Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback
type: Item
instance of: scholarly article
title: Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback
zbMATH Open document ID: 1232.37041
author: Li Zhang
author: Huailei Wang
author: Haiyan Hu
published in: Science China. Technological Sciences
publication date: 2010-08-13T00:00:00Z
review text: This paper deals with a van der Pol oscillator with delayed state feedback \[ x''(t)-\mu \omega_{0}(1-\delta x(t)^2)x'(t)+\omega_{0}^2 x(t)=u\omega_{0}^2x(t-\tau)+v\omega_{0}x'(t-\tau), \tag{1} \] where \(\mu, \omega_{0}\) and \(\delta\) are positive real constants. The study is confined to the case of small damping. The authors first consider the stability switches of the trivial equilibrium. Then center manifold reduction and the normal form method are used to study the Hopf bifurcation. To study the global diagrams of the bifurcated periodic motions as the delay varies to a large scale, they use the method of multiple scales to obtain the global view of Hopf bifurcations with respect to time delay. Both the analysis results of Hopf bifurcations and global view of those bifurcations are validated via a collocation scheme implemented on the DDE-Biftool.
reviewed by: Rong Yuan
Mathematics Subject Classification ID: 37L10
Mathematics Subject Classification ID: 37L15
Mathematics Subject Classification ID: 34K18
zbMATH DE Number: 5770135
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: global view
zbMATH Keywords: center manifold reduction
zbMATH Keywords: method of multiple scales
zbMATH Keywords: collocation scheme
describes a project that uses: DDE-BIFTOOL
MaRDI profile type: Publication
full work available at URL: s11431-010-0073-9
OpenAlex ID: W1969030481
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DOI: 10.1007/S11431-010-0073-9
description: scientific article; zbMATH DE number 5770135