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

López, J. L.; Abbasbandy, S.; López-Ruiz, R.

doi:https://doi.org/10.3814/2009/854060

Scholarly Research Exchange

On this page

Abstract References Copyright Related Articles

Research Article | Open Access

Volume 2009 | Article ID 854060 | https://doi.org/10.3814/2009/854060

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

J. L. López,¹S. Abbasbandy,^2,3and R. López-Ruiz⁴

Received01 Sept 2008

Revised17 Feb 2009

Accepted03 Apr 2009

Published22 Apr 2009

Abstract

The limit cycle of the van der Pol oscillator, x¨+ε(x2−1)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.

References

B. van der Pol, “On relaxation oscillations,” Philosophical Magazine, vol. 2, pp. 978–992, 1926.
View at: Google Scholar
A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators, Dover, New York, NY, USA, 1st edition, 1989.
B. van der Pol and M. van der Mark, “The beating of the heart considered as relaxation oscillation and an electric model of the heart,” L'Onde électrique, vol. 7, pp. 365–392, 1928.
View at: Google Scholar
R. López-Ruiz and Y. Pomeau, “Transition between two oscillation modes,” Physical Review E, vol. 55, no. 4, pp. R3820–R3823, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
K. Odani, “The limit cycle of the van der Pol equation is not algebraic,” Journal of Differential Equations, vol. 115, no. 1, pp. 146–152, 1995.
View at: Publisher Site | Google Scholar | MathSciNet
A. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, Oxford, UK, 1st edition, 1977.
Y. Ye, Theory of Limit Cycles, vol. 66 of Translations of Mathematical Monographs, American Mathematical Society, Boston, Mass, USA, 1986.
S. J. Liao, The proposed homotopy analysis technique for the solutions of non-linear problems, Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China, 1992.
S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2003.
S.-J. Liao, “An analytic approximate approach for free oscillations of self-excited systems,” International Journal of Non-Linear Mechanics, vol. 39, no. 2, pp. 271–280, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
W. Wu and S.-J. Liao, “Solving solitary waves with discontinuity by means of the homotopy analysis method,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 177–185, 2005.
View at: Publisher Site | Google Scholar
T. Hayat and M. Khan, “Homotopy solutions for a generalized second-grade fluid past a porous plate,” Nonlinear Dynamics, vol. 42, no. 4, pp. 395–405, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Hayat, M. Khan, and M. Ayub, “On non-linear flows with slip boundary condition,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, no. 6, pp. 1012–1029, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
T. Hayat, M. Khan, M. Sajid, and M. Ayub, “Steady flow of an Oldroyd 8-constant fluid between coaxial cylinders in a porous medium,” Journal of Porous Media, vol. 9, no. 8, pp. 709–722, 2006.
View at: Publisher Site | Google Scholar
T. Hayat, M. Khan, and M. Ayub, “The effect of the slip condition on flows of an Oldroyd 6-constant fluid,” Journal of Computational and Applied Mathematics, vol. 202, no. 2, pp. 402–413, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Khan, Z. Abbas, and T. Hayat, “Analytic solution for flow of Sisko fluid through a porous medium,” Transport in Porous Media, vol. 71, no. 1, pp. 23–37, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
S. Abbasbandy, “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Physics Letters A, vol. 361, no. 6, pp. 478–483, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Abbasbandy, “Homotopy analysis method for heat radiation equations,” International Communications in Heat and Mass Transfer, vol. 34, no. 3, pp. 380–387, 2007.
View at: Publisher Site | Google Scholar
S. Abbasbandy, Y. Tan, and S. J. Liao, “Newton-homotopy analysis method for nonlinear equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1794–1800, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Tan and S. Abbasbandy, “Homotopy analysis method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 539–546, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Abbasbandy, J. L. López, and R. López-Ruiz, “The homotopy analysis method and the Liénard equation,” http://arxiv.org/abs/0805.3916.
View at: Google Scholar
K. Odani, “On the limit cycle of the Liénard equation,” Archivum Mathematicum, vol. 36, no. 1, pp. 25–31, 2000.
View at: Google Scholar
J. L. López and R. López-Ruiz, “Approximating the amplitude and form of limit cycles in the weakly nonlinear regime of Liénard systems,” Chaos, Solitons & Fractals, vol. 34, no. 4, pp. 1307–1317, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
J. L. López and R. López-Ruiz, “The limit cycles of Liénard equations in the weakly nonlinear regime,” to appear in Far East Journal of Dynamical Systems (2009); also available at http://arxiv.org/abs/nlin/0605025.
View at: Google Scholar
M. L. Cartwright, “Van der Pol's equation for relaxation oscillation,” in Contributions to the Theory of Non-Linear Oscillations II, S. Lefschetz, Ed., vol. 29 of Annals of Mathematics Studies, pp. 3–18, Princeton University Press, Princeton, NJ, USA, 1952.
View at: Google Scholar
J. L. López and R. López-Ruiz, “The limit cycles of Liénard equations in the strongly nonlinear regime,” Chaos, Solitons & Fractals, vol. 11, no. 5, pp. 747–756, 2000.
View at: Google Scholar
R. López-Ruiz and J. L. López, “Bifurcation curves of limit cycles in some Liénard systems,” International Journal of Bifurcation and Chaos, vol. 10, no. 5, pp. 971–980, 2000.
View at: Google Scholar

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

1371

Downloads

0

Citations