多层单向耦合星形网络的特征值谱及同步能力分析

随着复杂网络同步的进一步发展,对复杂网络的研究重点由单层网络转向更加接近实际网络的多层有向网络.本文分别严格推导出三层、多层的单向耦合星形网络的特征值谱,并分析了耦合强度、节点数、层数对网络同步能力的影响,重点分析了层数和层间中心节点之间的耦合强度对多层单向耦合星形网络同步能力的影响,得出了层数对多层网络同步能力的影响至关重要.当同步域无界时,网络的同步能力与耦合强度、层数有关,同步能力随其增大而增强;当同步域有界时,对于叶子节点向中心节点耦合的多层星形网络,当层内耦合强度较弱时,层内耦合强度的增大会使同步能力增强,而层间叶子节点之间的耦合强度、层数的增大反而会使同步能力减弱;当层间中心节点之间的耦合强度较弱时,层间中心节点之间的耦合强度、层数的增大会使同步能力增强,层内耦合强度、层间叶子节点之间的耦合强度的增大反而会使同步能力减弱.对于中心节点向叶子节点耦合的多层星形网络,层间叶子节点之间的耦合强度、层数的增大会使同步能力增强,层内耦合强度、节点数、层间中心节点之间的耦合强度的增大反而会使同步能力减弱.     

Synchronization of complex community networks with nonidentical nodes and adaptive coupling strength

In this Letter, the complex dynamical networks with community structure and nonidentical nodes are considered. The globally asymptotical synchronization of the time-delayed complex community networks onto any uniformly smooth state is studied. Some simple and useful criteria are derived by constructing an effective control scheme and adjusting automatically the adaptive coupling strength. Finally, the developed techniques are applied to two complex community networks which are respectively synchronized to a chaotic trajectory and a periodic orbit, and numerical simulations are provided to show the feasibility of the developed methods.     

非线性耦合多重边赋权复杂网络的同步

研究了一类具有非线性耦合的多重边赋权复杂网络,基于网络拆分思想并运用Lyapunov稳定性理论给出了网络的同步准则,数值仿真验证了结论的有效性.     

Synchronization of oscillators in complex networks

Theory of identical or complete synchronization of identical oscillators in arbitrary networks is introduced. In addition, several graph theory concepts and results that augment the synchronization theory and a tie in closely to random, semirandom, and regular networks are introduced. Combined theories are used to explore and compare three types of semirandom networks for their efficacy in synchronizing oscillators. It is shown that the simplest k-cycle augmented by a few random edges or links are the most efficient network that will guarantee good synchronization.      

Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix

In this paper, the global synchronization for an array of nonlinearly coupled identical chaotic systems is investigated. A distinctive feature of this work is to address synchronization issues for nonlinearly coupled complex networks with an asymmetrical coupling matrix. By projecting the nonlinear coupling function onto a linear one and assuming the difference between them as a disturbing function, we give some criteria for the global synchronization in virtual of the left eigenvector corresponding to the zero eigenvalue of the coupling matrix. Numerical examples are also provided to demonstrate the effectiveness of the theory.     

Synchronization on coupled dynamical networks

In this paper, partial synchronization (PaS) in networks of coupled chaotic oscillator systems and synchronization in sparsely coupled spatiotemporal systems are explored. For the PaS, we reveal that the existence of PaS patterns depends on the symmetry property of the network topology, while the emergence of the PaS pattern depends crucially on the stability of the corresponding solution. An analytical criterion in judging the stability of PaS state on a given network are proposed in terms of a comparison between the Lyapunov exponent spectrum of the PaS manifold and that of the transversal manifold. The competition and selections of the PaS patterns induced by the presence of multiple topological symmetries of the network are studied in terms of the criterion. The phase diagram in distinguishing the synchronous and the asynchronous states is given. The criterion in judging PaS is further applied to the study of synchronization of two sparsely coupled spatiotemporal chaotic systems. Different synchronization regimes are distinguished. The present study reveals the intrinsic collective bifurcation of coupled dynamical systems prior to the emergence of global synchronization.     

Synchronization criteria for coupled Hopfield neural networks with time-varying delays

This paper proposes new delay-dependent synchronization criteria for coupled Hopfield neural networks with time-varying delays. By construction of a suitable Lyapunov-Krasovskii's functional and use of Finsler's lemma, novel synchronization criteria for the networks are established in terms of linear matrix inequalities (LMIs) which can be easily solved by various effective optimization algorithms. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Synchronization in complex clustered networks

