Nonlinear systems with fast parametric oscillations
type: Item
instance of: scholarly article
title: Nonlinear systems with fast parametric oscillations
zbMATH Open document ID: 0531.34024
DOI: 10.1016/0022-247X(83)90212-3
author: Richard Bellman
author: Joseph Bentsman
author: S. M. Meerkov
published in: Journal of Mathematical Analysis and Applications
publication date: 1983-01-01T00:00:00Z
review text: This paper discusses a fairly standard method using small parameters with the method of averaging [cf., for example Chapter 5 in \textit{J. K. Hale} ''Ordinary Differential Equations'' (1969; Zbl 0186.409] and applies it to study the effects of rapidly oscillatory time dependence in certain systems of ordinary differential equations on the stability properties of solutions of the equations without such oscillatory time dependence. Two types of specific examples are discussed; the first contains three socalled classical equations of the theory of oscillations: specifically, a Duffing equation, a Rayleigh equation, and a van der Pol equation. The second type are equations arising in certain chemical reactions. Since the Duffing equation discussed is of the form ẍ\(+a\dot x-bx+cx^ 3=0\), where a, b, and c are positive constants, it would be of interest to have mentioned an oscillatory system described by this of this type of equation; most standard examples would lead to an equation of this form but with \(b<0\).
reviewed by: G.Seifert
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 3842293
zbMATH Keywords: small parameters
zbMATH Keywords: method of averaging
zbMATH Keywords: Duffing equation
zbMATH Keywords: Rayleigh equation
zbMATH Keywords: van der Pol equation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3842293