The paper “Basins of attraction in forced systems with time-varying dissipation“, co-authored by James Wright, Jonathan Deane, Michele Bartuccelli, and Guido Gentile (Roma III, and VP at Surrey), has been published in Communications in Nonlinear Science and Numerical Simulation.  A link to the online version is here.  In the paper they consider periodically forced systems with damping coefficient growing very slowly in time towards a limiting value, and outline a method to reduce the computation time to estimate the relative areas of the basins of attraction of the coexisting attractors.