Synchronization in starlike networks of phase oscillators

We fully describe the mechanisms underlying synchronization in starlike networks of phase oscillators. In particular, the routes to synchronization and the critical points for the associated phase transitions are determined analytically. In contrast to the classical Kuramoto theory, we unveil that relaxation rates to each equilibrium state indeed exist and remain invariant under three levels of descriptions corresponding to different geometric implications. The special symmetry in the coupling determines a quasi-Hamiltonian property, which is further unveiled on the basis of singular perturbation theory. Since starlike coupling configurations constitute the building blocks of technological and biological real world networks, our paper paves the way towards the understanding of the functioning of such real world systems in many practical situations.