Yu. V. Kornienko, L. V. Stulova, D. S. Masalov


Subject and Purpose. The paper is concerned with the behavior of a nonlinear dynamic system that has two degrees of freedom and whose joint nonlinearity is established by all the nonlinear coupling between the degrees of freedom. The purpose is to find out if the Krylov—Bogolyubov—Mitropolsky (KBM) method is applicable to a system of partial differential equations.

Methods and Methodology. The consideration of the problem is by the Krylov—Bogolyubov—Mitropolsky method in the first approximation. Then the results are treated using numerical methods.

Results. An electromechanical system with two degrees of freedom and a known parametric resonance has been studied using the Krylov—Bogolyubov—Mitropolsky method in the first approximation. The phase space of the system has been described. It has been shown that the obtained solution covers an energy periodic transfer between the two degrees of freedom. The difference between the considered oscillation system and its analogs discussed in the literature lies in that the considered circuit is parametrically excited by an internal force rather than external one. In a similar system of two circuits connected through a diode, the coupling includes a linear component. In the system of present concern, the coupling is all-nonlinear.

Conclusion. The obtained results are of interest for the research into internal nonlinear resonances between degrees of freedom in an oscillation system that has two degrees of freedom and whose joint nonlinearity is due to all the nonlinear coupling between the degrees of freedom. The considered system can serve a test example in the development of programs implementing the Krylov—Bo- golyubov—Mitropolsky method as applied to an oscillation system with numerous degrees of freedom and a small nonlinearity.


Manuscript submitted 29.07.2021

Radio phys. radio astron. 2022, 27(1): 017-025



1. Bogolyubov, N.N., Mitropolskiiy, Yu.A., 1958. Asymptotic Method in the Nonlinear Oscillation Th eory. Moscow: Main Publishing House of Physical and Mathematical Literature (in Russian).

2. Mitropolskiiy, Yu.A., 1955. Non-Stationary Processes in Nonlinear Oscillation Systems. Kyiv: Publishing House of Academy of Sciences of UkrSSR (in Russian).

3. Kornienko, Yu.V., 1962. Construction of an Asymptotic Solution for a Wave Equation with Nonlinearity for a Tube. Th e Paper was introduced by Academician Yu.A. Mitropolskij. Reports of the National Academy of Sciences of Ukraine, 7, pp. 845—850 (in Ukrainian).

4. Mitropolskiiy, Yu.A., Mosyeyenkov, B.I., 1976. Asymptotic Solutions for Partial Diff erential Equations. Kyiv: Vyssha Shkola Publ. (in Russian).

5. Mosyeyenkov, B.I., 1955. Scientifi c Students Works of Kyiv State University. Mathematics, 16, p. 49 (in Ukrainian).

6. Kornienko, Yu.V., Masalov, D.S., 2014. Realization of the Krylov-Bogolyubov-Mitropoilskiiy Method in Computer Algebra System. Physical Bases of Instrumentation (Russia), 3(1), pp. 70—83 (in Russian). DOI:

7. Mandelshtam, L.I., 1955. Complete Collection of Works. Moscow: Publishing House of Academy Sciences of USSR (in Russian).

8. Gorelik, G.S., 1959. Oscillations and Waves. Moscow: State Publishing House of Physics and Mathematics (in Russian).

9. Andronov, A.A., Vitt, A.A., Khaikin, S.Ye., 1981. Th e Oscillation Th eory. Moscow: Nauka Publ. (in Russian).

10. Zabolotnov, Yu.M. ed., 1999. Th e Oscillation Th eory: Lecture Notes. Samara: Samara State Aerospace University Publ. (in Russian).

11. Aldoshin, G.P., Yakovlev, S.P., 2012. Oscillating Spring Dynamics. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 4, pp. 45—52 (in Russian).

12. Rabinovich, M.I., Trubetskov, D.I., 2000. Introduction to the Th eory of Oscillations and Waves [pdf]. Scientifi c Publishing Center "Regular and Chaotic Dynamics" (R&C Dymamics) (in Russian). Available from:

13. Kornienko, Yu.V., Stulova, L.V., Masalov, D.S., 2019. Internal Nonlinear Resonances in the Oscillatory System with Two Degrees of Freedom [pdf]. In: All-Russian open Armand readings. Modern problems of remote sensing, radar, wave propagation and diffraction. Materials of Russian open scientifi c conf. Murom. Publishing and Printing Center of MI VSU, 2019, pp. 226—235. ISSN 2304-0297 (CD-ROM) (in Russian). Available from:


nonlinear dynamic system with two degrees of freedom, internal nonlinear resonance, Krylov—Bogolyubov—Mitropolsky method, nonlinear oscillations

Full Text:


Creative Commons License

Licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0) .