Abstract
In this paper, the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol oscillator and the forced Van der Pol-Duffing oscillator are studied by combining the homotopy analysis method (HAM) with the multi-scale method analytically. Comparisons of the obtained solutions and numerical results show that this method is effective and convenient even when t is large enough, and the convergence of the approximate solutions is discussed by the so-called discrete square residual error. This method is a capable tool for solving this kind of nonlinear problems.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Lazer, A.C., McKenna, P.J.: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32(4), 537–578 (1990)
Kaplan, D., Glass, L.: Understanding Nonlinear Dynamics. Springer, New York (1995)
Richard, H.: It’s a Nonlinear World. Springer, New York (2011)
Pranay, P., Dhiman, M., Andreas, A., Saibal, R.: Influence of combined fundamental potentials in a nonlinear vibration energy harvester. Sci. Rep. 6, 1–13 (2016)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membranes. Biophys. 1, 445–466 (1961)
Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IEEE 50, 2061–2070 (1962)
Cartwright, J., Eguiluz, V., Hernandez-Garcia, E., Piro, O.: Dynamics of elastic excitable media. Int. J. Bifurc. Chaos 9, 2197–2202 (1999)
Lucero, J.C., Schoentgen, J.: Modeling vocal fold asymmetries with coupled van der Pol oscillators. Proc. Meetings Acoustics 19(1), 060165 (2013)
Cui, J.F., Liang, J.M., Lin, Z.L.: Stability analysis for periodic solutions of the Van der Pol-Duffing forced oscillator. Phys. Scripta 91(1), 7pp (2015)
Shukla, A.K., Ramamohan, T.R., Srinivas, S.: Analytical solutions for limit cycles of the forced Van der Pol-Duffing oscillator. AIP Conf. Proc. 1558, 2187–2192 (2013)
Stupnicka, W.S.: The coexistence of periodic, almost periodic and chaotic attractors in the Van der Pol-Duffing oscillator. J. Sound Vib. 199(2), 165–175 (1997)
Shukla, A.K., Ramamohan, T.R., Srinivas, S.: A new analytical approach for limit cycles and quasi-periodic solutions of nonlinear oscillators: the example of the forced Van der Pol-Duffing oscillator. J. Sound Vib. 89, 10pp (2014)
Chen, Y.M., Liu, J.K.: A study of homotopy analysis method for limit cycle of Van der Pol equation. Commun. Nonlinear Sci. Numer. Simul. 14(5), 1816–1821 (2009)
Chen, Y.M., Liu, J.K.: Uniformly valid solution of limit cycle of the Duffing-van der Pol equation. Mech. Res. Commun. 36, 845–850 (2009)
Kimiaeifar, A., Saidi, A.R., Bagheri, G.H., Rahimpour, M.D., Domairry, G.: Analytical solution for Van der Pol-Duffing oscillators. Chaos Soliton Fract. 42, 2660–2666 (2009)
Kartashova, E.: Nonlinear Resonance Analysis: Theory, Computation, Applications. Cambrige University Press, Cambrige (2011)
Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15(8), 2003–2016 (2010)
Liao, S.J.: Homotopy Analysis Method in Nonlinear Differential Equations. Higher Education Press, Beijing (2012)
Liao, S.J.: Advances in the Homotopy Analysis Method. World Scientific, New York (2013)
Liao, S.J.: On the reliability of computed chaotic solutions of non-linear differential equations. Tellus 6IA, 550–564 (2009)
Acknowledgements
The authors gratefully acknowledge the financial support from the Science Research Project of Inner Mongolia University of Technology (Approval No. ZD201613) and the science research project of Huazhong University of Science and Technology (Approval No. 0118140077 and 2006140115). This work was partly supported by the National Natural Science Foundation of China (NSFC) under Project Nos. 51609090 and 51679097.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cui, J., Zhang, W., Liu, Z. et al. On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator. Numer Algor 78, 1217–1231 (2018). https://doi.org/10.1007/s11075-017-0420-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-017-0420-z