Singular perturbation problems for a system of differential-difference equations. I
type: Item
instance of: scholarly article
title: Singular perturbation problems for a system of differential-difference equations. I
zbMATH Open document ID: 0814.34068
author: Shui-Nee Chow
author: Wenzhang Huang
published in: Journal of Differential Equations
publication date: 1994-10-24T00:00:00Z
review text: The a system of singularly perturbed delay differential-difference equations of the form \[ \varepsilon\dot x(t)= -Ax(t)+ g(y(t), \lambda),\;y(t)= -f(x(t- 1), y(t- 1), \lambda)\tag{1} \] is considered. The problem is to study the relation between (1) for \(\varepsilon\) small enough and the difference equation obtained formally setting \(\varepsilon= 0\), \[ y(t)= f(A^{-1} g(y(t- 1), \lambda), y(t- 1), \lambda).\tag{2} \] In the first part of the article is studied the relation between the stability and the eigenvalues of the infinitesimal generator of the linear system \[ \varepsilon\dot x(t)= -Ax(t)+ By(t),\;y(t)= Gx(t- 1)+ Fy(t- 1)\tag{3} \] and the results are applied to investigate the stability of the equilibria of the so-called ``ring cavity equation''. In the second part, a discussion of the existence of a heteroclinic orbit corresponding to the transition layers of the singularly perturbed equations (1) is done.
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 665714
zbMATH Keywords: ring cavity equation
zbMATH Keywords: system of singularly perturbed delay differential- difference equations
zbMATH Keywords: stability
zbMATH Keywords: heteroclinic orbit
zbMATH Keywords: transition layers
reviewed by: Vasile Dragan
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1105
OpenAlex ID: W1976615210
DOI: 10.1006/JDEQ.1994.1105
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description: scientific article; zbMATH DE number 665714