  In this thesis we study the evolution of systems of concentric shells
interacting gravitationally and in the process (1) propose and implement a
nearly energy-conserving numerical integration scheme for evolving the
concentric spherical shells systems with 1024 particles or less; (2) look at
the possibility of chaos in few shell systems; and (3) study the evolution of
many shell systems in the Vlasov limit.
  The proposed numerical integration scheme is a nearly energy conserving
hybrid of the Verlet and modified Euler-Cromer integration schemes.
  The rotational 2-shell spherical system is investigated in detail using the
hybrid numerical integration scheme. Plots of time-series, phase space
projections, Poincare sections, power spectra, and Lyapunov exponents are
obtained for the system. These diagnostic tools, taken together, clearly show
the chaotic nature of the rotational 2-shell system. Three types of periodic
orbits are observed: collapsed, one-point, and three-point periodic orbits. We
believe that the three-point periodic orbits result from a rotation-induced
bifurcation. Four types of quasiperiodic orbits are also observed. Three of
these are a result of slight changes in the initial conditions corresponding to
the three types of periodic orbits. The fourth type of quasiperiodic orbit
separates the chaotic region from the non-chaotic regions in phase space.
  The short-time evolution of collisionless spherical shells system is studied
using both numerical and analytical methods. Approximate expressions for the
short-time evolution of the collisionless rotational shells system are obtained
using Vlasov-Poisson perturbation theory in the high-virial limit. The
agreement between the analytical results and numerical results for finite
shells systems improves as the number of shells in the system increases.
