Skip to main content
Log in

Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case

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

Abstract

This paper concerns with the study of the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with 1-degree of freedom in the degenerate case H = q4 + H5 + H6 +.... Our main results complement the study initiated by Markeev in [9].

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. Bardin, B. and Lanchares, V., On the Stability of Periodic Hamiltonian Systewith One Degree of Freedom in the Case of Degeneracy, Regul. Chaoms tic Dyn., 2015, vol. 20, no. 6, pp. 627–648.

    Article  MATH  Google Scholar 

  2. Birkhoff, G.D., Dynamical Systems, Providence,RI: AMS, 1966.

    MATH  Google Scholar 

  3. Cabral, H.E. and Meyer, K. R., Stability of Equilibria and Fixed Points of Conservative Systems, Nonlinearity, 1999, vol. 12, no. 5, pp. 1351–1362.

    Article  MathSciNet  MATH  Google Scholar 

  4. Chetayev, N.G., The Stability of Motion, Oxford: Pergamon, 1961.

    Google Scholar 

  5. Deprit, A., Canonical Transformations Depending on a Small Parameter, Celest. Mech., 1969/1970, vol. 1, pp. 12–30.

    Article  MathSciNet  MATH  Google Scholar 

  6. Gustavson, F., On Constructing Formal Integrals of a Hamiltonian System Near an Equilibrium Point, Astron. J., 1966, vol. 71, no. 8, pp. 670–686.

    Article  Google Scholar 

  7. Krasovskii, N., Stability of Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay, Stanford,Calif.: Stanford Univ. Press, 1963.

    MATH  Google Scholar 

  8. Markeev, A.P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).

    Google Scholar 

  9. Markeev, A.P., Simplifying the Structure of the Third and Fourth Degree Forms in the Expansion of the Hamiltonian with a Linear Transformation, Nelin. Dinam., 2014, vol. 10, no. 4, pp. 447–464 (Russian).

    Article  MATH  Google Scholar 

  10. Markeev, A.P., On the Birkhoff Transformation in the Case of Complete Degeneracy of Quadratic Part of the Hamiltonian, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 309–316.

    Article  MathSciNet  MATH  Google Scholar 

  11. Markeev, A.P., On the Fixed Points Stability for the Area-Preserving Maps, Nelin. Dinam., 2015, vol. 11, no. 3, pp. 503–545 (Russian).

    Article  MATH  Google Scholar 

  12. Markeev, A.P., On the Problem of the Stability of a Hamiltonian System with One Degree of Freedom on the Boundaries of Regions of Parametric Resonance, J. Appl. Math. Mech., 2016, vol. 80, no. 1, pp. 1–6; see also: Prikl. Mat. Mekh., 2016, vol. 80, no. 1, pp. 3–10.

    Article  MathSciNet  Google Scholar 

  13. Meyer, K.R., Hall, G.R., and Offin, D., Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2nd ed., Appl. Math. Sci., vol. 90, New York: Springer, 2009.

  14. Siegel, C. L. and Moser, J. K., Lectures on Celestial Mechanics, Grundlehren Math. Wiss., vol. 187, New York: Springer, 1971.

  15. Sokol’skii, A. G., On the Stability of an Autonomous Hamiltonian System with Two Degrees of Freedom in the Case of Equal Frequencies, J. Appl. Math. Mech., 1974, vol. 38, no. 5, pp. 741–749; see also: Prikl. Mat. Mekh., 1974, vol. 38, no. 5, pp. 791–799.

    Article  MathSciNet  Google Scholar 

  16. Sokol’skii, A. G., On Stability of an Autonomous Hamiltonian System with Two Degrees of Freedom under First-Order Resonance, J. Appl. Math. Mech., 1977, vol. 41, no. 1, pp. 20–28; see also: Prikl. Mat. Mekh., 1977, vol. 41, no. 1, pp. 24–33.

    Article  MathSciNet  MATH  Google Scholar 

  17. Sokol’skiĭ, A. G., On Stability of Self-Contained Hamiltonian System with Two Degrees of Freedom in the Case of Zero Frequencies, J. Appl. Math. Mech., 1981, vol. 45, no. 3, pp. 321–327; see also: Prikl. Mat. Mekh., 1981, vol. 45, no. 3, pp. 441–449.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rodrigo Gutierrez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gutierrez, R., Vidal, C. Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case. Regul. Chaot. Dyn. 22, 880–892 (2017). https://doi.org/10.1134/S1560354717070097

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354717070097

Keywords

MSC2010 numbers

Navigation