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The influence of multiplicative stochastic perturbations on the class ofasymptotically Hamiltonian systems on the plane is investigated. It is assumedthat disturbances do not preserve the equilibrium of the corresponding limitingsystem and their intensity decays in time with power-law asymptotics. The paperdiscusses the long-term asymptotic behaviour of solutions and its dependence onthe structure and parameters of perturbations. In particular, it is shown thatperturbed trajectories can tend to the equilibrium of the limiting system ornew stochastically stable states can arise. The proposed analysis is based on acombination of the averaging method and the construction of stochastic Lyapunovfunctions. The results obtained are applied to the problem of capture intoparametric autoresonance in the presence of noise.

Long-term behaviour of asymptotically autonomous Hamiltonian systems with multiplicative noise