Numerical and analytical studies on model gravitating systems

TL;DR

This thesis investigates the dynamics of gravitating shell systems through numerical and analytical methods, revealing chaotic behavior, periodic orbits, and the evolution of large shell systems approaching the Vlasov limit.

Contribution

It introduces a nearly energy-conserving numerical scheme and analyzes chaos, periodic, and quasiperiodic orbits in shell systems, advancing understanding of their complex dynamics.

Findings

Chaotic nature of the rotational 2-shell system confirmed.

Identification of three types of periodic orbits and quasiperiodic orbits.

Analytical expressions for short-time evolution match numerical results as shells increase.

Abstract

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.