Skip to main content
SpringerLink
Log in

Nondegenerate Hamiltonian Hopf Bifurcations in \(\omega:3:6\) Resonance \((\omega=1\) or \(2)\)

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

This paper deals with the analysis of Hamiltonian Hopf bifurcations in three-degree-of-freedom systems, for which the frequencies of the linearization of the corresponding Hamiltonians are in \(\omega:3:6\) resonance (\(\omega=1\) or \(2\)). We obtain the truncated second-order normal form that is not integrable and expressed in terms of the invariants of the reduced phase space. The truncated first-order normal form gives rise to an integrable system that is analyzed using a reduction to a one-degree-of-freedom system. In this paper, some detuning parameters are considered and nondegenerate Hamiltonian Hopf bifurcations are found. To study Hamiltonian Hopf bifurcations, we transform the reduced Hamiltonian into standard form.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Abraham, R. and Marsden, J. E., Foundations of Mechanics, 2nd ed., rev. and enl., Reading, Mass.: Benjamin/Cummings, 1978.

    MATH  Google Scholar 

  2. Arnol’d, V. I., Kozlov, V. V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.

    Book  Google Scholar 

  3. Bates, L. and Zou, M., Degeneration of Hamiltonian Monodromy Cycles, Nonlinearity, 1993, vol. 6, no. 2, pp. 313–335.

    Article  MathSciNet  Google Scholar 

  4. Broer, H. and Takens, F., Dynamical Systems and Chaos, Appl. Math. Sci., vol. 172, New York: Springer, 2011.

    Book  Google Scholar 

  5. Buchanan, D., Trojan Satellites (Limiting Case), Trans. Roy. Soc. Canada Sect. III, 1941, vol. 35, pp. 9–25.

    MathSciNet  MATH  Google Scholar 

  6. Carrasco, D., Palacián, J. F., Vidal, C., Vidarte, J., and Yanguas, P., Dynamics of Axially Symmetric Perturbed Hamiltonians in \(1:1:1\) Resonance, J. Nonlinear Sci., 2018, vol. 28, no. 4, pp. 1293–1359.

    Article  MathSciNet  Google Scholar 

  7. Christov, O., Non-Integrability of First Order Resonances in Hamiltonian Systems in Three Degrees of Freedom, Celestial Mech. Dynam. Astronom., 2012, vol. 112, no. 2, pp. 149–167.

    Article  MathSciNet  Google Scholar 

  8. Crainic, M., Fernandes, R. L., and Mărcuţ, I., Lectures on Poisson Geometry, Utrecht: Univ. Utrecht, 2019.

    Google Scholar 

  9. Cushman, R. and van der Meer, J. C., The Hamiltonian Hopf Bifurcation in the Lagrange Top, in Géométrie Symplectique et Mécanique, C.Albert (Ed.), Lecture Notes in Math., vol. 1416, New York: Springer, 1988, pp. 26–38.

    Google Scholar 

  10. Dellnitz, M., Marsden, J. E., Melbourne, I., and Scheurle, J., Generic Bifurcations of Pendula, in Bifurcation and Symmetry (Marburg, 1991), Internat. Ser. Numer. Math., vol. 104, Basel: Birkhäuser, 1992, pp. 111–122.

    Chapter  Google Scholar 

  11. Deprit, A. and Henrard, J., A Manifold of Periodic Orbits, Adv. Astron. Astrophys., 1968, vol. 6, pp. 2–124.

    Google Scholar 

  12. Efstathiou, K., Metamorphoses of Hamiltonian Systems with Symmetries, Lecture Notes in Math., vol. 1864, New York: Springer, 2005.

    Book  Google Scholar 

  13. Efstathiou, K., Sadovskií, D. A., and Cushman, R. H., Linear Hamiltonian Hopf Bifurcation for Point-Group-Invariant Perturbations of the \(1:1:1\) Resonance, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 2003, vol. 459, no. 2040, pp. 2997–3019.

    Article  MathSciNet  Google Scholar 

  14. Egea, J., Ferrer, S., and van der Meer, J. C., Bifurcations of the Hamiltonian Fourfold \(1:1\) Resonance with Toroidal Symmetry, J. Nonlinear Sci., 2011, vol. 21, no. 6, pp. 835–874.

    Article  MathSciNet  Google Scholar 

  15. Groeniger, J. T. B. Oude, The \(1:3:4\) Resonance, Dynamics near a Resonant Equilibrium, Master’s Thesis, Utrecht University, Utrecht, 2018, 69 pp.

  16. Haller, G. and Wiggins, S., Geometry and Chaos near Resonant Equilibria of \(3\)-DOF Hamiltonian Systems, Phys. D, 1996, vol. 90, no. 4, pp. 319–365.

    Article  MathSciNet  Google Scholar 

  17. Hanßmann, H., Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems: Results and Examples, Lecture Notes in Math., vol. 1893, Berlin: Springer, 2007.

    MATH  Google Scholar 

  18. Hanßmann, H., Mazrooei-Sebdani, R., and Verhulst, F., The \(1:2:4\) Resonance in a Particle Chain, Indag. Math., 2020 (to appear).

  19. Hanßmann, H. and van der Meer, J.-C., Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems, J. Dynam. Differential Equations, 2005, vol. 17, no. 3, pp. 453–474.

    Article  MathSciNet  Google Scholar 

  20. Krein, M. G., Generalization of Certain Investigations of A. M. Liapunov on Linear Differential Equations with Periodic Coefficients, Dokl. Akad. Nauk SSSR (N.S.), 1950, vol. 73, pp. 445–448 (Russian).

