We investigate the behavior of one-dimensional non-autonomous dynamical systems obtained by slowly varying over time the bifurcation parameter of the corresponding autonomous systems, i.e., whose bifurcation parameter is constant. In this context, the study focuses on the case for which the time-varying parameter crosses the bistability domain of the corresponding autonomous system. Considering the time-varying parameter as a new (slow) state variable, the considered non-autonomous one-dimensional system becomes a two-dimensional fast-slow system. In the bistability domain, the latter has attracting manifolds (resp. a repelling manifold) associated with the two stable branches (resp. unstable branch) of the bifurcation diagram of the corresponding autonomous system with constant parameter. In the framework of the geometric singular perturbation theory, we define the separatrix in the phase space between what we call the dynamic basin of attraction of the attracting manifolds. Reverse time numerical integration is used to practically compute this separatrix. Finally the proposed methodology is illustrated on a simple musical reed instrument model. This highlights which asymptotic regime is reached (played note or silence) in the bistability domain, depending on how the musician's control is varied over time.