Periodic solutions of a generalized Van der Pol-Mathieu differential equation
type: Item
instance of: scholarly article
title: Periodic solutions of a generalized Van der Pol-Mathieu differential equation
zbMATH Open document ID: 1309.34068
author: Z. Kadeřábek
author: J. Kalas
published in: Applied Mathematics and Computation
publication date: 2014-11-13T00:00:00Z
reviewed by: Irena Rachůnková
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 6369140
zbMATH Keywords: Van der Pol-Mathieu equation
zbMATH Keywords: periodic solutions
zbMATH Keywords: quasiperiodic solutions
zbMATH Keywords: averaging method
zbMATH Keywords: complexification
zbMATH Keywords: autonomous equations
zbMATH Keywords: phase space analysis
MaRDI profile type: Publication
full work available at URL: j.amc.2014.01.161
OpenAlex ID: W1989942305
Wikidata QID: Q115361469
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DOI: 10.1016/J.AMC.2014.01.161
review text: The paper investigates the generalized Van der Pol-Mathieu equation NEWLINE\[NEWLINE \frac{d^2x}{dt^2}-\varepsilon(\alpha_0-\beta_0x^{2n})\frac{dx}{dt}+\omega_0^2(1+\varepsilon h_0 \cos \gamma t)x=0, \eqno(1) NEWLINE\]NEWLINE where \(n\in N\), \(\gamma=2\omega_0+2d_0\varepsilon\), \(\alpha_0>0\), \(\beta_0>0\), \(h_0>0\), \(\omega_0>0\), \(d_0\in R\), and \(\varepsilon>0\) is a small parameter. The authors prove the existence of nontrivial oscillatory periodic solutions of (1). Their proofs are based on the averaging method and the Bogoliubov theorem about the existence and stability of periodic solutions. They also use the method of complexification and the phase space analysis of a derived autonomous equation. In addition, the existence of oscillatory quasiperiodic solutions is discussed. It was shown that equation (1) has similar behaviour for \(n>1\) as for \(n=1\).
description: scientific article; zbMATH DE number 6369140