Table of Contents
Scholarly Research Exchange
Volume 2009, Article ID 854060, 7 pages
Research Article

Formulas for the Amplitude of the van der Pol Limit Cycle through the Homotopy Analysis Method

1Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain
2Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran 14778, Iran
3Department of Mathematics, Imam Khomeini International University, Ghazvin 34149-16818, Iran
4Department of Computer Science and BIFI, University of Zaragoza, 50009 Zaragoza, Spain

Received 1 September 2008; Revised 17 February 2009; Accepted 3 April 2009

Copyright © 2009 J. L. López et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The limit cycle of the van der Pol oscillator, x¨+ε(x21)x˙+x=0, is studied in the plane (x,x˙) by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter ε is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given.