\(C^ p\) singularity theory and heteroclinic bifurcation with a distinguished parameter
type: Item
instance of: scholarly article
title: \(C^ p\) singularity theory and heteroclinic bifurcation with a distinguished parameter
zbMATH Open document ID: 0762.34021
DOI: 10.1016/0022-0396(92)90025-I
author: Stephen Schecter
published in: Journal of Differential Equations
publication date: 1993-01-16T00:00:00Z
review text: A planar dynamical system \(\dot z=h(z,\lambda)\) with one distinguished real parameter \(\lambda\) having for \(\lambda=0\) a hyperbolic saddle \(p_ 0\) and a semihyperbolic equilibrium \(q_ 0\) (a fold) with one negative and one zero eigenvalue of \(Dh(q_ 0)\) is considered. The bifurcation being investigated is the existence of a heteroclinic orbit \(\Gamma\) connecting \(p_ 0\) and \(q_ 0\) tangential to the eigendirection of the negative eigenvalue of \(q_ 0\). Associated with this situation is a map \(g(x,\lambda)=(g_ 1(x,\lambda),g_ 2(x,\lambda))\) where \(x\) is a coordinate along the center subspace of \(q_ 0\), and where zeros of \(g_ 2\) correspond to equilibria near \(q_ 0\) while common zeros of \(g_ 1\) and \(g_ 2\) correspond to heteroclinic bifurcations. The aim are normal forms \(f=(f_ 1,f_ 2)\), recognition criteria and universal unfoldings of \(g=(g_ 1,g_ 2)\). Since even for \(h\in C^ \infty\), \(g\) is only \(C^ p\) with finite \(p\), a new \(C^ p\) singularity theory is stated which enables the author to present several normal forms and universal unfoldings for low codimensional heteroclinic bifurcations together with bifurcation diagrams and phase portraits. It is indicated that this kind of heteroclinic bifurcation being considered is of relevance for shock solutions for a system of two conservation laws in one space dimension.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37G05
zbMATH DE Number: 93319
zbMATH Keywords: planar dynamical system
zbMATH Keywords: heteroclinic orbit
zbMATH Keywords: heteroclinic bifurcations
zbMATH Keywords: normal forms
zbMATH Keywords: recognition criteria
zbMATH Keywords: universal unfoldings
zbMATH Keywords: \(C^ p\) singularity theory
zbMATH Keywords: bifurcation diagrams
zbMATH Keywords: phase portraits
zbMATH Keywords: shock solutions
zbMATH Keywords: system of two conservation laws in one space dimension
reviewed by: Bodo Werner
MaRDI profile type: Publication
cites work: Q3770650
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description: scientific article; zbMATH DE number 93319