The impact of inner-coupling and time delay on synchronization: From single-layer network to hypernetwork

Though synchronization of complex dynamical systems has been widely studied in the past few decades, few studies pay attention to the impact of network parameters on synchronization in hypernetworks. In this paper, we focus on a specific hypernetwork model consisting of coupled Rössler oscillators and investigate the impact of inner-coupling and time delay on the synchronized region (SR). For the sake of simplicity, the inner-coupling matrix is chosen from three typical forms, which result in classical bounded, unbounded, and empty SR in a single-layer network, respectively. The impact of inner-couplings or time delays on unbounded SR is the most interesting one among the three types of SR. Once the SR of one subnetwork is unbounded, the SR of the whole hypernetwork is also unbounded with a different inner-coupling matrix. In a hypernetwork with unbounded SR, the time delays change not only the size but also the type of SR. In a hypernetwork with bounded or empty SR, the time delays have almost no effect on the type of SR.