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Numerical and analytical studies on model gravitating systems

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.