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

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

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...

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Main Authors: Sandile S. Motsa, Precious Sibanda
Format: Article
Language:English
Published: Hindawi Limited 2012-01-01
Series:Mathematical Problems in Engineering
Online Access:http://dx.doi.org/10.1155/2012/693453
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spelling doaj-24efb566817b4405bad0681eaceeaee92020-11-25T00:13:18ZengHindawi LimitedMathematical Problems in Engineering1024-123X1563-51472012-01-01201210.1155/2012/693453693453A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation MethodSandile S. Motsa0Precious Sibanda1School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South AfricaSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South AfricaWe 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.http://dx.doi.org/10.1155/2012/693453
collection DOAJ
language English
format Article
sources DOAJ
author Sandile S. Motsa
Precious Sibanda
spellingShingle Sandile S. Motsa
Precious Sibanda
A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
Mathematical Problems in Engineering
author_facet Sandile S. Motsa
Precious Sibanda
author_sort Sandile S. Motsa
title A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
title_short A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
title_full A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
title_fullStr A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
title_full_unstemmed A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
title_sort note on the solutions of the van der pol and duffing equations using a linearisation method
publisher Hindawi Limited
series Mathematical Problems in Engineering
issn 1024-123X
1563-5147
publishDate 2012-01-01
description 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.
url http://dx.doi.org/10.1155/2012/693453
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AT sandilesmotsa noteonthesolutionsofthevanderpolandduffingequationsusingalinearisationmethod
AT precioussibanda noteonthesolutionsofthevanderpolandduffingequationsusingalinearisationmethod
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