  We study synchronization in large populations of coupled phase oscillators
with time-delays, higher order interactions. With each of these effects
individually giving rise to bistabiltiy between incoherence and synchronization
via a subcriticality at the onset of synchronization and the development of a
saddle node, we find that their combination yields another mechanism behind
bistability, where supercriticality at onset may be maintained and instead the
formation of two saddle nodes creates tiered synchronization, i.e., bistability
between a weakly synchronized state and a strongly synchronized state. We
demonstrate these findings by first deriving the low dimensional dynamics of
the system and examining the system bifurcations using a stability and
steady-state analysis.
