On the number of branches of bifurcation points
type: Item
instance of: scholarly article
title: On the number of branches of bifurcation points
zbMATH Open document ID: 0766.58039
author: Marek Izydorek
author: Sławomir Maciej Rybicki
published in: Kodai Mathematical Journal
publication date: 1993-01-16T00:00:00Z
review text: In this paper the branching behaviour for multiparameter bifurcation is investigated. The main issue is to give the precise number of bifurcating branches. The authors concentrate on the case with two parameters, i.e. let \(f:\mathbb{R}^ n\times\mathbb{R}^ 2\to\mathbb{R}^ n\). Define the nontrivial set of solutions to be those of the form \((x,\lambda)\) with nonzero \(x\). Bifurcation points are trivial zeros in the closure of the set of nontrivial zeros. Using methods of Aoki et al. they compute the precise number of emanating curves. An auxiliary function \(H:\mathbb{R}^ 2\to\mathbb{R}^ 2\) is introduced. It is defined as \(H=(\nabla,\Phi)\), where \(\Phi(\lambda)=\text{det}(D_ xf(0,\lambda)\), and \(\nabla=\text{det} \left({\partial(\lambda^ 2_ 1+\lambda^ 2_ 2,\Phi)\over\partial (\lambda_ 1,\lambda_ 2)}\right)\). The local Brouwer degree of this mapping \(H\) at zero is related to number of branches: 2 deg \(H\) gives this number. The authors define a special class of \(f\)'s for which they are able to compute the degree of \(H\). This class of functions is related to what the authors call regular matrices. They show that (finite sequences of) regular matrices are a big set among (finite sequences of) all matrices. Finally the results are applied to boundary value problems of ODE's and PDE's.
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 92936
zbMATH Keywords: topological degree
zbMATH Keywords: boundary value problems
zbMATH Keywords: multiparameter bifurcation
reviewed by: Reiner Lauterbach
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039531
OpenAlex ID: W2086067258
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DOI: 10.2996/KMJ/1138039531
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description: scientific article; zbMATH DE number 92936