Abstract
We describe an asymptotic method for the integration of m-frequency oscillation systems of order 2n, analyze averaged equations in nonresonance and resonance cases, prove a theorem on the preservation of smooth p-dimensionai invariant tori under perturbation for any 0 ≤ p ≤ n, and indicate forms of decomposable m-frequency oscillation systems.
Similar content being viewed by others
References
Yu. A. Mitropol’skii and A. M. Samoilenko, “On the problem of asymptotic expansions in nonlinear mechanics,” Ukr. Mat. Zh., 31, No. 1, 42–53 (1979).
Yu. A. Mitropol’skii and A. M. Samoilenko, General Problems of the Theory of Asymptotic Integration of Systems in Nonlinear Mechanics [in Russian], Preprint No. 87.41, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987).
F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1967).
A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations [in Russian], Nauka, Moscow (1987).
A. D. Bryuno, Local Method of Nonlinear Analysis of Differential Equations [in Russian], Nauka, Moscow (1979).
A. M. Samoilenko, “On asymptotic expansions of solutions of systems in nonlinear mechanics,” in: Proceedings of the IX International Conference on Nonlinear Oscillations [in Russian], Vol. 1, Naukova Dumka, Kiev (1984), pp. 323–333.
A. M. Samoilenko, “On some problems in the theory of perturbations of smooth invariant tori of dynamical systems,” Ukr. Mat. Zh., 46, No. 12, 1665–1669(1994).
A. M. Samoilenko, “N. N. Bogolyubov and nonlinear mechanics,” Usp. Mat. Nauk, 49, Issue 12, 103–146 (1994).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 10, pp. 1366–1387, October, 1998.
Rights and permissions
About this article
Cite this article
Samoilenko, A.M. Asymptotic method for the investigation of m-frequency oscillation systems. Ukr Math J 50, 1559–1585 (1998). https://doi.org/10.1007/BF02513504
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02513504