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

A parametric resonance in a nonlinear system of ordinary differential equations, which is a mathemetical model of a water–oil gas containing layer, is considered. The Krylov–Bolgoliubov–Mitropolsky averaging method is applied to investigate the instability of zero solution of the system and deduce averaged equations for time evolution of the amplitude of oscillations in the cases of main and combinational resonances. The original and averaged equations are also integrated numerically with a high-order strong stability preserving Runge–Kutta scheme. By comparing the numerical solutions it is shown that the averaged equations enable us to predict correctly the maximum amplitude of oscillations and the time moment when it is achieved. The dependence of resonance characteritics on the small parameter is also studied.