Universal phase transitions to synchronization in Kuramoto-like models with heterogeneous coupling

We reveal a class of universal phase transitions to synchronization in Kuramoto-like models with both in- and out-coupling heterogeneity. By analogy with metastable states, an oscillatory state occurs as a high-order coherent phase accompanying explosive synchronization in the system. The critical points of synchronization transition and the stationary solutions are obtained analytically, by the use of mean-field theory. In particular, the stable conditions for the emergence of phase-locked states are determined analytically, consistently with the analysis based on the Ott-Antonsen manifold. We demonstrate that the in- or out-coupling heterogeneity have influence on both the dynamical properties (eigen'spectrum) and the synchronizability of the system.