Geometric analysis of a two-body problem with quick loss of mass

We consider a two-body problem with quick loss of mass which was formulated by Verhulst in [14]. The corresponding dynamical system is singularly perturbed due to the presence of a small parameter in the governing equations which corresponds to the reciprocal of the initial rate of loss of mass, resulting in a boundary layer in the asymptotics. Here, we showcase a geometric approach which allows us to derive asymptotic expansions for the solutions of that problem via a combination of geometric singular perturbation theory [5] and the desingularisation technique known as \blow-up" [4]. In particular, we justify the unexpected dependence of those expansions on fractional powers of the singular perturbation parameter; moreover, we show that the occurrence of logarithmic (\switchback") terms therein is due to a resonance phenomenon that arises in one of the coordinate charts after blow-up.