Asymptotic regimes in oscillatory systems with damped non-resonant perturbations

An autonomous system of ordinary differential equations describing nonlinear oscillations on the plane is considered. The influence of time-dependent perturbations decaying at infinity in time is investigated. It is assumed that the perturbations satisfy the non-resonance condition and do not vanish at the equilibrium of the limiting system. Possible long-term asymptotic regimes for perturbed solutions are described. In particular, we show that the perturbed system can behave like the corresponding limiting system or new asymptotically stable regimes may appear. The proposed analysis is based on the combination of the averaging technique and the construction of Lyapunov functions.