  The effect of multiplicative stochastic perturbations on Hamiltonian systems
on the plane is investigated. It is assumed that perturbations fade with time
and preserve a stable equilibrium of the limiting system. The paper
investigates bifurcations associated with changes in the stability of the
equilibrium and with the appearance of new stochastically stable states in the
perturbed system. It is shown that depending on the structure and the
parameters of the decaying perturbations the equilibrium can remain stable or
become unstable. In some intermediate cases, a practical stability of the
equilibrium with estimates for the length of the stability interval is
justified. The proposed stability analysis is based on a combination of the
averaging method and the construction of stochastic Lyapunov functions.
