Global asymptotic stability for differentiable vector fields of \(\mathbb R^2\)
type: Item
instance of: scholarly article
title: Global asymptotic stability for differentiable vector fields of \(\mathbb R^2\)
zbMATH Open document ID: 1126.37306
author: D. Kharzeev
published in: Journal of Differential Equations
publication date: 2005-01-11T00:00:00Z
review text: (a) Let \(X:\mathbb R^2\to\mathbb R^2\) be a differentiable map (not necessarily \(C^1\)) and let \(\text{Spec}(X)\) be the set of (complex) eigenvalues of the derivative \(DX_p\) when \(p\) varies in \(\mathbb E^2\). If, for some \(\varepsilon>0\), \(\text{Spec}(X)\cap [0,\varepsilon)= \emptyset\) then \(X\) is injective. (b) Let \(X:\mathbb R^2\to\mathbb R^2\) be a differentiable vector field such that \(X(0)=0\) and \(\text{Spec}(X)\subset \{z\in\mathbb C:\Re(z)<0\}\). Then, for all \(p\in\mathbb R^2\), there is a unique positive trajectory starting at \(p\); moreover the \(\omega\)-limit set of \(p\) is equal to \(\{0\}\).
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 2126465
zbMATH Keywords: Planar vector fields
zbMATH Keywords: Global injectivity
zbMATH Keywords: Asymptotic stability
MaRDI profile type: Publication
full work available at URL: j.jde.2004.04.015
OpenAlex ID: W2038166261
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DOI: 10.1016/J.JDE.2004.04.015
description: scientific article; zbMATH DE number 2126465