    MATH  Google Scholar 

  21. Marsden, J. E., Lectures on Mechanics, London Math. Soc. Lect. Note Ser., vol. 174, Cambridge: Cambridge Univ. Press, 1992.

    Book  Google Scholar 

  22. Marsden, J. E. and Scheurle, J., Lagrangian Reduction and the Double Spherical Pendulum, Z. Angew. Math. Phys., 1993, vol. 44, no. 1, pp. 17–43.

    Article  MathSciNet  Google Scholar 

  23. Mazrooei-Sebdani, R. and Yousefi, Z., The Coupled \(1:2\) Resonance in a Symmetric Case and Parametric Amplification Model, Discrete Contin. Dyn. Syst. Ser. B, 2020 (to appear).

  24. Mazrooei-Sebdani, R. and Hakimi, E., On detuned \(1:1:3\) Hamiltonian resonance with cases of symmetric cubic and quartic potentials, Chaos, 2020, vol. 30, no. 9, 093119, 12 pp.

    Article  MathSciNet  Google Scholar 

  25. Meyer, K. R., Generic Bifurcation in Hamiltonian Systems, in Dynamical Systems (Warwick, 1974), A.Manning (Ed.), Lecture Notes in Math., vol. 468, Berlin: Springer, 1975, pp. 62–70.

    Chapter  Google Scholar 

  26. Meyer, K. R. and Offin, D. C., Introduction to Hamiltonian Dynamical Systems and the \(N\)-Body Problem, 3rd ed., Appl. Math. Sci., vol. 90, Cham: Springer, 2017.

    Book  Google Scholar 

  27. Meyer, K. R., Palacián, J. F., and Yanguas, P., Singular Reduction of Resonant Hamiltonians, Nonlinearity, 2018, vol. 31, no. 6, pp. 2854–2894.

    Article  MathSciNet  Google Scholar 

  28. Meyer K. R. and Schmidt D. S., Periodic Orbits near \({\cal L}_{4}\) for Mass Ratios near the Critical Mass Ratio of Routh, Celestial Mech., 1971, vol. 4, pp. 99–109.

    Article  MathSciNet  Google Scholar 

  29. Palacián, J. F., Vidal, C., Vidarte, J., and Yanguas, P., Periodic Solutions, KAM Tori and Bifurcations in a Cosmology-Inspired Potential, Nonlinearity, 2019, vol. 32, no. 9, pp. 3406–3444.

    Article  MathSciNet  Google Scholar 

  30. Palmore J. I., Numerical Experimentation into Effects of Varying the Mass Ratio on Periodic Solutions of the Restricted Problem of Three Bodies, PhD Thesis, Yale: Yale Univ., 1967.

  31. Sanders, J. A., Verhulst, F., and Murdock, J., Averaging Methods in Nonlinear Dynamical Systems, 2nd ed., Appl. Math. Sci., vol. 59, New York: Springer, 2007.

    Google Scholar 

  32. Verhulst, F., Integrability and Non-Integrability of Hamiltonian Normal Forms, Acta Appl. Math., 2015, vol. 137, pp. 253–272.

    Article  MathSciNet  Google Scholar 

  33. van der Aa, E. and Verhulst, F., Asymptotic Integrability and Periodic Solutions of a Hamiltonian System in \(1:2:2\)-Resonance, SIAM J. Math. Anal., 1984, vol. 15, no. 5, pp. 890–911.

    Article  MathSciNet  Google Scholar 

  34. van der Meer, J. C., On the Geometry of Hamiltonian Systems: Lecture Notes Seminar GISDA (Universidad del Bío-Bío, Concepción, Chile), Eindhoven: Eindhoven Univ. of Technology, 2017.

    Google Scholar 

  35. van der Meer, J. C., Degenerate Hamiltonian Hopf Bifurcations, in Conservative Systems and Quantum Chaos, L. M. Bates, D. L. Rod (Eds.), Fields Institute Communications, vol. 8, Providence, R.I.: AMS, 1996, pp. 159–176.

    Google Scholar 

  36. van der Meer, J. C., The Hamiltonian Hopf Bifurcation, Lecture Notes in Math., vol. 1160, Berlin: Springer, 1985.

    Book  Google Scholar 

  37. van der Meer, J. C., Hamiltonian Hopf Bifurcation with Symmetry, Nonlinearity, 1990, vol. 3, no. 4, pp. 1041–1056.

    Article  MathSciNet  Google Scholar 

  38. Weinstein, A., Normal Modes for Nonlinear Hamiltonian Systems, Invent. Math., 1973, vol. 20, pp. 47–57.

    Article  MathSciNet  Google Scholar 

  39. Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed., Texts Appl. Math., vol. 2, New York: Springer, 2003.

    MATH  Google Scholar 

Download references

ACKNOWLEDGMENTS

We deeply appreciate suggestions and comments of the referee which have contributed to the improvement in clarity of the text.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran

    Reza Mazrooei-Sebdani & Elham Hakimi

Authors
  1. Reza Mazrooei-Sebdani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elham Hakimi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Reza Mazrooei-Sebdani or Elham Hakimi.

Ethics declarations

The authors have no conflict of interest.

Additional information

MSC2010

70K30, 37J35, 70H06, 70H33, 70K45

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mazrooei-Sebdani, R., Hakimi, E. Nondegenerate Hamiltonian Hopf Bifurcations in \(\omega:3:6\) Resonance \((\omega=1\) or \(2)\). Regul. Chaot. Dyn. 25, 522–536 (2020). https://doi.org/10.1134/S1560354720060027

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354720060027

Keywords

Search

Navigation