Exploring Rigidly Rotating Vortex Configurations and their Bifurcations in Atomic Bose-Einstein Condensates

In the present work, we consider the problem of a system of few vortices $N \leq 5$ as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion, we use a two-pronged approach in order to reveal the configuration space of the system's preferred dynamical states. On the one hand, we use a Monte-Carlo method parametrizing the vortex "particles" by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for $N=2, ..., 5$ and different vortex particle angular momenta. We then complement this picture with a dynamical systems analysis of the possible rigidly rotating states. The latter reveals all the supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise exposing the full wealth of the problem even at such low dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte-Carlo approach selects for different values of the angular momentum which is used as a bifurcation parameter.