  In the present work, we explore the existence, stability and dynamics of
single and multiple vortex ring states that can arise in Bose-Einstein
condensates. Earlier works have illustrated the bifurcation of such states, in
the vicinity of the linear limit, for isotropic or anisotropic
three-dimensional harmonic traps. Here, we extend these states to the regime of
large chemical potentials, the so-called Thomas-Fermi limit, and explore their
properties such as equilibrium radii and inter-ring distance, for multi-ring
states, as well as their vibrational spectra and possible instabilities. In
this limit, both the existence and stability characteristics can be partially
traced to a particle picture that considers the rings as individual particles
oscillating within the trap and interacting pairwise with one another. Finally,
we examine some representative instability scenarios of the multi-ring dynamics
including breakup and reconnections, as well as the transient formation of
vortex lines.
