Skip to main content
SpringerLink
Log in

Nonlinear dynamics of a \(\varvec{\phi ^6}-\) modified Duffing oscillator: resonant oscillations and transition to chaos

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The nonlinear dynamics of a hybrid Rayleigh–Van der Pol–Duffing oscillator includes pure and impure quadratic damping are investigated. The multiple timescales method is used to study exhaustively various resonances states. It is noticed that the system presents nine resonances states. The frequency response curves of quintic, third and second superharmonic, and subharmonic resonances states are obtained. Bistability, hysteresis, and jump phenomenon are also obtained. It is pointed out that these resonance phenomena are strongly related to the nonlinear cubic and quadratic damping and to the external force. The numerical simulations are used to make bifurcation sequences displayed by the model for each type of oscillatory. It is noticed that the pure quadratic, impure cubic damping, and external excitation affect regular and chaotic states.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  2. Hayashi, C.: Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  3. Strogatz, S.H.: Nonlinear Dynamics and Chaos with Applications to Physics, Chemistry and Engineering. Westview Press, Cambridge Sec. 1.2 (1994)

  4. Warminski, J., Lenci, S., Cartmell, P.M., Giuseppe Rega, G., Wiercigroch, M.: Nonlinear Dynamic Phenomena in Mechanics. Solid Mechanics and its Applications, vol. 181. Springer, Berlin (2012)

    Book  Google Scholar 

  5. Carrol, T.L.: Communicating with use of filtered, synchronized, chaotic signals. IEEE Trans. Circuits syst. I Fundam. Theory Appl. 42, 105–110 (1995)

    Article  Google Scholar 

  6. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)

    MATH  Google Scholar 

  7. Enjieu Kadji, H.G., Nana Nbendjo, B.R., Chabi Orou, J.B., Talla, P.K.: Nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator. Phys. Plasmas 15, 032308 (2008)

    Article  Google Scholar 

  8. Miwadinou, C.H., Monwanou, A.V., Hinvi, A.L., Koukpemedji, A.A., Ainamon, C., Chabi Orou, J.B.: Melnikov chaos in a modified Rayleigh–Duffing oscillator with \( \phi ^6\) potential. Int. J. Bifurc. Chaos 26, 1650085 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Pandey, M., Rand, R., Zehnder, A.: Perturbation Analysis of Entrainment in a Micromechanical Limit Cycle Oscillator. Communications in Nonlinear Science and Numerical Simulation, available online (2006)

  10. Yamapi, R., Aziz-Alaoui, M.A.: Vibration analysis and bifurcations in the self-sustained electromechanical system with multiple functions. Commun. Nonlinear Sci. Numer. Simul. 12, 1534–1549 (2007)

    Article  MATH  Google Scholar 

  11. Rand, R.H., Ramani, D.V., Keith, W.L., Cipolla, K.M.: The quadratically damped Mathieu equation and its application to submarine dynamics. Control Vib. Noise New Millenn. 61, 39–50 (2000)

    Google Scholar 

  12. Verveyko, D.V., Verisokin, A.Y.: Application of He’s method to the modified Rayleigh equation. Discrete and Continuous Dynamical Systems, Supplement 1423–1431, (2011)

  13. Miwadinou, C.H., Monwanou, A.V., Chabi Orou, J.B.: Active control of the parametric resonance in the modified Rayleigh-Duffing oscillator. Afr. Rev. Phys. 9, 227–235 (2014)

    MATH  Google Scholar 

  14. Miwadinou, C.H., Monwanou, A.V., Chabi Orou, J.B.: Effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh-Duffing oscillator. Int. J. Bifurc. Chaos 25, 1550024 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Blagoveshchensky, S.N.: Theory of Ship Motion. The Seakeeping Symposium Commemorating the 20th Anniversay of the St. Denis-Pierson Paper (1962)

  16. Bhattacharyya, R.: Dynamics of Marine Vehicles. Ocean Engineering Series. Wiley, New York (1978)

  17. de Kat, J.O., Paulling, J.R.: The simulation of ship motions and capsizing in severe seas. Trans. Soc. Archit. Mar. Eng. 97, 139–168 (1989)

