  The influence of time-dependent perturbations on an autonomous Hamiltonian
system with an equilibrium of center type is considered. It is assumed that the
perturbations decay at infinity in time and vanish at the equilibrium of the
unperturbed system. In this case the stability and the long-term behaviour of
trajectories depend on nonlinear and non-autonomous terms of the equations. The
paper investigates bifurcations associated with a change of Lyapunov stability
of the equilibrium and the emergence of new attracting or repelling states in
the perturbed asymptotically autonomous system. The dependence of bifurcations
on the structure of decaying perturbations is discussed.
