Controllable Hopf bifurcations of codimension 1 and 2 in nonlinear control systems
type: Item
instance of: scholarly article
title: Controllable Hopf bifurcations of codimension 1 and 2 in nonlinear control systems
zbMATH Open document ID: 1222.37045
author: Denis De Carvalho Braga
author: Luis Fernando Mello
author: Carmen Rocşoreanu
author: Mihaela Sterpu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-04-19T00:00:00Z
review text: Suppose that \(F(x)\) and \(G(x)\) are smooth vector fields with \(F(0)=0\) and \(G(0)=B \neq 0\). Assume that \(DF(0)\) has eigenvalues \(\lambda, \bar{\lambda}\), with \(\lambda = i \omega_0\), where \(\omega_0> 0\). All the other eigenvalues have nonzero real parts. Let \(q\) be an eigenvector corresponding to \(\lambda\) and \(p\) an adjoint eigenvector corresponding to \(\bar{\lambda}\). The vector \(p\) is normalized with respect to \(B\) such that \(\langle p, B \rangle = 1\); \(q\) is normalized with respect to \(p\) such that \(\langle p, q \rangle =1\). Let \( \omega= \langle p, x \rangle\), and \(\mu=(\mu_0,\mu_1,\mu_2,\mu_3)\) be the real control vector. The authors consider the system \[ \dot{x}=F(x)+G(x)\big[2 \operatorname{Re}(\omega)(\mu_0\mu_1+\mu_2\omega \bar{\omega}+\mu_3\omega^2\bar{\omega}^2)\big] . \] It is shown that there exists \(\mu_{2}^{0}\) such that if \(\mu_{2} \neq \mu_{2}^{0}\), then \((0,\mu_0,0,\mu_{2},\mu_{3})\) is a Hopf bifurcation point of codimension 1; it is a transversal point when \(\mu_0\neq 0\). Moreover, there exists \(\mu_3^0\) such that if \(\mu_0\neq 0\) and \(\mu_3\neq \mu_3^0\), then \((0,\mu_0,0,\mu_2^0,\mu_3)\) is a transversal Hopf bifurcation point of codimension 2.
reviewed by: M. A. M. Alwash
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 47N70
zbMATH DE Number: 5880260
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: Bautin bifurcation
zbMATH Keywords: stability
zbMATH Keywords: periodic orbit
zbMATH Keywords: control system
MaRDI profile type: Publication
full work available at URL: j.na.2011.01.033
OpenAlex ID: W2040534670
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DOI: 10.1016/J.NA.2011.01.033
description: scientific article; zbMATH DE number 5880260