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Mathematical Problems in Engineering
Volume 2012, Article ID 693453, 10 pages
http://dx.doi.org/10.1155/2012/693453
Research Article

A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa

Received 17 March 2012; Accepted 4 July 2012

Academic Editor: Zoran Gajic

Copyright © 2012 Sandile S. Motsa and Precious Sibanda. 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.

Abstract

We present a novel application of the successive linearisation method to the classical Van der Pol and Duffing oscillator equations. By recasting the governing equations as nonlinear eigenvalue problems we obtain accurate values of the frequency and amplitude. We demonstrate that the proposed method can be used to obtain the limit cycle and bifurcation diagrams of the governing equations. Comparison with exact and other results in the literature shows that the method is accurate and effective in finding solutions of nonlinear equations with oscillatory solutions, nonlinear eigenvalue problems, and other nonlinear problems with bifurcations.