Generic bifurcation of refracted systems
type: Item
instance of: scholarly article
title: Generic bifurcation of refracted systems
zbMATH Open document ID: 1266.34024
author: C. A. Buzzi
author: J. C. Medrado
author: Marco A. Teixeira
published in: Advances in Mathematics
publication date: 2013-02-11T00:00:00Z
review text: Let \(f\) be a smooth function on \(\mathbb{R}^3\), with \(f(0)=0\), and assume that \(0\) is a regular value. Define the sets \(\Sigma=f^{-1}(0)\), \(\Sigma_{+}=\{q|f(q) \geq 0\}\), \(\Sigma_{-}=\{q|f(q) \leq 0\}\). Consider the space of all vector fields \(Z(x,y,z)=X(x,y,z)\) if \((x,y,z)\epsilon \Sigma_{+}\), and \(Z(x,y,z)=Y(x,y,z)\) if \((x,y,z)\epsilon \Sigma_{-}.\) These systems are called refracted systems. The authors consider the space of all fold-fold elliptic systems. These are refracted systems satisfying \(X f(0)=0\), \(X^{2}f(0)<0\) and \(Y^{2}f(0) >0\); the notation \(X f(p)=(\nabla f(p),X(p))\) is used. The authors prove that the set of all structurally stable systems is an open dense set. Moreover, the normal forms of structurally stable systems are given. The authors give a classification of co-dimension one singularities. The main tools are results on embedding diffeomorphism in flows and properties of reversible mappings.
reviewed by: M. A. M. Alwash
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C26
Mathematics Subject Classification ID: 34D30
zbMATH DE Number: 6135762
zbMATH Keywords: discontinuous vector field
zbMATH Keywords: bifurcation
zbMATH Keywords: refracted system
zbMATH Keywords: structurally stable
MaRDI profile type: Publication
full work available at URL: j.aim.2012.11.008
OpenAlex ID: W2002695016
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DOI: 10.1016/J.AIM.2012.11.008
description: scientific article; zbMATH DE number 6135762