Bifurcations via singular value decompositions
type: Item
instance of: scholarly article
title: Bifurcations via singular value decompositions
zbMATH Open document ID: 0674.65028
DOI: 10.1016/0096-3003(88)90139-7
published in: Applied Mathematics and Computation
publication date: 1988-01-01T00:00:00Z
review text: The paper deals with an equation (1) \(F(a,\lambda)=0\) where \(\lambda\in {\mathbb{R}}\), \(a\in {\mathbb{R}}^ m\), F: \({\mathbb{R}}^ m\times {\mathbb{R}}\to {\mathbb{R}}^ m\) and \(\lambda\) has a meaning of a bifurcation parameter. The authors, considering the equation (1) as an equation (2) \(f(x)=0\) where \(f:{\mathbb{R}}^{m+1}\to {\mathbb{R}}\), \(x=(a,\lambda)\), present a method of applying a singular value decomposition to solve the equation (2) at any case: if x is a regular point, a turning point or a bifurcation point. The local analysis of solutions is provided with the help of Lyapunov- Schmidt reduction via the singular value decomposition to decompose the equation (2) into two equations, an auxiliary equation and a bifurcation equation. A method for determining all the tangent vectors at a bifurcation point is proposed. The global numerical analysis of solutions of (2) gives a numerical method of going from one solution of (2) to another solution. The main results are illustrated by an example.
Mathematics Subject Classification ID: 65H17
zbMATH DE Number: 4104271
zbMATH Keywords: singular value decomposition
zbMATH Keywords: regular point
zbMATH Keywords: turning point
zbMATH Keywords: bifurcation point
zbMATH Keywords: Lyapunov-Schmidt reduction
author: Shui-Nee Chow
author: Yun-Qiu Shen
describes a project that uses: IMSL Numerical Libraries
describes a project that uses: EISPACK
describes a project that uses: LINPACK
MaRDI profile type: Publication
full work available at URL: 0096-3003(88)90139-7
OpenAlex ID: W2079177133
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description: scientific article; zbMATH DE number 4104271