Asymptotic solution of the autoresonance problem

We study the problem of forced oscillations near a stable equilibrium of a two-dimensional nonlinear Hamiltonian system of equations. A given exciting force is represented as rapid oscillations with a small amplitude and a slowly varying frequency. We study the conditions under which such a perturbation makes the phase trajectory of the system recede from the original equilibrium point to a distance of the order of unity. To study the problem, we construct asymptotic solutions using a small amplitude parameter. We present the solution for not-too-small values of time outside the original boundary layer.