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Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

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Abstract

The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of1:2 and 1:3 are investigated. The Multiple Scale Method is employedto derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the criticalstate. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter ε, of the order ofthe amplitude of motion. However, while resonant terms only appear atthe ε3-order in the 1:3 case, they already arise at theε2-order in the 1:2 case. Thus, by truncating theanalysis at the ε3-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of thethree control parameters and the asymptotic results are compared withexact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and theirstability analyzed.

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References

  1. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.

    Google Scholar 

  2. Luongo, A. and Paolone, A., ‘On the reconstitution problem in the multiple time scale method’, Nonlinear Dynamics 14, 1999, 133–156.

    Google Scholar 

  3. Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics, Wiley, New York, 1995.

    Google Scholar 

  4. Pignataro, M., Rizzi, N., and Luongo, A., Bifurcation, Stability and Postcritical Behaviour of Elastic Structures, Elsevier, Amsterdam, 1990 (first Italian edition ESA, Roma, 1983).

    Google Scholar 

  5. Luongo, A. and Paolone, A., ‘Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues’, Nonlinear Dynamics 14, 1997, 193–210.

    Google Scholar 

  6. Luongo, A. and Paolone, A., ‘Multiple scale analysis for divergence-Hopf bifurcation of imperfect symmetric systems’, Journal of Sound and Vibration 218, 1998, 527–539.

    Google Scholar 

  7. Luongo, A., Paolone, A., and Di Egidio, A., ‘Multiple time scale analysis for bifurcation from a double-zero eigenvalue’, ASME, Las Vegas, NV, 1999.

    Google Scholar 

  8. Paolone, A., Di Egidio A., and Luongo, A., ‘Post-critical analysis of self-excited structures in 1:1 resonance condition’, in Proceedings of the XIV Italian Congress of Theoretical and Applied Mechanics, Structural Mechanics Section, Como, Italy, 1999 [in Italian].

    Google Scholar 

  9. Luongo, A., ‘Eigensolutions sensitivity for nonsymmetric matrices with repeated eigenvalues’, AIAA Journal 31, 1993, 1321–1328.

    Google Scholar 

  10. Luongo, A., Paolone, A., and Di Egidio, A., ‘Sensitivities and linear stability analysis around a double zero eigenvalue’, AIAA Journal 38(4), 2000, 702–710.

    Google Scholar 

  11. Natsiavas, S., ‘Free vibration of two coupled nonlinear oscillators’, Nonlinear Dynamics 6, 1994, 69–86.

    Google Scholar 

  12. Natsiavas, S., ‘Free vibration in a class of self-excited oscillators with 1:3 internal resonance’, Nonlinear Dynamics 12, 1997, 109–128.

    Google Scholar 

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

    Google Scholar 

  14. Luongo, A., Di Egidio, A., and Paolone, A., ‘On the proper form of the amplitude modulation equations for resonant systems’, Nonlinear Dynamics 27(3), 2002, 237–254.

    Google Scholar 

  15. Doedel, E. and Kernevez, J. P., ‘AUTO: Software for continuation and bifurcations problem in ordinary differential equations’, Applied Mathematics Report, California Institute of Technology, 1986.

  16. Piccardo, G., ‘A methodology for the study of coupled aeroelastic phenomena’, Journal of Wind Engineering and Industrial Aerodynamics 48, 1993, 241–252.

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno, Università di L'Aquila, I-67040, Monteluco di Roio (L'Aquila), Italy

    Angelo Luongo & Angelo Di Egidio

  2. Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma ‘La Sapienza’, via Eudossiana 18, I-00184, Roma, Italy

    Achille Paolone

Authors
  1. Angelo Luongo
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  2. Achille Paolone
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  3. Angelo Di Egidio
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Luongo, A., Paolone, A. & Di Egidio, A. Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations. Nonlinear Dynamics 34, 269–291 (2003). https://doi.org/10.1023/B:NODY.0000013508.50435.39

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  • DOI: https://doi.org/10.1023/B:NODY.0000013508.50435.39

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