Newtonian core-shell systems, as limiting cases of relativistic core-shell models under the two conditions of weak field and slow motion, could account for massive circumstellar dust shells and rings around certain types of star remnants. Because this kind of systems have Hamiltonians that can be split into a main part and a small perturbing part, a good choice of the numerical tool is the pseudo 8th order symplectic integrator of Laskar & Robutel, and, to match the symplectic calculations, a good choice of chaos indicator is the fast Lyapunov indicator (FLI) with two nearby trajectories proposed by Wu, Huang & Zhang. Numerical results show that the FLI is very powerful when describing not only the transition from regular motion to chaos but also the global structure of the phase space of the system.