Transition to synchronization in the adaptive Sakaguchi-Kuramoto model with higher-order interactions

We investigate the phenomenon of transition to synchronization in the Sakaguchi-Kuramoto model in the presence of higher-order interactions and global order parameter adaptation. The investigation is done by performing extensive numerical simulations and low-dimensional modeling of the system. Numerical simulations of the full system show both continuous (second-order) as well as discontinuous transitions. The discontinuous transitions can either be associated with explosive (first-order) or tiered synchronization states depending on the choice of parameters. To develop an in depth understanding of the transition scenario in the parameter space we derive a reduced order model (ROM) using the Ott-Antonsen ansatz, the results of which closely match with those of the numerical simulations of the full system. The simplicity and analytical accessibility of the ROM help to conveniently unfold the transition scenario in the system having complex dependence on the parameters. Simultaneous analysis of the full system and the ROM clearly identifies the regions of the parameter space exhibiting different types of transitions. It is observed that the second-order transition is connected with a supercritical pitchfork bifurcation (PB) of the ROM. On the other hand, the discontinuous tiered transition is associated with multiple saddle-node (SN) bifurcations along with a supercritical PB and the first-order transition involves a subcritical PB alongside a SN bifurcation. Finally, the stability analysis of the different synchronization states of the system is performed analytically.