Stability and bifurcation phenomena in asymptotically Hamiltonian systems

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.