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, k 1 , k 2 and k 3 , 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 k 2 and k 3 , and fully connected networks, the behavior of the order parameter r (k 1) is described by a universal curve given by its value at k 2 = k 3 = 0 , shifted along the k 1 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. • Higher order asymmetric interactions are extended to the multidimensional Kuramoto model. • We show that three body interactions shift the critical coupling, except in two dimensions. • After the critical point, three and four body interactions facilitate synchronization. • Bi-stability and hysteresis develop for large enough higher order interactions. • We show simulations in D = 3 and 4 to illustrate the dynamics. [ABSTRACT FROM AUTHOR]