Ukrainian Mathematical Journal

, Volume 60, Issue 1, pp 59–65 | Cite as

On one mathematical problem in the theory of nonlinear oscillations

  • E. A. Grebenikov


We consider one mathematical problem that was discussed by the author and A. M. Samoilenko at the Third International Conference on the Theory of Nonlinear Oscillations (Transcarpathia, 1967).


Hamiltonian System Mathematical Problem Nonlinear Oscillation Celestial Mechanic Integrable Hamiltonian System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Poincaré, New Methods of Celestial Mechanics, Vols. 1–3, NASA (1967).Google Scholar
  2. 2.
    A. N. Kolmogorov, “On preservation of quasiperiodic motions under a small variation in the Hamiltonian function,” Dokl. Akad. Nauk SSSR, 98, No. 4, 527–530 (1954).zbMATHMathSciNetGoogle Scholar
  3. 3.
    V. I. Arnol’d, Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow (1974).Google Scholar
  4. 4.
    V. I. Arnol’d, “Small denominators and the problem of stability in classical and celestial mechanics,” Usp. Mat. Nauk, 18, Issue 6, 91–192 (1963).Google Scholar
  5. 5.
    J. K. Moser, Integrable Hamiltonian Systems and Spectral Theory [Russian translation], Izhevsk, Izhevskaya Respublikanskaya Tipografiya (1999).zbMATHGoogle Scholar
  6. 6.
    E. A. Grebenikov, Averaging Method in Applied Problems [in Russian], Nauka, Moscow (1986).Google Scholar
  7. 7.
    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Academy of Sciences of the USSR, Moscow (1963).Google Scholar
  8. 8.
    N. N. Bogolyubov, Yu. A. Mitropol’skii, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).Google Scholar
  9. 9.
    E. A. Grebenikov and Yu. A. Ryabov, New Qualitative Methods in Celestial Mechanics [in Russian], Nauka, Moscow (1971).Google Scholar
  10. 10.
    A. Zygmund, Trigonometric Series, Vol. 1, Cambridge University Press, Cambridge (1959).zbMATHGoogle Scholar
  11. 11.
    E. A. Grebenikov and Yu. A. Ryabov, Resonances and Small Denominators in Celestial Mechanics [in Russian], Nauka, Moscow (1979).Google Scholar
  12. 12.
    E. Grebenikov, D. Kozak-Skoworodkin, and M. Jakubiak, “The algebraic problems of the normalization in Hamiltonian theory,” in: Mathematical Systems in Teaching and Research (2000), pp. 73–90.Google Scholar
  13. 13.
    C. L. Siegel, “Iterations of analytic functions,” Ann. Math., 43, No. 4, 807–812 (1942).Google Scholar
  14. 14.
    D. V. Anosov, S. Kh. Aranson, I. U. Bronshtein, and V. Z. Grines, “Smooth dynamical systems,” in: VINITI Series in Contemporary Problems of Mathematics, Fundamental Trends [in Russian], Vol. 1, VINITI, Moscow (1985), pp. 151–242.Google Scholar
  15. 15.
    A. V. Bolsinov and A. T. Fomenko, Introduction to Topology of Integrable Hamiltonian Systems [in Russian], Nauka, Moscow (1997).zbMATHGoogle Scholar
  16. 16.
    H. Poincaré, On Curves Defined by Differential Equations [Russian translation], Nauka, Moscow (1978).Google Scholar
  17. 17.
    G. E. Shilov, Mathematical Analysis [in Russian], Fizmatgiz, Moscow (1961).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • E. A. Grebenikov
    • 1
  1. 1.Computing CenterRussian Academy of SciencesMoscowRussia

Personalised recommendations