On the efficiency of pairwise Hamiltonian control to desynchronize the higher-order Kuramoto model

Synchronization of coupled oscillators is observed in many natural and engineered systems and emerges due to the interactions within the system. It can be both beneficial, e.g., in power grids, and harmful, e.g., in epileptic seizures. In the latter case, efficient control methods to desynchronize the systems are crucial. Recent studies have shown that interactions are not always pairwise, but higher-order, i.e., many-body, and this greatly affects the dynamics. For instance, higher-order interactions increase the linear stability of synchronized states but simultaneously shrink their attraction basin, with potentially opposite effects on control methods. Here, we use a minimally invasive pairwise control based on Hamiltonian control theory and investigate its efficiency on phase oscillators with higher-order interactions. We show that, if the initial phases are close to the synchronized state, higher-order interactions make desynchronization more difficult to achieve. Otherwise, a non-monotonic effect appears: intermediate strengths of higher-order interactions impede desynchronization while larger ones facilitate it. In all cases, the control can desynchronize the system with a sufficient number of controlled nodes and intensity.