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

摘要

A parametric resonance in a nonlinear system of ordinarydifferential equations, which is a mathemetical model of a water–oilgas containing layer, is considered. The Krylov–Bolgoliubov–Mitropolskyaveraging method is applied to investigate the instability of zero solutionof the system and deduce averaged equations for time evolution of theamplitude of oscillations in the cases of main and combinational resonances.The original and averaged equations are also integrated numericallywith a high-order strong stability preserving Runge–Kutta scheme. Bycomparing the numerical solutions it is shown that the averaged equationsenable us to predict correctly the maximum amplitude of oscillationsand the time moment when it is achieved. The dependence of resonancecharacteritics on the small parameter is also studied.