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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.

We study a singularly perturbed fast-slow system of two partial differential equations (PDEs) of reaction-diffusion type on a bounded domain via Galerkin discretisation. We assume that the reaction kinetics in the fast variable realise a generic fold singularity, whereas the slow variable takes the role of a dynamic bifurcation parameter, thus extending the classical analysis of the singularly perturbed fold. Our approach combines a spectral Galerkin discretisation with techniques from geometric singular perturbation theory which are applied to the resulting high-dimensional systems of ordinary differential equations. In particular, we show the existence of invariant slow manifolds in the phase space of the original system of PDEs away from the fold singularity, while the passage past the singularity of the Galerkin manifolds obtained after discretisation is described by geometric desingularisation, or blow-up. Finally, we discuss the relation between these Galerkin manifolds and the underlying slow manifolds.

Marshal Michel Ney, commanding the VI Corps and the French right wing, was tasked with securing Friedland and the Alle River bridges in order to cut off the Russians' escape routes. Gen. Alexandre-Antoine Sénarmont, commanding the artillery of a reserve corps at the center of the French line, obtained permission to move up independently and test his theories on the use of massed artillery. Bennigsen later claimed he'd believed Lannes' corps was isolated, and he hadn't expected French reinforcements to march the same distance in 12 hours his own force had taken 24 to cover.