Skip to main content
SpringerLink
Log in

On Resonances in Hamiltonian Systems with Three Degrees of Freedom

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

Abstract

We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus and is generically non-integrable. Methods of differential topology are applied to full description of equilibrium states and phase foliations of the average system. The results are illustrated by a simple model combining the non-degeneracy and non-integrability of the isoenergetically reduced system.

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

Similar content being viewed by others

References

  1. Poincaré, H., Les méthodes nouvelles de la mécanique céleste: Vol. 1. Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotique, Paris: Gauthier-Villars, 1892.

    MATH  Google Scholar 

  2. Arnol’d, V. I. and Avez, A., Ergodic Problems of Classical Mechanics, New York: Benjamin, 1968.

    MATH  Google Scholar 

  3. Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.

    Book  MATH  Google Scholar 

  4. Chirikov, B., A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep., 1979, vol. 52, no. 5, pp. 263–379.

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  6. Kozlov, V. V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996.

    Book  Google Scholar 

  7. Lichtenberg, A. J. and Lieberman, M. A., Regular and Chaotic Dynamics, 2nd ed., Appl. Math. Sci., vol. 38, New York: Springer, 1992.

    Book  MATH  Google Scholar 

  8. Wiggins, S., Chaotic Transport in Dynamical Systems, Interdiscip. Appl. Math., vol. 2, New York: Springer, 1992.

    Book  MATH  Google Scholar 

  9. Hamiltonian Systems with Three or More Degrees of Freedom (S’Agaró, 1995), C. Simó (Ed.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 533, Dordrecht: Kluwer, 1999.

    MATH  Google Scholar 

  10. Zaslavsky, G. M., Chaos, Fractional Kinetics, and Anomalous Transport, Phys. Rep., 2002, vol. 371, no. 6, pp. 461–580.

    Article  MathSciNet  MATH  Google Scholar 

  11. Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci., vol. 42, New York: Springer, 1983.

    Book  MATH  Google Scholar 

  12. Morozov, A. D., Quasi-Conservative Systems: Cycles, Resonances and Chaos, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 30, River Edge, N.J.: World Sci., 1999.

    MATH  Google Scholar 

  13. Karabanov, A. A. and Morozov, A. D., On Degenerate Resonances in Hamiltonian Systems with Two Degrees of Freedom, Chaos Solitons Fractals, 2014, vol. 69, pp. 201–208.

    Article  MathSciNet  MATH  Google Scholar 

  14. Karabanov, A. A. and Morozov, A. D., Degenerate Resonances in Hamiltonian Systems: From Poincaré-Birkhoff Chains to Vortex Pairs and Kármán Vortex Streets, J. Math. Sci. (N. Y.), 2016, vol. 219, no. 1, pp. 155–159; see also: Problemy Matem. Anal., 2016, vol. 85, pp. 147–151.

    Article  MathSciNet  MATH  Google Scholar 

  15. Treshchev, D. V., The Mechanism of Destruction of Resonant Tori in Hamiltonian Systems, Math. USSR Sb., 1991, vol. 68, no. 1, pp. 181–203; see also: Mat. Sb., 1989, vol. 180, no. 10, pp. 1325–1346.

    Article  MathSciNet  MATH  Google Scholar 

  16. Haller, G., Diffusion at Intersecting Resonances in Hamiltonian Systems, Phys. Lett. A, 1995, vol. 200, no. 1, pp. 34–42.

    Article  MathSciNet  MATH  Google Scholar 

  17. 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  MATH  Google Scholar 

  18. Lerman, L., Isoenergetical Structure of Integrable Hamiltonian Systems in Extended Neighborhoods of Simple Singular Points: Three Degrees of Freedom, in Methods of Qualitative Theory of Differential Equations and Related Topics: Supplement, L. M. Lerman, G. Polotovskii, L. P. Shilnikov (Eds.), Amer. Math. Soc. Transl. Ser. 2, vol. 200, Providence, R.I.: AMS, 2000, pp. 219–242.

    Google Scholar 

  19. Honjo, S. and Kaneko, K., Structure of Resonances and Transport in Multidimensional Hamiltonian Dynamical Systems, in Geometric Structures of Phase Space in Multidimensional Chaos: Applications to Chemical Reaction Dynamics in Complex Systems: Part 2, M. Toda, T. Komatsuzaki, T. Konishi, R. S. Berry, S. A. Rice (Eds.), Adv. Chem. Phys., vol. 130, Hoboken, N.J.: Wiley, 2005, pp. 437–463.

    Chapter  Google Scholar 

  20. Gelfreich, V., and Simó, C., and Vieiro, A., Dynamics of 4D Symplectic Maps near a Double Resonance, Phys. D, 2013, vol. 243, pp. 92–110.

