Asymptotic stability of linearizations of a planar vector field with a singular point implies global stability
type: Item
instance of: scholarly article
title: Asymptotic stability of linearizations of a planar vector field with a singular point implies global stability
zbMATH Open document ID: 0855.34057
DOI: 10.1007/BF01077471
author: A. A. Glutsuk
publication date: 1997-02-03T00:00:00Z
review text: The author presents a proof of a result already announced in [Usp. Mat. Nauk 49, No. 3, 179-180 (1994; Zbl 0828.34041)] which gives a positive solution to the famous ``global asymptotic stability Jacobian conjecture'' [cf. \textit{G. Meisters} and \textit{C. Olech}, Analyse mathématique et applications, Contrib. Honneur Jacques-Louis Lions, 373-381 (1988; Zbl 0668.34048)]) for \(n = 2\) and reads as follows: Theorem. Let a vector field in \(\mathbb{R}^2\) have a singular point, and let its linearization be asymptotically stable at every point of the phase space. Then the singular point is unique and globally attractive.
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 923586
zbMATH Keywords: global asymptotic stability Jacobian conjecture
zbMATH Keywords: vector field
MaRDI profile type: Publication
cites work: Q3820959
cites work: Q5330830
cites work: On the Kalman problem
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 923586