Computing multiple pitchfork bifurcation points
type: Item
instance of: scholarly article
title: Computing multiple pitchfork bifurcation points
zbMATH Open document ID: 0890.65050
DOI: 10.1007/BF02684441
published in: Computing
publication date: 1998-07-06T00:00:00Z
review text: For the parameter dependent nonlinear equation \(F(x,\lambda) =0\), \(F: \mathbb{R}^n \times \mathbb{R}^1 \to\mathbb{R}^n\), the generically important case \(\text{rank} \partial_x E(x^*, \lambda^*) =n-1\) is investigated. In a neighborhood of such pitchfork bifurcation point \((x^*, \lambda^*)\) of multiplicity \(p\geq 1\) the Lyapunov-Schmidt branching equation has the normal form \(g(\xi,\mu) =\pm \xi^{2+p} \pm \mu \xi=0\). It is shown that such points satisfy a minimally extended system \(G(y) =0\), \(G: \mathbb{R}^{n+2} \to \mathbb{R}^{n+2}\), where the dimension \(n+2\) is independent of \(p\). For solving this system a two-stage Newton-type method is suggested. The method is illustrated by numerical experiments.
Mathematics Subject Classification ID: 65H17
zbMATH DE Number: 1112331
zbMATH Keywords: singular points
zbMATH Keywords: minimally extended systems
zbMATH Keywords: parameter dependent nonlinear equation
zbMATH Keywords: pitchfork bifurcation point
zbMATH Keywords: Lyapunov-Schmidt branching equation
zbMATH Keywords: two-stage Newton-type method
zbMATH Keywords: numerical experiments
author: G. Pönisch
author: U. Schnabel
author: Hubert Schwetlick
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02684441
OpenAlex ID: W1968629662
description: scientific article; zbMATH DE number 1112331