New Approach to Cluster Synchronization in Complex Dynamical Networks

In this paper, a distributed control strategy is proposed to make a complex dynamical network achieve cluster synchronization, which means that nodes in the same group achieve the same synchronization state, while nodes in different groups achieve different synchronization states. The local and global stability of the cluster synchronization state are analyzed. Moreover, simulation results verify the effectiveness of the new approach.

Cluster synchronization in a network of non-identical dynamic systems

Cluster synchronization in a network of non-identical dynamic systems is studied in this paper, using two-cluster synchronization for detailed analysis and discussion. The results show that the common intercluster coupling condition is not always needed for the diffusively coupled network. Several sufficient conditions are obtained by using the Schur unitary triangularization theorem, which extends previous results. Some numerical examples are presented for illustration.

Cluster synchronization of uncertain complex networks with desynchronizing impulse

Cluster synchronization of nonlinear uncertain complex networks with desynchronizing impulse is explored. First of all, a feedback controller is employed, based on the Lyapunov stability theorem and Lipschitz condition, to guarantee that the uncertain complex networks with desynchronizing impulse synchronize with an object trajectory. Furthermore, a synchronizing impulse controller is presented, which is more efficiently and directly used to achieve the cluster synchronization. Finally, numerical examples are examined to show the effectiveness of the proposed methods.

Cluster synchronization in community network with nonidentical nodes via intermittent pinning control

In this paper, cluster synchronization in community network with nonidentical nodes is investigated. By combining intermittency with a pinning control scheme, some effective controllers are designed. In the control scheme, only one node in each community is controlled and coupling weights of a spanning tree in each community are enhanced. Based on the Lyapunov function method and mathematical analysis technique, two results for achieving cluster synchronization are obtained. Noticeably, by introducing an adaptive strategy, some universal adaptive intermittent pinning controllers are designed for different networks. Finally, two numerical simulations are performed to verify the correctness of the derived results.

Eigenvector-based analysis of cluster synchronization in general complex networks of coupled chaotic oscillators

Whereas topological symmetries have been recognized as crucially important to the exploration of synchronization patterns in complex networks of coupled dynamical oscillators,the identification of the symmetries in largesize complex networks remains as a challenge.Additionally,even though the topological symmetries of a complex network are known,it is still not clear how the system dynamics is transited among different synchronization patterns with respect to the coupling strength of the oscillators.We propose here the framework of eigenvector-based analysis to identify the synchronization patterns in the general complex networks and,incorporating the conventional method of eigenvalue-based analysis,investigate the emergence and transition of the cluster synchronization states.We are able to argue and demonstrate that,without a prior knowledge of the network symmetries,the method is able to predict not only all the cluster synchronization states observable in the network,but also the critical couplings where the states become stable and the sequence of these states in the process of synchronization transition.The efficacy and generality of the proposed method are verified by different network models of coupled chaotic oscillators,including artificial networks of perfect symmetries and empirical networks of non-perfect symmetries.The new framework paves a way to the investigation of synchronization patterns in large-size,general complex networks.

CLUSTER SYNCHRONIZATION IN A COMPLEX DYNAMICAL NETWORK WITH TWO NONIDENTICAL CLUSTERS

This paper further investigates cluster synchronization in a complex dynamical network with two-cluster.Each cluster contains a number of identical dynamical systems,however,the sub- systems composing the two clusters can be different,i.e.,the individual dynamical system in one cluster can differ from that in the other cluster.Complete synchronization within each cluster is possible only if each node from one cluster receives the same input from nodes in other cluster.In this case,the stability condition of one-cluster synchronization is known to contain two terms:the first accounts for the contribution of the inner-cluster coupling structure while the second is simply an extra linear term,which can be deduced by the'same-input'condition.Applying the connection graph stability method,the authors obtain an upper bound of input strength for one cluster if the first account is known,by which the synchronizability of cluster can be scaled.For different clusters,there are different upper bound of input strength by virtue of different dynamics and the corresponding cluster structure.Moreover,two illustrative examples are presented and the numerical simulations coincide with the theoretical analysis.

Cluster Synchronization in Hybrid Coupled Discrete-Time Delayed Complex Networks

This paper studies the exponential cluster synchronization in arrays of coupled discrete-time dynamical networks with time-varying delay, in which the hybrid coupling is involved. Through choosing two improved Lyapunov-Krasovskii functionals, some delay-dependent sufficient conditions are presented based on reciprocal convex technique and Kronecker product. These criteria are presented in terms of LMIs and their feasibility can be easily checked by resorting to Matlab LMI Toolbox. Moreover, the addressed system can include some famous network models as its special cases and the effective techniques are used, which can extend some earlier reported results. Finally, the effectiveness of the proposed methods can be further illustrated with the help of two numerical examples.

