Third order interactions shift the critical coupling in multidimensional Kuramoto models

The study of higher order interactions in the dynamics of Kuramoto oscillators has been a topic of intense research. Previous works have demonstrated that such interactions can give rise to interesting new phenomena such as multi-stability and synchronization even if the interaction between oscillators is repulsive. Here we consider higher order interactions in the multidimensional Kuramoto model where pairs (1-simplex), triplets (2-simplex) and quadruples (3-simplex) of oscillators interact simultaneously with different coupling strengths, k1, k2 and k3, respectively. For the types of asymmetric interactions considered, we show that three body terms shift the critical coupling for synchronization towards higher values, except in 2 dimensions where a cancellation occurs. However, after the transition, three and four body interactions combine to facilitate synchronization. We also show that, for fixed values of k2 and k3, and fully connected networks, the behavior of the order parameter r(k1) is described by a universal curve given by its value at k2=k3=0, shifted along the k1 axis. Similar to the 2-dimensional case with asymmetric interactions, bi-stability and hysteresis develop for large enough higher order interactions. Multi-stability, typical of symmetric higher order interactions, is not found. We show simulations in three and four dimensions to illustrate the dynamics.