Abstract
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).
Similar content being viewed by others
References
H. Poincaré, New Methods of Celestial Mechanics, Vols. 1–3, NASA (1967).
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).
V. I. Arnol’d, Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow (1974).
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).
J. K. Moser, Integrable Hamiltonian Systems and Spectral Theory [Russian translation], Izhevsk, Izhevskaya Respublikanskaya Tipografiya (1999).
E. A. Grebenikov, Averaging Method in Applied Problems [in Russian], Nauka, Moscow (1986).
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).
N. N. Bogolyubov, Yu. A. Mitropol’skii, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).
E. A. Grebenikov and Yu. A. Ryabov, New Qualitative Methods in Celestial Mechanics [in Russian], Nauka, Moscow (1971).
A. Zygmund, Trigonometric Series, Vol. 1, Cambridge University Press, Cambridge (1959).
E. A. Grebenikov and Yu. A. Ryabov, Resonances and Small Denominators in Celestial Mechanics [in Russian], Nauka, Moscow (1979).
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.
C. L. Siegel, “Iterations of analytic functions,” Ann. Math., 43, No. 4, 807–812 (1942).
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.
A. V. Bolsinov and A. T. Fomenko, Introduction to Topology of Integrable Hamiltonian Systems [in Russian], Nauka, Moscow (1997).
H. Poincaré, On Curves Defined by Differential Equations [Russian translation], Nauka, Moscow (1978).
G. E. Shilov, Mathematical Analysis [in Russian], Fizmatgiz, Moscow (1961).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 56–62, January, 2008.
Rights and permissions
About this article
Cite this article
Grebenikov, E.A. On one mathematical problem in the theory of nonlinear oscillations. Ukr Math J 60, 59–65 (2008). https://doi.org/10.1007/s11253-008-0041-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-008-0041-8