Stirring N-body systems II: Necessary conditions for the dark matter
  attractor
  We study the evolution of the phase-space of collisionless N-body systems
under repeated stirrings or perturbations, which has been shown to lead to a
convergence towards a limited group of end states. This so-called attractor was
previously shown to be independent of the initial system and environmental
conditions. However the fundamental reason for its appearance is still unclear.
It has been suggested that the origin of the attractor may be either radial
infall (RI), the radial orbit instability (ROI), or energy exchange which, for
instance, happens during violent relaxation. Here we examine the effects of a
set of controlled perturbations, referred to as `kicks', which act in addition
to the standard collisionless dynamics by allowing pre-specified instantaneous
perturbations in phase-space. We first demonstrate that the attractor persists
in the absence of RI and ROI by forcing the system to expand. We then consider
radial velocity kicks in a rigid potential and isotropic velocity kicks, since
there are no energy exchanges in these two recipes of kicks. We find that these
kicks do not lead to the attractor, indicating that the energy exchange in a
dynamic potential is indeed the physical mechanism responsible for the
attractor.
