On necessary conditions for global asymptotic stability of equilibrium for the Liénard equation
type: Item
instance of: scholarly article
title: On necessary conditions for global asymptotic stability of equilibrium for the Liénard equation
zbMATH Open document ID: 1270.34159
DOI: 10.1134/S0001434613010082
author: A. O. Ignat'ev
author: V. V. Kirichenko
published in: Mathematical Notes
publication date: 2013-04-26T00:00:00Z
review text: The authors consider the Liénard equation \[ x''(t) + f({x}'(t)){x}'(t) + g(x(t)) = 0,\;g(0) = 0. \tag{1} \] In [\textit{C. Qian}, Nonlinear Anal., Theory Methods Appl. 22, No. 7, 823--833 (1994; Zbl 0801.34055)], necessary and sufficient conditions for the global asymptotic stability of the zero solution of equation (1) were obtained under the condition \[ x\int_0^x f(s)\,ds \geq 0. \tag{2} \] In [loc. cit.], also, the following problem was posed. Determine whether condition (2) is a necessary condition for the global asymptotic stability of the zero solution of the Liénard equation (1). The present paper answers this question, and the answer is negative, i.e., condition (2) is not necessary.
reviewed by: Cemil Tunç
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 6158163
zbMATH Keywords: Liénard differential equation
zbMATH Keywords: global asymptotic stability
MaRDI profile type: Publication
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full work available at URL: s0001434613010082
OpenAlex ID: W2135950109
description: scientific article; zbMATH DE number 6158163