On the periodic orbits of the third-order differential equation \(x^{\prime\prime\prime}-\mu x^{\prime\prime}+x^{\prime }-\mu x=\varepsilon F(x, x^{\prime },x^{\prime\prime})\)
type: Item
instance of: scholarly article
title: On the periodic orbits of the third-order differential equation \(x^{\prime\prime\prime}-\mu x^{\prime\prime}+x^{\prime }-\mu x=\varepsilon F(x, x^{\prime },x^{\prime\prime})\)
zbMATH Open document ID: 1263.34056
author: Jaume Llibre
author: Luci Any Roberto
published in: Applied Mathematics Letters
publication date: 2013-03-07T00:00:00Z
review text: Consider the differential equation \[ x'''-\mu x''+ x'-\mu x=\varepsilon F(x,x',x''),\tag{\(*\)} \] where \(\mu\) and \(\varepsilon\) are real parameters, \(|\varepsilon|\) small, \(F\) is a \(C^2\)-function. After some transformations and introducing cylindrical coordinates, equation \((*)\) is equivalent to the system \[ r'=\varepsilon G(r,\theta,Z)\cos\theta,\quad \theta'= 1-\varepsilon G(r,\theta,Z)/r,\quad Z'= \mu Z-\varepsilon G(r,\theta,Z) \] which is for sufficiently small \(\varepsilon\) equivalent to the non-autonomous system \[ \begin{aligned} {dr\over d\theta} &=\varepsilon G(r,\theta,Z)\cos\theta+ \theta(\varepsilon^2),\\ {dZ\over d\theta} &=\mu Z+\varepsilon {\mu Z\sin\theta- r\over r} G(r,\theta,Z)+ 0(\varepsilon^2).\end{aligned}\tag{\(**\)} \] Applying averaging theory to \((**)\), the authors derive results on the existence of periodic solutions for sufficiently small \(\varepsilon\) bifurcating from special periodic solutions of the unperturbed \((\varepsilon= 0)\) system. They distinguish the cases \(\mu= 0\) and \(\mu\neq 0\) in the unperturbed system.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6142811
zbMATH Keywords: periodic solution
zbMATH Keywords: averaging method
zbMATH Keywords: third-order differential equation
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: j.aml.2012.10.017
OpenAlex ID: W1489823878
Wikidata QID: Q115360853
DOI: 10.1016/J.AML.2012.10.017
description: scientific article; zbMATH DE number 6142811