Synchronization transition via rhythmic states in the four-dimensional Kuramoto model with isoclinic rotations

In this work, we are devoted to examining the phase transition to synchronization in the four-dimensional Kuramoto model with isoclinic rotations, where the rotation rate 4×4 antisymmetric matrix of each uncoupled agent is generated by a real three-dimensional vector. We uncover that the transition from incoherence to partial synchronization is mediated by time-dependent rhythmic states as the strength of coupling increases. The incoherent state is observed for a coupling strength below a certain threshold. Subsequently, a time-dependent rhythmic state appears as further increasing the strength of coupling, which persists for a pronounced interval of coupling strength. For a sufficiently large coupling strength, the system finally transits to partially locked states, where the generalized order parameter goes to a nontrivial fixed point. Via employing a higher-dimensional Ott-Antonsen ansatz in the thermodynamic limit of infinite system size, we theoretically establish that the uniformly incoherent state loses its stability via Hopf bifurcation at the critical coupling strength, which signals the emergence of a rhythmic state. We also obtain a self-consistency equation of the order parameter of the model undergoing partially locked states, from which the degree of coherence of the system at a sufficiently large coupling strength is theoretically predicted by two parametrized expressions. Our theoretical results agree very well with the results from our numerical simulations of the model with a sufficiently large but finite number of agents.