Maximal hysteretic range for explosive synchronization

Traditionally, phase oscillators on a frequency-degree correlated star network provides a building motif for understanding explosive transitions to synchronization and the associated hysteresis behavior. Here we show that a transition from explosive to continuous synchronization is resulted from the frustration term when implementing α_c=π/4 in the Kuramoto-Sakaguchi models on a star. Interestingly, the existence condition for phase locking manifold does not coincide with the backward critical threshold for desynchronization. In addition, the nonlinear effects of the phase shifts are derived analytically, showing a maximal hysteresis range when the frequency-degree correlation is strong enough, i.e., β>β_c. On the other hand, the hysteresis range decreases monotonically when β<β_c. The maximal hysteresis ranges are not found in other models. Furthermore, numerical results precisely confirm the theoretical predictions. Therefore, the phase shift provides a natural way to control explosive synchronization.