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

Abstract

We consider a two-body problem with quick loss of mass which was formulated by Verhulst (Verhulst in J Inst Math Appl 18: 87–98, 1976). 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 (Fenichel in J Differ Equ 31: 53–98, 1979) and the desingularization technique known as “blow-up” (Dumortier, in: Bifurcations and Periodic Orbits of Vector Fields, Springer, Dordrecht, 1993). 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.