau_i\in \mathbb{R}^+_0\), \(j= 1,\dots, m\). The authors present a robust numerical method to compute the rightmost roots of the nonlinear characteristic equation to the fixed point under consideration. These roots determine the stability of the fixed point, provide inside into the system's behaviour, and can be used for bifurcation detection and indirect calculation of bifurcation points. Examples are presented to demonstrate the usefulness of the method. (One of the delay differential systems is a discrete form of a model equation for the flow of a viscoelastic fluid with fading memory).
Mathematics Subject Classification ID: 65L15
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 34L16
Mathematics Subject Classification ID: 34K12
zbMATH DE Number: 1309405
zbMATH Keywords: Hopf bifurcations
zbMATH Keywords: stability
zbMATH Keywords: steady state solutions
zbMATH Keywords: delay differential equations
zbMATH Keywords: bifurcation
reviewed by: Waldemar Velte
describes a project that uses: JDQR
describes a project that uses: JDQZ
MaRDI profile type: Publication
full work available at URL: a:1018986817622
OpenAlex ID: W305513724
description: scientific article; zbMATH DE number 1309405