Continuous and explosive synchronization of phase oscillators on star network: Effect of degree-frequency correlations and time-delays

Abstract

It is crucial to find time-delay intervals that control and mediate the onset and suppression of explosive synchronization in complex systems. To address this, we study the desynchrony-to-synchrony transitions in delay-coupled phase oscillators with degree-frequency correlation on a star network. We study how delay affects the transitions from desynchrony to synchrony. We observe that the system first undergo remote synchronization and the routes to synchrony are either continuous or discontinuous, depending on the delay intervals. We provide numerical results to establish a fundamental relation between delay intervals and time-period induced by the system’s average frequency of intrinsic oscillators. In the regime of global synchrony, all oscillators operate with a common frequency, and a phase shift develops between the hub and peripheral oscillators. We derive the analytical expression of common frequency, the phase difference between oscillators, and the threshold coupling strength for the onset of global synchronization. We perform a linear stability analysis of the common frequency at the point of threshold coupling and examine the series of saddle–node bifurcations present in the system. Estimates of linearly stable solutions of frequency are found to be in good agreement with numerical results. Hence, this study elucidates the role of delay on the routes to synchrony, stability of synchronization solutions, and multistability regimes.