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In nature and human society, successive lag synchronization (SLS) is an important synchronization phenomenon. Compared with other synchronization patterns, the control theory of SLS is very lacking. To this end, we first introduce a complex dynamical network model with distributed delayed couplings, and design both the linear feedback pinning control and adaptive feedback pinning control to push SLS to the desired trajectories. Second, we obtain a series of sufficient conditions to achieve SLS to a desired trajectory with global stability. What is more, the control flow of SLS is given to show how to pick the pinned nodes accurately and set the feedback gains as well. Finally, since time-varying delay is common, we extend the constant time delay in SLS to be time varying. We find that the proposed pinning control schemes are still feasible if the coupling terms are appropriately adjusted. The theoretical results are verified on a neural network and the coupled Chua’s circuits. [ABSTRACT FROM AUTHOR]

Firstly, the definition of clustering component synchronization is given (that is, all nodes in each cluster realize partial component synchronization). Then, both stability theory and matrix analysis are used to study the group dynamics of a class of unconnected networks via pinning control. Furthermore, a sufficient condition on clustering component synchronization in the network is derived, and the correctness of the theoretical results is verified by numerical simulation. The advantage of the control scheme is that, by adjusting the dynamics of a very small number of pinning nodes, some clusters can be merged in the network, and then the presetting synchronous patterns can emerge. Compared with cluster synchronization and partial component synchronization, clustering component synchronization is a weaker group dynamics behavior. • The definition of clustering component synchronization is given for the first time. • Clustering component synchronization is a weaker group dynamics behavior. • A sufficient condition for clustering component synchronization is derived. [ABSTRACT FROM AUTHOR]

Abstract As for the dynamical networks which consist of some high-dimensional nonlinear systems, the problems that researchers are concerned with are usually the asymptotic convergence on some components (rather than all components) of node's state variables under certain condition. This means that partial component synchronization is more meaningful than identical synchronization in some cases. In this paper, the definition of partial component synchronization is given, and then the problem of partial component synchronization on a class of chaotic dynamical networks is investigated. By using matrix theory, stability theory and the hypothesis that several components in the solution vector of a single uncoupled node are ultimately dissipative, some sufficient conditions on partial component synchronization in the chaotic dynamical networks are derived. Finally, numerical simulations are shown to demonstrate the correctness of the theoretical results. Highlights • Partial component synchronization is a kind of group dynamics behavior weaker than identical synchronization. • In this paper, the definition of partial component synchronization is given, and the stability theory of partial variables is applied to study it. • Several sufficient conditions for partial component synchronization to be realized on the network are derived. [ABSTRACT FROM AUTHOR]

In this article, the problem of cluster synchronization in the complex networks with nonidentical nonlinear dynamics is considered. By Lyapunov functional and M-matrix theory, some sufficient conditions for cluster synchronization are obtained. Moreover, the least number of nodes which should be pinned is given. It is shown that when the root nodes of all the clusters are pinning-controlled, cluster synchronization with adaptive coupling strength can be achieved. Different from the constraints of many literatures, the assumption is that each row sum for all diagonal submatrices of the Laplacian matrix is equal to zero. Finally, a numerical simulation in the network with three scale-free subnetwork is provided to demonstrate the effectiveness of the theoretical results. © 2016 Wiley Periodicals, Inc. Complexity 21: 380-387, 2016 [ABSTRACT FROM AUTHOR]

Many realistic epidemic networks display statistically synchronous behavior which we will refer to as epidemic synchronization. However, to the best of our knowledge, there has been no theoretical study of epidemic synchronization. In fact, in many cases, synchronization and epidemic behavior can arise simultaneously and interplay adaptively. In this paper, we first construct mathematical models of epidemic synchronization, based on traditional dynamical models on complex networks, by applying the adaptive mechanisms observed in real networks. Then, we study the relationship between the epidemic rate and synchronization stability of these models and, in particular, obtain the conditions of local and global stability for epidemic synchronization. Finally, we perform numerical analysis to verify our theoretical results. This work is the first to draw a theoretical bridge between epidemic transmission and synchronization dynamics and will be beneficial to the study of control and the analysis of the epidemics on complex networks. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of cluster synchronization in networks with asymmetric negative couplings. By decomposing the coupling matrix into three matrices, and employing Lyapunov function method, sufficient conditions are derived for cluster synchronization. The conditions show that the couplings of multi-node clusters from one-node clusters have beneficial effects on cluster synchronization. Based on the effects of the one-node clusters, an effective and universal control scheme is put forward for the first time. The obtained results may help us better understand the relation between cluster synchronization and cluster structures of the networks. The validity of the control scheme is confirmed through two numerical simulations, in a network with no cluster structure and in a scale-free network. [ABSTRACT FROM AUTHOR]

Successive lag synchronization (SLS) is a novel and important synchronization pattern, whose control problem has been deeply investigated in the past several years. However, the impact of noise perturbation on the control of SLS is still unclear. To this end, this paper focuses on the intermittent pinning control of SLS on a dynamical network with noise perturbation. In order to push the SLS to a desired trajectory, we design an intermittent pinning control scheme that is active only on controlled intervals and a small proportion of nodes. By applying the stability theory of stochastic differential equation, we obtain sufficient conditions under which the network can realize the SLS with exponential stability in mean square. According to these conditions, we can determine which nodes should be controlled or not, the minimum control ratio and the impact of noise perturbation on the SLS. An interesting finding is that performing the control on at least half of each subinterval can ensure the stochastic stability of SLS, which is independent of any network parameters. The effectiveness of proposed intermittent pinning control scheme is verified by the network of coupled Chua's circuits. [ABSTRACT FROM AUTHOR]

In recent years, the successive lag synchronization (SLS) has been deeply analyzed, however, the control theory of SLS on complex networks with noise perturbation is lacking. To this end, this paper focuses on the pinning control of SLS on a dynamical network with noise perturbation. Both the constant pinning control law and adaptive pinning control law are designed respectively to push the network to achieve the desired SLS. By utilizing the Lyapunov stability theory of stochastic differential equations, several sufficient conditions for the controlled networks to achieve the SLS are obtained. The influences of network structure, noise strength and coupling strength on the realization of SLS are also discussed. We find that the noise can suppress or even destroy the realization of SLS. Meanwhile, we attain the index set of pinned nodes, which can tell us whether a node should be controlled or not. In particular, those nodes with zero in-degree must be controlled to achieve SLS, while those nodes with non-zero in-degree need not to be controlled if the coupling strength is large enough. All theoretical results are verified by numerical simulations. • A dynamical network with noise perturbation is introduced to realize the SLS. • Stochastic stability conditions of SLS on the controlled networks are obtained. • The noise can suppress or even destroy the SLS. • The pinned nodes are determined by solving 1-dimensional inequalities. [ABSTRACT FROM AUTHOR]