Skip to main content
SpringerLink
Log in

On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Kaplan, D., Glass, L.: Understanding Nonlinear Dynamics. Springer, New York (1995)

    Book  MATH  Google Scholar 

  3. Richard, H.: It’s a Nonlinear World. Springer, New York (2011)

    MATH  Google Scholar 

  4. 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)

    Article  Google Scholar 

  5. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membranes. Biophys. 1, 445–466 (1961)

    Google Scholar 

  6. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IEEE 50, 2061–2070 (1962)

    Google Scholar 

  7. Cartwright, J., Eguiluz, V., Hernandez-Garcia, E., Piro, O.: Dynamics of elastic excitable media. Int. J. Bifurc. Chaos 9, 2197–2202 (1999)

    Article  MATH  Google Scholar 

  8. Lucero, J.C., Schoentgen, J.: Modeling vocal fold asymmetries with coupled van der Pol oscillators. Proc. Meetings Acoustics 19(1), 060165 (2013)

    Article  Google Scholar 

  9. 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)

    Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Google Scholar 

  13. 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)

    Article  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

    Article  MATH  Google Scholar 

  16. Kartashova, E.: Nonlinear Resonance Analysis: Theory, Computation, Applications. Cambrige University Press, Cambrige (2011)

    MATH  Google Scholar 

  17. Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15(8), 2003–2016 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liao, S.J.: Homotopy Analysis Method in Nonlinear Differential Equations. Higher Education Press, Beijing (2012)

    Book  MATH  Google Scholar 

  19. Liao, S.J.: Advances in the Homotopy Analysis Method. World Scientific, New York (2013)

    Google Scholar 

  20. Liao, S.J.: On the reliability of computed chaotic solutions of non-linear differential equations. Tellus 6IA, 550–564 (2009)

    Google Scholar 

Download references

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

  1. College of Science, Inner Mongolia University of Technology, Hohhot, 010051, China

    Jifeng Cui & Wenyu Zhang

  2. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Zeng Liu & Jianglong Sun

  3. Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Jianglong Sun

  4. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, China

    Jianglong Sun

Authors
  1. Jifeng Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenyu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zeng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianglong Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianglong Sun.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0420-z

Keywords

Search

Navigation