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Nonlinear Dynamics

, Volume 39, Issue 4, pp 411–421 | Cite as

Parametric Resonance of Hopf Bifurcation

  • Richard Rand
  • Albert Barcilon
  • Tina Morrison
Article

Abstract

We investigate the dynamics of a system consisting of a simple harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation). Using perturbation methods and numerical integration, we investigate the changes which occur in long-time behavior as the damping parameter is varied. We show that for large positive damping, the origin is stable, whereas for large negative damping a quasi-periodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.

Key words

parametric excitation resonance Hopf bifurcation 

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Referencesy

  1. 1.
    Rand, R. H., Lecture Notes on Nonlinear Vibrations (Version 45), Published on-line by The Internet-First University Press, Ithaca, New York, 2004, http://dspace.library.cornell.edu/handle/1813/79Google Scholar
  2. 2.
    Wang, B. and Fang, Z., ‘Chaotic oscillations of tropical climate: A dynamic system theory for ENSO’, Journal of Atmospheric Sciences 53, 1996, 2786–2802.Google Scholar
  3. 3.
    Wang, B., Barcilon, A., and Fang, Z., ‘Stochastic dynamics of El Nino-Southern Oscillation’, Journal of Atmospheric Sciences 56, 1999, 5–23.Google Scholar
  4. 4.
    Zalalutdinov, M., Aubin, K. L., Pandey, M., Zehnder, A. T., Rand, R. H., Craighead, H. T., and Parpia, J. M., ‘Frequency entrainment for micromechanical oscillator’, Applied Physics Letters 83, 2003, 3281–3283.Google Scholar
  5. 5.
    Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.Google Scholar
  6. 6.
    Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Theoretical & Applied MechanicsCornell UniversityIthacaU.S.A.
  2. 2.Department of Meteorology and GFDIFlorida State UniversityTallahasseeU.S.A.

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