  A class of asymptotically autonomous systems on the plane with oscillatory
coefficients is considered. It is assumed that the limiting system is
Hamiltonian with a stable equilibrium. The effect of damped multiplicative
stochastic perturbations of white noise type on the stability of the system is
discussed. It is shown that different long-term asymptotic regimes for
solutions are admissible in the system and the stochastic stability of the
equilibrium depends on the realized regime. In particular, we show that stable
phase locking is possible in the system due to decaying stochastic
perturbations. The proposed analysis is based on a combination of the averaging
technique and the construction of stochastic Lyapunov functions.
