Tiered synchronization in coupled oscillator populations with interaction delays and higher-order interactions

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.