Period doubling in singularly perturbed delay equations
type: Item
instance of: scholarly article
title: Period doubling in singularly perturbed delay equations
zbMATH Open document ID: 0817.34040
author: Jack K. Hale
author: Wenzhang Huang
published in: Journal of Differential Equations
publication date: 1995-08-07T00:00:00Z
review text: The authors consider periodic solutions of the delay differential equation \((\epsilon> 0)\) \(\epsilon x'(t)= -x(t)+ f(x(t- 1),\lambda)\) assuming that \(\lambda= 0\) corresponds to a generic period doubling for the map \(x\to f(x, 0)\). Specifically, it is assumed that \[ f(x, \lambda)= -(1+ \lambda) x+ ax^ 2+ bx^ 3+ o(x^ 3),\quad x\to 0, \] where the constant \(a\), \(b\) satisfy \(\beta= a^ 2+ b\neq 0\). Under this assumption on \(f\), and if \(\lambda\beta> 0\) there are periodic points \(d_{1\lambda}\), \(d_{2\lambda}\) of period 2 of the map \(f(.,\lambda)\) and such that \(d_{1\lambda}, d_{2\lambda}\to 0\) as \(\lambda\to 0\). If \(\beta> 0\) \((<0)\) the bifurcation is said to be supercritical (subcritical). The supercritical case with \(a= 0\) has earlier been investigated by \textit{S.-N. Chow} and the authors [Proc. Soc. Edinb., Sect. A (to appear)]. In the present paper the supercritical case with \(a\neq 0\) and the subcritical case are analyzed. The result proved is the same: Theorem: Let \(f(x,\lambda)\) be as above. Then there is a neighborhood \(U\) of \((0,0)\) in the \((\lambda, \epsilon)\) plane and a sectorial region \(S\) in \(U\) such that, if \((\lambda, \epsilon)\in U\), then there is a periodic solution \(x_{\lambda, \epsilon}\) of the above equation with period \(2\tau(\lambda, \epsilon)= 2+ 2\epsilon+ O(|\epsilon| (|\lambda|+ |\epsilon|))\) as \((\lambda, \epsilon)\to (0,0)\) if and only if \((\lambda, \epsilon)\in S\). This solution is unique.
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 696878
zbMATH Keywords: periodic solutions
zbMATH Keywords: delay differential equation
zbMATH Keywords: period doubling
zbMATH Keywords: bifurcation
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1138
OpenAlex ID: W2002214649
DOI: 10.1006/JDEQ.1994.1138
description: scientific article; zbMATH DE number 696878