Phase transitions in an adaptive network with the global order parameter adaptation

We consider an adaptive network of Kuramoto oscillators with purely dyadic coupling, where the adaption is proportional to the degree of the global order parameter. We find only the continuous transition to synchronization via the pitchfork bifurcation, an abrupt synchronization (desynchronization) transition via the pitchfork (saddle-node) bifurcation resulting in the bistable region ${R}_{1}$. This is a smooth continuous transition to a weakly synchronized state via the pitchfork bifurcation followed by a subsequent abrupt transition to a strongly synchronized state via a second saddle-node bifurcation along with an abrupt desynchronization transition via the first saddle-node bifurcation resulting in the bistable region ${R}_{2}$ between the weak and strong synchronization. The transition goes from the bistable region ${R}_{1}$ to the bistable region ${R}_{2}$, and transition from the incoherent state to the bistable region ${R}_{2}$ as a function of the coupling strength for various ranges of the degree of the global order parameter and the adaptive coupling strength. We also find that the phase-lag parameter enlarges the spread of the weakly synchronized state and the bistable states ${R}_{1}$ and ${R}_{2}$ to a large region of the parameter space. We also derive the low-dimensional evolution equations for the global order parameters using the Ott-Antonsen ansatz. Further, we also deduce the pitchfork, first and second saddle-node bifurcation conditions, which is in agreement with the simulation results.