Synchronization in the network-frustrated coupled oscillator with attractive-repulsive frequencies

We investigate the synchronization behavior of a generalized useful mode of the emergent collective behavior in sets of interacting dynamic elements. The network-frustrated Kuramoto model with interaction-repulsion frequency characteristics is presented, and its structural features are crucial to capture the correct physical behavior, such as describing steady states and phase transitions. Quantifying the effect of small-world phenomena on the global synchronization of the given network, the impact of the random phase-shift and their mutual behavior shows particular challenges. In this paper, we derive the phase-locked states and identify the significant synchronization transition points analytically with exact boundary conditions for the correlated and uncorrelated degree-frequency distributions and their full stability analysis. We find that a supercritical to subcritical bifurcation transition occurs depending on the synchronic transition points, characterized by the power scale of the network for the correlated degree frequency and the largest eigenvalue of the network in the uncorrelated case. Furthermore, our frustrated degree-frequency distribution brings us to the classical Kuramoto model with all-to-all coupling, with β=1/2 for the correlated case and λ_{N}=1 for the uncorrelated distribution. In addition, the interplay between the network topology and the frustration forms a powerful alliance, where they control the synchronization ability of the generalized model without affecting its stability.