Instability of a nonlinear system of two oscillators under main and combination resonances

A nonlinear reversible system of two oscillators depending on a small parameter $q>0$ is considered. The instability of the zero equilibrium of this system under a nonautonomous periodic perturbation is analyzed using the Krylov–Bogolyubov averaging method. In the case of main and combination resonances, independent integrals of the averaged autonomous nonlinear system are found, which are used to determine the maximum amplitude of oscillations of solutions to the original system for small $q$. In the case of the main resonance, the averaged system is reduced to a completely integrable Hamiltonian system by making a change of variables. In the case of combination resonance, the averaged system is integrated by applying the integrals found.