Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation
type: Item
instance of: scholarly article
title: Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation
zbMATH Open document ID: 0617.34071
DOI: 10.1007/BF01790539
author: John Mallet-Paret
author: Roger D. Nussbaum
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1986-01-01T00:00:00Z
review text: An extensive treatment of the parametrized differential-delay equation \((1)\quad \dot x(t)=-\lambda x(t)+\lambda f(x(t-1)),\lambda >0\) is presented with particular emphasis on the bifurcation behaviour as \(\lambda\) is varied from 0 to \(\infty\). In the first section topological fixed point index theory is used to show that the equation (1) has a global continuum of ''slowly oscillating'' periodic solutions (for \(\lambda\in [0,\infty))\). Such solutions are characterized by the fact that the period of the solution is greater than twice the delay. In section 2 similar results are obtained for rapidly oscillating periodic solutions. The period of oscillation of solutions as \(\lambda\to \infty\) is discussed in section 3 and, in particular, it is shown that, under certain conditions, we have \(1<q<1+C/\lambda\), \(2<\bar q<2(1+\bar C/\lambda)\) for some constants \(C,\bar C\) where \(q,\bar q\) are the first two positive zeros of the solution. In the final section it is shown that as \(\lambda\to \infty\) the periodic solutions converge uniformly to a square wave away from the points 0,1,2,.... At these points the nature of the solution is derived by considering the ''transition layer equations'' \[ \dot y(t)=y(t)-f(z(t-r)),\quad \dot z(t)=z(t)-f(y(t-r)), \] and it is shown that, under certain circumstances, one can obtain finite overshoot in any neighbourhood of 0 or 1 as \(\lambda\to \infty\).
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 4000467
zbMATH Keywords: parametrized differential-delay equation
zbMATH Keywords: bifurcation
zbMATH Keywords: rapidly oscillating periodic solutions
zbMATH Keywords: transition layer equations
Wikidata QID: Q115393171
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01790539
OpenAlex ID: W2058319209
description: scientific article; zbMATH DE number 4000467