    Article  MathSciNet  MATH  Google Scholar 

  21. Medvedev, A. G., Neishtadt, A. I., and Treschev, D. V., Lagrangian Tori near Resonances of Near-Integrable Hamiltonian Systems, Nonlinearity, 2015, vol. 28, no. 7, pp. 2105–2130.

    Article  MathSciNet  MATH  Google Scholar 

  22. Melnikov, V. K., On Some Cases of Conservation of Conditionally Periodic Motions under a Small Change of the Hamiltonian Function, Soviet Math. Dokl., 1965, vol. 6, no. 6, pp. 1592–1596; see also: Dokl. Akad. Nauk SSSR, 1965, vol. 165, no. 6, pp. 1245–1248.

    MATH  Google Scholar 

  23. Moser, J., Convergent Series Expansions for Quasi-Periodic Motions, Math. Ann., 1976, vol. 169, no. 1, pp. 136–176.

    Article  MathSciNet  MATH  Google Scholar 

  24. Arnold, V. I., On the Nonstability of Dynamical Systems with Many Degrees of Freedom, Soviet Math. Dokl., 1964, vol. 5, no. 3, pp. 581–585; see also: Dokl. Akad. Nauk SSSR, 1964, vol. 156, no. 1, pp. 9–12.

    Google Scholar 

  25. Nekhoroshev, N. N., An Exponential Estimate of the Stability Time of Near-Integrable Hamiltonian Systems, Russian Math. Surveys, 1977, vol. 32, no. 6, pp. 1–65; see also: Uspekhi Mat. Nauk, 1977, vol. 32, no. 6(198), pp. 5–66.

    Article  MATH  Google Scholar 

  26. Benettin, G. and Gallavotti, G., Stability of motions near resonances in quasi-integrable Hamiltonian systems, J. Stat. Phys., 1986, vol. 44, nos. 3–4, pp. 293–338.

    Article  MathSciNet  MATH  Google Scholar 

  27. Bogoliubov, N. N. and Mitropolsky, Yu. A., Asymptotic Methods in the Theory of Non-Linear Oscillations, New York: Gordon & Breach, 1961.

    Google Scholar 

  28. Gantmacher, F. R., The Theory of Matrices: In 2 Vols., New York: Chelsea, 1959.

    MATH  Google Scholar 

  29. Takens, F., The Minimal Number of Critical Points of a Function on a Compact Manifold and the Lusternik — Schnirelman Category, Invent. Math., 1968, vol. 6, pp. 197–244.

    Article  MathSciNet  MATH  Google Scholar 

  30. Arnold, V. I., Topological Classification of Morse Functions and Generalisations of Hilbert’s 16th Problem Math. Phys. Anal. Geom., 2007, vol. 10, no. 3, pp. 227–236.

    Article  MathSciNet  MATH  Google Scholar 

  31. Hirsch, M. W., Differential Topology, Grad. Texts in Math., vol. 33, New York: Springer, 1976.

    Book  MATH  Google Scholar 

  32. Meiss, J. D., Differential Dynamical Systems, rev. ed., Math. Model. Comput., vol. 22, Philadelphia, Pa.: SIAM, 2017.

    MATH  Google Scholar 

  33. Giorgilli, A., Delshams, A., Fontich, E., Galgani, L., and Simó, C., Effective Stability for a Hamiltonian System near an Elliptic Equilibrium Point, with an Application to the Restricted Three-Body Problem, J. Differential Equations, 1989, vol. 77, no. 1, pp. 167–198.

    Article  MathSciNet  MATH  Google Scholar 

  34. Devaney, R. L., Homoclinic Orbits in Hamiltonian Systems, J. Differential Equations, 1976, vol. 21, no. 2, pp. 431–438.

    Article  MathSciNet  MATH  Google Scholar 

  35. Korolev, S. A., On Resonances in a System of Two Loosely Coupled Pendulums, Vestn. Nizhegorodsk. Univ., 2010, vol. 5, no. 1, pp. 149–157 (Russian).

    Google Scholar 

Download references

Acknowledgements

This work has been partially supported by the Russian Foundation for Basic Research under grants no. 18-01-00306 and by the Ministry of Education and Science of the Russian Federation (project no. 1.3287.2017/PCh). The authors are sincerely grateful to A. V. Zhubr for useful discussions.

Author information

Authors and Affiliations

  1. Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, University Park, NG7 2RD, UK

    Alexander A. Karabanov

  2. Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia

    Albert D. Morozov

Authors
  1. Alexander A. Karabanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Albert D. Morozov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Alexander A. Karabanov or Albert D. Morozov.

Ethics declarations

The authors declare that they have no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karabanov, A.A., Morozov, A.D. On Resonances in Hamiltonian Systems with Three Degrees of Freedom. Regul. Chaot. Dyn. 24, 628–648 (2019). https://doi.org/10.1134/S1560354719060042

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354719060042

Keywords

MSC2010 numbers

Search

Navigation