A stability criterion for systems with delay in one critical case
type: Item
instance of: scholarly article
title: A stability criterion for systems with delay in one critical case
zbMATH Open document ID: 0920.34063
DOI: 10.1007/BF02674133
author: L. Z. Fishman
published in: Siberian Mathematical Journal
publication date: 1999-05-10T00:00:00Z
review text: The author studies the stability of equilibria for systems with delay described by the first-order equation with delay \[ \dot x(t) = ax(t) + b(t)x(t-\tau) + F\bigl(x(t),x(t-\tau)\bigr), \] where \(x\) is a scalar, \(a\) and \(b\) are constants, and \(\tau >0\). It is supposed that the analytic function \(F\) expands in a neighborhood at the point \(x_1=x_2=0\) in a series starting with terms of at least second order. The value similar to the first Lyapunov value for the equation is assumed to be equal to \(0\). The author obtains an expression for the value similar to the second Lyapunov value for the equation in terms of the coefficients of the equation. Using this expression, the author gives a criterion for the stability of the equilibrium \(x=0\) of the equation.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1285588
zbMATH Keywords: system of differential equations with delay
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: stability criterion
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
cites work: Q3937701
cites work: Q3810932
cites work: Q3992752
cites work: Q3789450
cites work: Q3497433
cites work: Q4349202
cites work: Q4350303
full work available at URL: https://doi.org/10.1007/bf02674133
OpenAlex ID: W2313193782
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description: scientific article; zbMATH DE number 1285588