Phase transitions in three-dimensional Kuramoto model under nonlinear mean-field coupling

We investigate a three-dimensional Kuramoto model with nonlinear mean-field coupling, where the interaction strength depends on the macroscopic coherence through nonlinear feedback from the global order parameter. In the thermodynamic limit, a self-consistency analysis reveals multiple solution branches, including an additional branch between partially coherent and completely synchronized states, leading to bistability, controlled by the feedback parameters. This is a distinctive feature of the three-dimensional Kuramoto model with nonlinear mean-field coupling, which is absent in the classical three-dimensional Kuramoto model. Using the reduced dynamics based on the Ott-Antonsen ansatz and direct numerical simulation, we show that the system exhibits a two-step synchronization transition. While the transition from incoherence to partial coherence remains discontinuous at zero coupling for all parameter values, the transition from partial coherence to completely synchronized states can be continuous or discontinuous depending upon the feedback parameters. In the discontinuous regime, the coexistence of steady states leads to hysteresis. The observed hysteresis remains robust over the range of system sizes investigated. We further show that the hysteresis begins only above a threshold value of the feedback strength. These results demonstrate that nonlinear adaptation of the coupling provides a mechanism for controlling synchronization transitions and bistability in higher-dimensional systems.