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Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance

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Abstract

In this work we study the existence of Silnikov homoclinicorbits in the averaged equations representing the modal interactionsbetween two modes with zero-to-one internal resonance. The fast mode isparametrically excited near its resonance frequency by a periodicforcing. The slow mode is coupled to the fast mode when the amplitude ofthe fast mode reaches a critical value so that the equilibrium of theslow mode loses stability. Using the analytical solutions of anunperturbed integrable Hamiltonian system, we evaluate a generalizedMelnikov function which measures the separation of the stable and theunstable manifolds of an annulus containing the resonance band of thefast mode. This Melnikov function is used together with the informationof the resonances of the fast mode to predict the region of physicalparameters for the existence of Silnikov homoclinic orbits.

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References

  1. Nayfeh, A. H. and Balachandran, B., ‘Modal interactions in dynamical and structural systems’, Applied Mechanics Review 42, 1989, S175-S201.

    Google Scholar 

  2. Anderson, T. J., Balachandran, B., and Nayfeh, A. H., ‘Observations of nonlinear interactions in a flexible cantilever beam’, AIAA Paper 92–2332, 1992.

  3. Nayfeh, S. A. and Nayfeh, A. H., ‘Nonlinear interactions between two widely spaced modes-External excitation’, International Journal of Bifurcation and Chaos 3, 1993, 417–427.

    Google Scholar 

  4. Nayfeh, S. A. and Nayfeh, A. H., ‘Energy transfer from high-to-low-frequency modes in a flexible structure via modulation’, Journal of Vibration and Acoustics 116, 1994, 203–207.

    Google Scholar 

  5. Feng, Z. C., ‘Instability caused by the coupling between nonresonant shape oscillation modes of a charged conducting drop’, Journal of Fluid Mechanics 333, 1997, 1–21.

    Google Scholar 

  6. Feng, Z. C., ‘Nonresonant modal interactions’, in Proceedings of 1995 Design Engineering Technical Conference, Vol. 3-Part A, A. K. Bajaj (ed.), ASME Press, New York, 1995, pp. 511–517.

    Google Scholar 

  7. Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.

    Google Scholar 

  8. Nayfeh, A. H. and Chin, C-M., ‘Nonlinear interactions in a parametrically excited system with widely space frequencies’, Nonlinear Dynamics 7, 1995, 1–22.

    Google Scholar 

  9. Kovacic, G. and Wiggins, S., ‘Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped Sine-Gordon equation’, Physica D 57, 1992, 185–225.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, U.S.A.

    Z. C. Feng & K. M. Liew

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  1. Z. C. Feng
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  2. K. M. Liew
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Feng, Z.C., Liew, K.M. Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance. Nonlinear Dynamics 21, 249–263 (2000). https://doi.org/10.1023/A:1008364914446

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