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A class of asymptotically autonomous systems on the plane with oscillatorycoefficients is considered. It is assumed that the limiting system isHamiltonian with a stable equilibrium. The effect of damped multiplicativestochastic perturbations of white noise type on the stability of the system isdiscussed. It is shown that different long-term asymptotic regimes forsolutions are admissible in the system and the stochastic stability of theequilibrium depends on the realized regime. In particular, we show that stablephase locking is possible in the system due to decaying stochasticperturbations. The proposed analysis is based on a combination of the averagingtechnique and the construction of stochastic Lyapunov functions.

Stability of asymptotically Hamiltonian systems with damped oscillatory and stochastic perturbations