  18. Zborowski, A., Taylan, M.: Evaluation of Small Vessels Roll Motion Stability Reserve for Resonance Conditions. SNAME Spring Meeting/STAR Symposium, (S1-2), New Orleans, USA, 1– (1989)

  19. Witz, J.A., Ablett, C.B., Harrison, J.H.: Roll response of semisubmersibles with nonlinear restoring moment characteristics. Appl. Ocean Res. 11, 153–166 (1989)

    Article  Google Scholar 

  20. Denise, J.-P.F.: On the roll motion of barges. Trans. R. Inst. Nav. Archit. 125, 255–268 (1983)

    Google Scholar 

  21. Francescutto, A., Contento, G.: Bifurcations in ship rolling: experimental results and parameter identification technique. Ocean Eng. 26, 1095–1123 (1999)

    Article  Google Scholar 

  22. Wu, W., McCue, L.: Application of the extended Melnikovs method for single-degree-of-freedom vessel roll motion. Ocean Eng. 35, 1739–1746 (2008)

    Article  Google Scholar 

  23. Soliman, M.S., Thompson, J.M.T.: The effect of damping on the steady state and basin bifurcation patterns of a nonlinear mechanical oscillator. Int. J. Bifurc. Chaos 2, 81–91 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Spyrou, K.J., Thompson, J.M.T.: The nonlinear dynamics of ship motions. Phil. Trans. R. Soc. Lond. A (Theme Issue) 358, 1731–1981 (2000)

    MATH  Google Scholar 

  25. El-Bassiouny, A.F.: Nonlinear rolling of a biased ship in a regular beam wave under external and parametric excitations. Z. Naturforsch. 62a, 573–586 (2007)

    MATH  Google Scholar 

  26. Holappa, K.W., Falzarano, J.M.: Application of extended state space to nonlinear ship rolling. Ocean Eng. 26, 227–240 (1999)

    Article  Google Scholar 

  27. Cardo, A., Francescutto, A., Nabergoj, R.: Ultraharmonics and subharmonics in the rolling motion of a ship: steady-state solution. Int. Shipbuild. Prog. 28, 234–251 (1981)

    Google Scholar 

  28. Cardo, A., Francescutto, A., Nabergoj, R.: Subharmonic oscillations in nonlinear rolling. Ocean Eng. 11, 663–669 (1984)

    Article  Google Scholar 

  29. Cardo, A., Francescutto, A., Nabergoj, R.: The excitation threshold and the onset of subharmonic oscillations in nonlinear rolling. Int. Shipbuild. Prog. 32, 210–214 (1985)

    Google Scholar 

  30. Mook, D.T., Marshall, L.R., Nayfeh, A.H.: Subharmonic and superharmonic resonances in the pitch and roll modes of ship motions. J. Hydronaut. 8, 32–40 (1974)

    Article  Google Scholar 

  31. Contento, G., Francescutto, A., Piciullo, M.: On the effectiveness of constant coefficients roll motion equation. Ocean Eng. 23, 597–618 (1996)

    Article  Google Scholar 

  32. Ainamon, C., Miwadinou, C.H., Monwanou, A.V., Chabi Orou, J.B.: Analysis of multiresonance and chaotic behavior of the polarization in materials modeled by a Duffing equation with multifrequency excitations. Appl. Phys. Res. 6, 74–86 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank very much Drs. Enjieu Kadji, Victor Kamdoum, and Peguy Roussel Nwagour for their collaborations. We also thank very much the anonymous referees whose useful criticisms, comments, and suggestions have helped strengthen the content and the quality of the paper.

Author information

Authors and Affiliations

  1. Institut de Mathématiques et de Sciences Physiques, BP 613, Porto-Novo, Benin

    C. H. Miwadinou, L. A. Hinvi, A. V. Monwanou & J. B. Chabi Orou

Authors
  1. C. H. Miwadinou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. A. Hinvi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. V. Monwanou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. B. Chabi Orou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Monwanou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Miwadinou, C.H., Hinvi, L.A., Monwanou, A.V. et al. Nonlinear dynamics of a \(\varvec{\phi ^6}-\) modified Duffing oscillator: resonant oscillations and transition to chaos. Nonlinear Dyn 88, 97–113 (2017). https://doi.org/10.1007/s11071-016-3232-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3232-0

Keywords

Search

Navigation