  The reversible-irreversible (RI) transition of particle trajectories in
athermal colloidal suspensions under cyclic shear deformation is an archetypal
nonequilibrium phase transition which attracts much attention recently. In the
low-density limit, the RI transition is predicted to belong to a universality
class of the absorbing state transitions, whereas at the high densities well
above the jamming transition density, $\varphi_{\rm J}$, the transition is
discontinuous and is closely related to the yielding transition. The transition
between the two limiting cases is largely unexplored. In this paper, we study
the RI transition of athermal frictionless colloidal particles over a wide
range of densities, with emphasis on the region below $\varphi_{\rm J}$, by
using oscillatory sheared molecular dynamics simulation. We reveal that the
nature of the RI transitions in the intermediate densities is very rich. As
demonstrated by the previous work by Schreck $et\ al.$ [Phys. Rev. E. ${\bf
88}$, 052205 (2013)], there exist the point-reversible and the loop-reversible
phases depending on the density and the shear strain amplitude. We find that,
between the two reversible phases, a quasi-irreversible phase where the
particles' trajectories are highly non-affine and diffusive. The averaged
number of contacts of particles is found to characterize the phase boundaries.
We also find that the system undergoes the yielding transition below but in the
vicinity of $\varphi_{\rm J}$ when the strain with a small but finite strain
rate is applied. This yielding transition line matches with the RI transition
line separating the loop-reversible from the irreversible phases. Surprisingly,
the nonlinear rheological response called "softening" has been observed even
below $\varphi_{\rm J}$. These findings imply that geometrical properties
encoded in the sheared configurations control the dynamical transitions.