Impulsive Cluster Synchronization in Community Network with Nonidentical Nodes

In this paper,cluster synchronization in community network with nonidentical nodes and impulsive effects is investigated.Community networks with two kinds of topological structure are investigated.Positive weighted network is considered first and external pinning controllers are designed for achieving cluster synchronization.Cooperative and competitive network under some assumptions is investigated as well and can achieve cluster synchronization with only impulsive controllers.Based on the stability analysis of impulsive differential equation and the Lyapunov stability theory,several simple and useful synchronization criteria are derived.Finally,numerical simulations are provided to verify the effectiveness of the derived results.

Cluster exponential synchronization of a class of complex networks with hybrid coupling and time-varying delay

This paper deals with the cluster exponential synchronization of a class ot complex networks wlm nyorm coupm^g and time-varying delay. Through constructing an appropriate Lyapunov-Krasovskii functional and applying the theory of the Kronecker product of matrices and the linear matrix inequality （LMI） technique, several novel sufficient conditions for cluster exponential synchronization are obtained. These cluster exponential synchronization conditions adopt the bounds of both time delay and its derivative, which are less conservative. Finally, the numerical simulations are performed to show the effectiveness of the theoretical results.

Cluster projective synchronization of complex networks with nonidentical dynamical nodes

We investigate a new cluster projective synchronization （CPS） scheme in time-varying delay coupled complex dynamical networks with nonidentical nodes. Based on the community structure of the networks, the controllers are designed differently for the nodes in one community, which have direct connections to the nodes in the other communities and the nodes without direct connections to the nodes in the other communities. Some sufficient criteria are derived to ensure the nodes in the same group projectively synchronize and there is also projective synchronization between nodes in different groups. Particularly, the weight configuration matrix is not assumed to be symmetric or irreducible. The numerical simulations are performed to verify the effectiveness of the theoretical results.

Synchronization clusters located on epileptic onset zones in neocortical epilepsy

Background Brain function is thought to rely on complex interactions of dynamic neural systems,which depend on the integrity of structural and functional networks.Focal epilepsy is considered to result from excessive focal synchronization in the network.Synchronization analysis of multichannel electrocorticography(ECoG)contributes to the understanding of and orientation of epilepsy.The aim of this study was to explore the synchronization in multichannel ECoG recordings from patients with neocortical epilepsy and characterize neural activity inside and outside the onset zone.Methods Four patients with neocortical epilepsy,who became seizure-free for more than 1 year after surgery guided by ECoG monitoring,were included in this study.ECoG data recorded during pre-surgical evaluation were analyzed.Synchronizations in phase and amplitude of different frequency bands between ECoG channels was analyzed using MATLAB.We generated 100 surrogate data from the original ECoG data using Amplitude Adjusted Fourier Transform to calculate the enhanced synchronization.The relationship between synchronization characteristics and seizure onset zone was analyzed.Results We found synchronization clusters in the 14–30 Hz and 30–80 Hz bands around the onset areas during both interictal and the beginning of ictal periods in all four patients.Conclusions The enhanced-synchronization clusters play a central role in epilepsy,and may activate the onset areas and contribute to the spreading of epileptiform activity.

A RESEARCH ON PHASE SYNCHRONIZATION FOR EXTENDED KURAMOTO MODEL

This paper discuses three types of phase synchronization phenomena in extended Kuramoto model. A certain quadratic form is applied to analyze the stability of phase synchronization manifolds without any stability knowledge for error systems. Some simple and convenient criteria are obtained for these types of phase synchronization. Also, the effectiveness of the proposed criteria is illustrated successfully by an example.

Oscillatory Dynamics and Oscillation Death in Complex Networks Consisting of Both Excitatory and Inhibitory Nodes

Zn neural networks, both excitatory and inhibitory cells play important roles in determining the functions of systems. Various dynamical networks have been proposed as artificial neural networks to study the properties of biological systems where the influences of excitatory nodes have been extensively investigated while those of inhibitory nodes have been studied much less. In this paper, we consider a model of oscillatory networks of excitable Boolean maps consisting of both excitatory and inhibitory nodes, focusing on the roles of inhibitory nodes. We find that inhibitory nodes in sparse networks （smM1 average connection degree） play decisive roles in weakening oscillations, and oscillation death occurs after continual weakening of oscillation for sufficiently high inhibitory node density. In the sharp contrast, increasing inhibitory nodes in dense networks may result in the increase of oscillation amplitude and sudden oscillation death at much lower inhibitory node density and the nearly highest excitation activities. Mechanism under these peculiar behaviors of dense networks is explained by the competition of the duplex effects of inhibitory nodes.