Multiple bifurcation analysis in a NDDE arising from van der Pol's equation with extended delay feedback
type: Item
instance of: scholarly article
title: Multiple bifurcation analysis in a NDDE arising from van der Pol's equation with extended delay feedback
zbMATH Open document ID: 1276.34059
author: Ben Niu
author: Weihua Jiang
published in: Nonlinear Analysis. Real World Applications
publication date: 2012-12-28T00:00:00Z
review text: Consider the neutral differential delay equation \[ \begin{multlined} \ddot x-\mu\ddot x(t- \tau)+ \varepsilon(x^2- 1)\dot x- \mu\varepsilon[x^2(t-\tau)- 1]\dot x(t-\tau)+\\ x-\mu x(t-\tau)=\varepsilon k(1-\mu) x(t)+ g(x)-\mu g(x(t-\tau))\end{multlined}\tag{\(*\)} \] with \(0<\mu< 1\), \(\varepsilon> 0\), \(k> 0\), \(g(0)= g'(0)= 0\), \(g''(0)= 2\kappa_1\), \(g'''(0)= 6\kappa_2\). The authors study fold bifurcation and Hopf bifurcation in \((*)\). Their main focus is on codimension two bifurcations (Bogdanov-Takens, Hopf-fold, Hopf-Hopf). The main tools are center manifold reduction and normal form theory. They establish three periodic solutions near the Bogdanov-Takens point. Numerical solutions show three stable limit cyclies near the Hopf-pitchfork bifurcation point and a stable torus near the Hopf-Hopf bifurcation point. This paper is an important contribution to codimension-two bifurcations in neutral delay-differential equations.
Mathematics Subject Classification ID: 34K18
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K17
Mathematics Subject Classification ID: 34K19
zbMATH DE Number: 6118980
zbMATH Keywords: neutral equation
zbMATH Keywords: codimension-two bifurcation
zbMATH Keywords: periodic solution
zbMATH Keywords: extended delay feedback
zbMATH Keywords: van der Pol's equation
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2012.07.028
OpenAlex ID: W2011338968
DOI: 10.1016/J.NONRWA.2012.07.028
description: scientific article; zbMATH DE number 6118980