Synchronization in duplex networks of coupled Rössler oscillators with different inner-coupling matrices

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.

Synchronizability of double-layer dumbbell networks

Synchronization of multiplex networks has been a topical issue in network science. Dumbbell networks are very typical structures in complex networks which are distinguished from both regular star networks and general community structures, whereas the synchronous dynamics of a double-layer dumbbell network relies on the interlink patterns between layers. In this paper, two kinds of double-layer dumbbell networks are defined according to different interlayer coupling patterns: one with the single-link coupling pattern between layers and the other with the two-link coupling pattern between layers. Furthermore, the largest and smallest nonzero eigenvalues of the Laplacian matrix are calculated analytically and numerically for the single-link coupling pattern and also obtained numerically for the two-link coupling pattern so as to characterize the synchronizability of double-layer dumbbell networks. It is shown that interlayer coupling patterns have a significant impact on the synchronizability of multiplex systems. Finally, a numerical example is provided to verify the effectiveness of theoretical analysis. Our findings can facilitate company managers to select optimal interlayer coupling patterns and to assign proper parameters in terms of improving the efficiency and reducing losses of the whole team.

Rössler-network with time delay: Univariate impulse pinning synchronization

Rössler had a brilliant and successful life as a scientist during which he published a benchmark dynamical system by using an electronic circuit interpreting chemical reactions. This is our contribution to honor his splendid erudite career. It is a hot topic to regulate a network behavior using the pinning control with respect to a small set of nodes in the network. Besides pinning to a small number of nodes, small perturbation to the node dynamics is also demanded. In this paper, the pinning synchronization of a coupled Rössler-network with time delay using univariate impulse control is investigated. Using the Lyapunov theory, a theorem is proved for the asymptotic stability of synchronization in the network. Simulation is given to validate the correctness of the analysis and the effectiveness of the proposed univariate impulse pinning controller.