Periodic solutions of second-order nonlinear difference equations containing a small parameter. III: Perturbation theory
type: Item
instance of: scholarly article
title: Periodic solutions of second-order nonlinear difference equations containing a small parameter. III: Perturbation theory
zbMATH Open document ID: 0592.39005
DOI: 10.1016/0016-0032(86)90054-2
author: Ronald E. Mickens
published in: Journal of the Franklin Institute
publication date: 1986-01-01T00:00:00Z
review text: The paper is the continuation of two previous ones, concerning the same subject and by the same author [ibid. 316, 273-277 (1983; Zbl 0529.39002) and ibid. 320, 169-174 (1985; Zbl 0589.39004)]. Here the purpose is to construct first and second approximations to the solutions of the following class of nonlinear difference equations \[ x_{k+1}-(2 \cos \phi)x_ k+x_{k-1}=\epsilon f, \] where \(\phi\) is a given constant; f is a polynomial function of \((x_{k+1},x_ k,x_{k-1})\); \(\epsilon\) is a small positive parameter \((0<\epsilon \ll 1)\) and \(\{\) \(k\}\) is the set of positive integers. The techniques exploited are the perturbation techniques of the Lindstedt-Poincaré method.
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 3951177
zbMATH Keywords: periodic solutions
zbMATH Keywords: second-order
zbMATH Keywords: small parameter
zbMATH Keywords: nonlinear difference equations
zbMATH Keywords: perturbation techniques
zbMATH Keywords: Lindstedt-Poincaré method
reviewed by: Vinicio C. Boffi
MaRDI profile type: Publication
full work available at URL: 0016-0032(86)90054-2
OpenAlex ID: W1990181498
cites work: Periodic solutions of second-order nonlinear difference equations containing a small parameter
cites work: Periodic solutions of second-order nonlinear difference equations containing a small parameter. II: Equivalent linearization
cites work: Generalization of the method of slowly varying amplitude and phase to non-linear oscillatory systems with two degrees of freedom
cites work: Perturbation methods and non-linear hyperbolic waves
cites work: Second-order nonlinear difference equations containing small parameters
cites work: Krylov–Bogoljubov Method for Difference Equations
cites work: Q5604158
cites work: Q5591849
cites work: Q5539404
description: scientific article; zbMATH DE number 3951177