Skip to main content
SpringerLink
Log in

Two models for the parametric forcing of a nonlinear oscillator

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

Abstract

Instabilities associated with 2:1 and 4:1 resonances of two models for the parametric forcing of a strictly nonlinear oscillator are analyzed. The first model involves a nonlinear Mathieu equation and the second one is described by a 2 degree of freedom Hamiltonian system in which the forcing is introduced by the coupling. Using averaging with elliptic functions, the threshold of the overlapping phenomenon between the resonance bands 2:1 and 4:1 (Chirikov’s overlap criterion) is determined for both models, offering an approximation for the transition from local to global chaos. The analytical results are compared to numerical simulations obtained by examining the Poincaré section of the two systems.

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. Chirikov, B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263–379 (1979)

    Article  Google Scholar 

  2. Belhaq, M., Lakrad, F.: Prediction of homoclinic bifurcation: the elliptic averaging method. Chaos Solit. Fractals 11, 2251–2258 (2000)

    Article  MATH  Google Scholar 

  3. Belhaq, M., Lakrad, F.: Analytics of homoclinic bifurcations in three-dimensional systems. Int. J. Bif. Chaos 12, 2479–2486 (2002)

    Article  MATH  Google Scholar 

  4. Zounes, R.S., Rand, R.H.: Global behavior of a nonlinear quasiperiodic Mathieu equation. Nonlinear Dyn. 27, 87–105 (2002)

    Article  MATH  Google Scholar 

  5. Holmes, C.A., Rand, R.H.: Coupled oscillators as a model for nonlinear parametric excitation. Mech. Res. Comm. 8(5), 263–268 (1981)

    Article  MATH  Google Scholar 

  6. Rand, R.H.: Lecture Notes on Nonlinear Vibrations, version 52 (2005). Available on-line at http://www.tam.cornell.edu/ randdocs/nlvibe52.pdf

  7. Byrd, P., Friedman, M.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1954)

    Google Scholar 

  8. Coppola, V.T., Rand, R.H.: Averaging using elliptic functions: approximation of a limit cycles. Acta Mech. 81, 125–142 (1990)

    Article  Google Scholar 

  9. Vakakis, A.F., Rand, R.H.: Nonlinear dynamics of a system of coupled oscillators with essential stifness non-linearities. Int. J. Non-linear Mech. 39, 1079–1091 (2004)

    Article  MATH  Google Scholar 

  10. Rand, R.H.: Topics in Nonlinear Dynamics with Computer Algebra. Gordon and Breach, Langhorne, PA (1994)

    MATH  Google Scholar 

  11. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    MATH  Google Scholar 

  12. Minorsky, N.: Nonlinear Oscillations. van Nostrand, Princeton, NJ (1962)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Sciences Aïn Chock, BP 5366, Maârif, Casablanca, Morocco

    Nazha Abouhazim & Mohamed Belhaq

  2. Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14853, USA

    Richard H. Rand

Authors
  1. Nazha Abouhazim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohamed Belhaq
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Richard H. Rand
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard H. Rand.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abouhazim, N., Belhaq, M. & Rand, R.H. Two models for the parametric forcing of a nonlinear oscillator. Nonlinear Dyn 50, 147–160 (2007). https://doi.org/10.1007/s11071-006-9148-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-006-9148-3

Keywords

Search

Navigation