Skip to main content
SpringerLink
Log in

Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

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

Abstract

Bifurcations and route to chaos of the Mathieu–Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.

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. Seydel, R.: Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, 2nd edn. Springer, New York (1994)

    MATH  Google Scholar 

  2. Belhaq, M., Houssni, M.: Symmetry breaking bifurcation and first period-doubling following a Hopf bifurcation in a three-dimensional system. Mech. Res. Commun. 22(3), 221–231 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Belhaq, M., Houssni, M., Freire, E., Rodriguez-Luis, A.J.: Analytical prediction of the two first period-doublings in a three-dimensional system. Int. J. Bifur. Chaos 10, 1497–1508 (2000)

    Article  Google Scholar 

  4. Awrejcewicz, J., Krysko, V.A.: Feigenbaum scenario excited by thin plate dynamics. Nonlinear Dyn. 24, 373–398 (2001)

    Article  MATH  Google Scholar 

  5. Musielak, D.E., Musielak, Z.E., Benner, J.W.: Chaos and route to chaos in coupled Duffing oscillators with multiple degrees of freedom. Chaos Solitons Fractals 24, 907–922 (2005)

    Article  MATH  Google Scholar 

  6. Kenfack, A.: Bifurcation structure of two coupled periodically driven double-well Duffing oscillators. Chaos Solitons Fractals 15, 205–218 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ravindra, B., Mallik, A.K.: Chaotic response of a harmonically excited mass on an isolator with nonlinear stiffness and damping characteristics. J. Sound Vib. 182(3), 345–353 (1995)

    Article  MathSciNet  Google Scholar 

  8. Kim, S.Y., Hu, B.: Bifurcations and transitions to chaos in an inverted pendulum. Phys. Rev. E 58(3), 3028–3035 (1998)

    Article  MathSciNet  Google Scholar 

  9. Thamilmaran, K., Lakshmanan, M.: Classification of bifurcation and routes to chaos in a variant of Murali–Lakshmanan–Chua circuit. Int. J. Bifurc. Chaos 12(4), 783–813 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Awrejcewicz, J.: Bifurcation portrait of the human vocal cord oscillations. J. Sound Vib. 136(1), 151–156 (1990)

    Article  MathSciNet  Google Scholar 

  11. Awrejcewicz, J.: Numerical analysis of the oscillation of human vocal cords. Nonlinear Dyn. 2, 35–52 (1991)

    Article  Google Scholar 

  12. Awrejcewicz, J., Delfs, J.: Dynamics of a self-excited stick-slip oscillator with two degree of freedom Part I. Investigation of equilibria. Eur. J. Mech. A/Solid 9(4), 269–282 (1990)

    MATH  MathSciNet  Google Scholar 

  13. Awrejcewicz, J., Delfs, J.: Dynamics of a self-excited stick-slip oscillator with two degree of freedom Part II. Slip–stick, slip–slip, stick–slip, periodic and chaotic orbits. Eur. J. Mech. A/Solid 9(5), 397–418 (1990)

    MATH  MathSciNet  Google Scholar 

  14. Leung, A.Y.T., Fung, T.C.: Construction of the chaotic regions. J. Sound Vib. 131(3), 445–455 (1989)

    Article  MathSciNet  Google Scholar 

  15. Ge, Z.M., Chen, H.H.: Bifurcations and chaotic motions in a rate gyro with a sinusoidal velocity about the spin axis. J. Sound Vib. 200(2), 121–137 (1997)

    Article  MathSciNet  Google Scholar 

  16. Xu, L., Lu, M.W., Cao, Q.: Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise nonlinearity by incremental harmonic balance method. J. Sound Vib. 264, 873–882 (2003)

    Article  Google Scholar 

  17. Raghothama, A., Narayanan, S.: periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dyn. 27, 341–365 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Raghothama, A., Narayanan, S.: Bifurcation and chaos in escape equation model by incremental harmonic balancing. Chaos Solitons Fractals 11, 1349–1363 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Raghothama, A., Narayanan, S.: Bifurcation and chaos of an articulated loading platform with piecewise nonlinear stiffness using the incremental harmonic balance method. Ocean Eng., 27, 1087–1107 (2000)

    Article  Google Scholar 

  20. Testa, J., Perez, J., Jeffries, G.: Evidence for universal chaotic behavior of a driven nonlinear oscillator. Phys. Rev. Lett. 48(11), 714–720 (1982)

    Article  MathSciNet  Google Scholar 

  21. Lau, S.L., Cheung, Y.K.: Amplitude incremental variational principle for nonlinear vibration of elastic systems. ASME J. Appl. Mech. 48, 959–964 (1981)

    Article  MATH  Google Scholar 

  22. Lau, S.L.: The incremental harmonic balance method and its application to nonlinear vibrations. In: Proceeding of International Conference on Structure Dynamics, Vibration, Noise and Control, Hong Kong, pp. 50–57 (1995)

  23. Cheung, Y.K., Chen, S.H., Lau, S.L.: Application of the incremental harmonic balance method to cubic nonlinearity systems. J. Sound Vib. 140, 273–286 (1990)

    Article  MathSciNet  Google Scholar 

  24. Dowell, E.H.: Chaotic oscillation in mechanical systems. Comput. Mech. 3, 199–216 (1988)

    Article  Google Scholar 

  25. Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ravindra, B., Zhu, W.D.: Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime. Arch. Appl. Mech. 68, 195–205 (1998)

    Article  MATH  Google Scholar 

  27. Friedmann, P., Hammond, C.E.: Efficient numerical treatment of periodic systems with application to stability problems. Int. J. Numer. Method Eng. 11, 1117–1136 (1977)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou, 510275, China

    J. H. Shen, K. C. Lin & S. H. Chen

  2. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong SAR, China

    K. Y. Sze

Authors
  1. J. H. Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. C. Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. H. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. K. Y. Sze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. H. Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, J.H., Lin, K.C., Chen, S.H. et al. Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method. Nonlinear Dyn 52, 403–414 (2008). https://doi.org/10.1007/s11071-007-9289-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-007-9289-z

Keywords

Search

Navigation