Abrupt desynchronization and abrupt transition to π-state in globally coupled oscillator simplexes with contrarians and conformists

We study the phase transitions in a generalized Kuramoto model with contrarians and conformists along with a higher order coupling encoded simplex. Incoherent state persists for a large fraction of contrarians, while there is a continuous transition from the former to π-state and traveling wave state as the fraction of conformists increases under the high proportion of the pair-wise coupling and for a higher coupling strength of the conformists than contrarians. Hong and Strogatz (2011) showed the abrupt transition to the π-state when the contrarians coupling strength supersede in a purely pair-wise coupling. A large proportion of higher order coupling induces the abrupt transition to the π-state even when the contrarians coupling strength is feeble than that of the conformists, destabilizes the traveling wave state, and stabilizes the incoherent state in the entire phase diagram despite the presence of a large fraction of the conformists, which coexists with the π-state. Consequently, one can observe an abrupt desynchronization transition to the incoherent state during the backward trace, while the system remains in the incoherent state during the forward trace. The range of the bistable region between the incoherent state and the π-state increases with the proportion of the higher order coupling to a maximum until the abrupt transition to the π-state ceases to exist. We deduce the pitchfork and saddle–node bifurcation curves across which the incoherent state and the π-state lose their stability from the evolution equation for the low-dimensional dynamics corresponding to the macroscopic order parameters.