In the case of weakly damped Hamiltonian systems with one saddle point with a double homoclinic orbit, the boundaries of the basin of attraction (that is, the stable manifolds of the saddle point) have a simple topology. However, in the case of weakly damped Hamiltonian systems with two saddle points and two double heteroclinic orbits, there are a countable infinity of possible topological situations. It is shown that these may be described by sequences of 4 parameters, namely the values of the associated Melnikov integrals. This seminar describes work by Haberman and Rand.