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Conference publications

Abstracts

XXI conference

Asymptotic and numerical analysis of parametric resonance in a nonlinear system of two ocsillators

Lyulko N.A., Kudryavtsev A.N.1

Sobolev Institute of Mathematics SB RAS, 630090, Novosibirsk, 4 Koptyg pr.

1Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, 630090, Novosibirsk, 4/1 Institutskaya st.

1 pp. (accepted)

A mathematical model of water-oil gas-containing layered systems has been developed in [1]. It has been shown that a periodic external excitation causes a parametric resonance in the linearized distributed system, which leads to a breakdown of the entire system. In the present paper, an initial value problem for a nonlinear system of two oscillators

(1)

which models the nonlinear system in [1] is considered. Here q > 0 is a small parameter, ω is the frequency of the external excitation, ε > 0 is the amplitude of the external excitation, are some parameters of the model.

It has been shown in [2] that the main resonance occurs in (1) at the frequency and the combinational resonance takes place at the frequency . The Krylov-Bogolyubov method is used to investigate the instability of zero solution of the system (1). It allows us to reduce the problem to analysis of a nonlinear autonomous averaged system corresponding to (1). In [2] phase portraits were built and independent integrals of averaged systems were found, which enable to determine the maximum resonance amplitude of oscillations. The goal of the present paper is to solve the original system numerically and compare the numerical solutions with the solutions found with the averaging method. The 6-stage strong stability preserving (SSP) Runge-Kutta scheme of the 5th order, which allows one to reproduce the exact solution precisely even at very large times, is used for numerical integration of the original system.

References

1. Belonosov V.S., Dorovsky V.N. et al. Hydrodynamics of gas-containing layered systems. Uspechi mechaniki 3, 2, 2005, pp. 37-70 (in Russian). 2. Lyulko, N.A. The main and combinational resonances in a nonlinear system of two oscillators. Preprint No 281, Institute of Mathematics SB RAS, Novosibirsk, 2012, 33 p. (in Russian).

The work was supported by Siberian Branch of Russian Academy of Sciences (Presidium Program of Basic Research No 15 and Interdisciplinary Integration Project No 30).



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