Explosive or Continuous: Incoherent state determines the route to synchronization

Abstract

Abrupt and continuous spontaneous emergence of collective synchronization of coupled oscillators have attracted much attention. In this paper, we propose a dynamical ensemble order parameter equation that enables us to grasp the essential low-dimensional dynamical mechanism of synchronization in networks of coupled oscillators. Different solutions of the dynamical ensemble order parameter equation build correspondences with diverse collective states, and different bifurcations reveal various transitions among these collective states. The structural relationship between the incoherent state and the synchronous state leads to different routes of transitions to synchronization, either continuous or discontinuous. The explosive synchronization is determined by the bistable state where the measure of each state and the critical points are obtained analytically by using the dynamical ensemble order parameter equation. Our method and results hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, provide an effective approach in understanding the routes to synchronization in more general complex networks.

Effects of frustration on explosive synchronization

“…It should be noted that when the frustration α = 0, the same results were obtained in previous studies [12,26,36]. Eq.…”

“…When the frustration α = 0, the result obtained above is Ω = 2k/(k + 1), which is independent of the coupling strength λ and it is also a general result of synchronization [12,26,36]. According to the panels in Fig.…”

“…As shown in Fig. 6(b)-(d), there is an upper branch inside the boundary, which is nonphysical and unstable [36]. Analogous to the two-oscillator system, a possible effect of frustration on the effective frequency is that the effective frequency increases when α < 0.5π; otherwise, the effective frequency declines when α > 0.5π.…”

“…By substituting this condition into Eq. (26), the critical coupling strength for α < 0.5π is λ = √ c − b 2 /a and the limit case is obtained for small frustration [12,26,36] …”

“…Previous investigations of explosive synchronization have focused on the topological configuration of networks and the coupled forms among oscillators [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35], whereas the effects of frustration on the synchronization process have not been considered in detail [36,41]. Frustration has important effects on the spin Heisenberg XY model [42], coupled Josephson-junction arrays or ladders, and other systems with potentially useful applications [43,44].…”