Nonlinear Dynamics

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

Parametric Resonance of Hopf Bifurcation

  • Richard Rand
  • Albert Barcilon
  • Tina Morrison


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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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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