Reduction of Hopf bifurcation problems with symmetries
type: Item
instance of: scholarly article
title: Reduction of Hopf bifurcation problems with symmetries
zbMATH Open document ID: 0976.34037
author: Jürgen Knobloch
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2000-09-28T00:00:00Z
review text: Here, the Hopf bifurcation is studied in the following family of \(\Gamma\)-invariant differential equations: \(\dot x=f(x,\lambda)\), where \(f: \mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}^n\) is of class \(C^\infty\), \(f(0,\lambda)=0\) for all \(\lambda\) and the matrix \(A=D_1f(0,0)\) has nonzero eigenvalues of the form \(i\mu\) for some integers \(\mu\in Z\). Further \(\Gamma\subset 0(n)\). This bifurcation problem is formulated as an operator equation in spaces of \(2\pi\)-periodic functions \(F(x, \sigma,\lambda)=0\) for \((x,\sigma,\lambda)\) close to \((0,1,0)\) with \(F:C^1_{2\pi}\times \mathbb{R}^1\times \mathbb{R}^m\to C^0_{2\pi}\), with \(F(x,\sigma,\lambda)(t)=-\sigma\dot x(t)+f(x(t),\lambda)\). The paper is well written and organized.
reviewed by: A. Klíč
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 37G10
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 37G40
zbMATH DE Number: 1512044
zbMATH Keywords: equivariant vector fields
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: Lyapunov-Schmidt reduction
zbMATH Keywords: normal forms
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1512044