Low dimensional Watanabe-Strogatz approach for Kuramoto oscillators with higher-order interactions

Watanabe-Strogatz theory provides a low-dimensional description of identical Kuramoto oscillators via the framework of the M\"obius transformation. Here, using the Watanabe-Strogatz theory, we provide a unifying description for a broad class of identical Kuramoto oscillator models with pairwise and higher-order interactions and their corresponding higher harmonics. We show that the dynamics of the Watanabe-Strogatz parameters are the same as those of the mean-field parameters. Additionally, the poles of the M\"obius transformation serve as basin boundaries for both global and cluster synchronization in the models discussed here. We present numerical simulations that illustrate how the basins boundaries evolve for these extended models.