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We have examined the synchronization and de-synchronization transitionsobservable in the Kuramoto model with a standard pair-wise first harmonicinteraction plus a higher order (triadic) symmetric interaction for unimodaland bimodal Gaussian distributions of the natural frequencies $\{ \omega_i \}$.These transitions have been accurately characterized thanks to aself-consistent mean-field approach joined with extensive numericalsimulations. The higher-order interactions favour the formation of two clusterstates, which emerge from the incoherent regime via continuous (discontinouos)transitions for unimodal (bimodal) distributions. Fully synchronized initialstates give rise to two symmetric equally populated bimodal clusters, eachcharacterized by either positive or negative natural frequencies. These bimodalclusters are formed at an angular distance $\gamma$, which increases fordecreasing pair-wise couplings until it reaches $\gamma=\pi$ (corresponding toan anti-phase configuration), where the cluster state destabilizes via anabrupt transition: the $\pi$-transition. The uniform clusters that reformimmediately after (with a smaller angle $\gamma$) are composed by oscillatorswith positive and negative $\{ \omega_i \}$. For bimodal distributions we haveobtained detailed phase diagrams involving all the possible dynamical states interms of standard and novel order parameters. In particular, the clusteringorder parameter, here introduced, appears quite suitable to characterize thetwo cluster regime. As a general aspect, hysteretic (non hysteretic)synchronization transitions, mostly mediated by the emergence of standingwaves, are observable for attractive (repulsive) higher-order interactions.

Cluster States and $\pi$-Transition in the Kuramoto Model with Higher Order Interactions