Abstract Hopf bifurcation theorem and further extensions via second variation
type: Item
instance of: scholarly article
title: Abstract Hopf bifurcation theorem and further extensions via second variation
zbMATH Open document ID: 1218.47084
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-08-26T00:00:00Z
review text: The paper is devoted to the bifurcation problem in Banach spaces of the form \(F(\mu, x)=0\), where \(F\) is a \(C^2\) map and \(\frac{\partial F}{\partial x}(\mu _0, x_0)\) is a Fredholm operator of index \(0\). While the abstract results are mainly cited from the earlier article [\textit{A. V. Arutyunov, A. F. Izmailov} and \textit{V. Jaćimović}, ``New bifurcation theorems via the second-order optimality conditions'', J. Math. Anal. Appl. 359, No.~2, 752--764 (2009; Zbl 1168.47050)], the author considers especially one corollary and gives three examples of its applications in ordinary and partial differential problems. More precisely, he investigates a special type of bifurcation and assumes that the parameter space and the kernel of \(\frac{\partial F}{\partial x}(\mu _0, x_0)\) have the same (finite) dimensions. As he proves, such methods are possible only when these dimensions are equal to \(1\), \(2\), \(4\) or \(8\), which is related to the Hopf fibrations of spheres.
Mathematics Subject Classification ID: 47J15
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 55Q25
zbMATH DE Number: 5774957
zbMATH Keywords: bifurcation
zbMATH Keywords: Fredholm operator
zbMATH Keywords: Hopf map
zbMATH Keywords: periodic solution
author: Vladimir Jacimovic
reviewed by: Dorota Gabor
MaRDI profile type: Publication
full work available at URL: j.na.2010.06.014
OpenAlex ID: W2041381875
cites work: Dual variational methods in critical point theory and applications
cites work: Q4774620
cites work: Bifurcation from simple eigenvalues
cites work: Q4002249
cites work: New bifurcation theorems via the second-order optimality conditions
cites work: Vector fields on spheres
DOI: 10.1016/J.NA.2010.06.014
description: scientific article; zbMATH DE number 5774957