Synchronization in complex networks has been an active area of research in recent years. While much effort has been devoted to networks with the small-world and scale-free topology, structurally they are often assumed to have a single, densely connected component. Recently it has also become apparent that many networks in social, biological, and technological systems are clustered, as characterized by a number (or a hierarchy) of sparsely linked clusters, each with dense and complex internal connections. Synchronization is fundamental to the dynamics and functions of complex clustered networks, but this problem has just begun to be addressed. This paper reviews some progress in this direction by focusing on the interplay between the clustered topology and network synchronizability. In particular, there are two parameters characterizing a clustered network: the intra-cluster and the inter-cluster link density. Our goal is to clarify the roles of these parameters in shaping network synchronizability. By using theoretical analysis and direct numerical simulations of oscillator networks, it is demonstrated that clustered networks with random inter-cluster links are more synchronizable, and synchronization can be optimized when inter-cluster and intra-cluster links match. The latter result has one counterintuitive implication: more links, if placed improperly, can actually lead to destruction of synchronization, even though such links tend to decrease the average network distance. It is hoped that this review will help attract attention to the fundamental problem of clustered structures/synchronization in network science.      

Adaptive synchronization in an array of asymmetric coupled neural networks

This paper investigates the global synchronization in an array of linearly coupled neural networks with constant and delayed coupling. By a simple combination of adaptive control and linear feedback with the updated laws, some sufficient conditions are derived for global synchronization of the coupled neural networks. The coupling configuration matrix is assumed to be asymmetric, which is more coincident with the realistic network. It is shown that the approaches developed here extend and improve the earlier works. Finally, numerical simulations are presented to demonstrate the effectiveness of the theoretical results.     

时滞离散神经网络的同步控制

利用既有效又便于实施的时滞状态反馈控制器,根据所给定的条件构造相应的不等式,研究了带有时滞的离散神经网络模型的同步控制问题,给出了该离散系统指数同步的充分条件.在设计同步控制的时候,没有假设激励函数的有界性、可微性和单调性,给出的条件简便易实施.数值结果进一步证明了该控制方法的有效性. 关键词： 离散神经网络 时滞 同步     

Synchronization analysis of delayed complex networks via adaptive time-varying coupling strengths

In this Letter, we study the synchronization for delayed complex networks by adjusting time-varying coupling strengths. Under some assumptions, the update laws of the coupling strengths are obtained to realize the synchronization based on Lassalle-Yoshizawa theorem. For the given delayed complex network, we can always find appropriate coupling strengths to achieve the synchronization. Compared with the existing results, the update laws don't need the information of the characteristics of the identical node and the coupling matrix. The state-dependencies of coupling strengths coupled to the dynamics of the nodes in a way to enhance synchronization. An example shows the proposed theoretical result is feasible and effective.     

Synchronization in weighted complex networks: Heterogeneity and synchronizability

Synchronization in different types of weighted networks based on a scale-free weighted network model is investigated. It has been argued that heterogeneity suppresses synchronization in unweighted networks [T. Nishikawa, A.E. Motter, Y.C. Lai, F.C. Hoppensteadt, Phys. Rev. Lett. 91 (2003) 014101]. However, it is shown in this work that as the network becomes more heterogeneous, the synchronizability of Type I symmetrically weighted networks, and Type I and Type II asymmetrically weighted networks is enhanced, while the synchronizability of Type II symmetrically weighted networks is weakened.     

Synchronization in complex dynamical networks with nonsymmetric coupling

Based on the work of Nishikawa and Motter, who have extended the well-known master stability framework to include non-diagonalizable cases, we develop another extension of the master stability framework to obtain criteria for global synchronization. Several criteria for global synchronization are provided which generalize some previous results. The Jordan canonical transformation method is used in stead of the matrix diagonalization method. Especially, we show clearly that, the synchronizability of a dynamical network with nonsymmetric coupling is not always characterized by its second-largest eigenvalue, even though all the eigenvalues of the nonsymmetric coupling matrix are real. Furthermore, the effects of the asymmetry of coupling on synchronizability of networks with different structures are analyzed. Numerical simulations are also done to illustrate and verify the theoretical results on networks in which each node is a dynamical limit cycle oscillator consisting of a two-cell cellular neural network.     

Synchronization of coupled logistic maps on random community networks

The collective synchronization of a system of coupled logistic maps on random community networks is investigated. It is found that the synchronizability of the community network is affected by two factors when the size of the network and the number of connections are fixed. One is the number of communities denoted by the parameter rn, and the other is the ratio σ of the connection probability p of each pair of nodes within each community to the connection probability q of each pair of nodes among different communities. Theoretical analysis and numerical results indicate that larger rn and smaller σ are the key to the enhancement of network synchronizability. We also testify synchronous properties of the system by analysing the largest Lyapunov exponents of the system.     

Synchronization of stochastically hybrid coupled neural networks with coupling discrete and distributed time-varying delays

A general model of linearly stochastically coupled identical connected neural networks with hybrid coupling is proposed, which is composed of constant coupling, coupling discrete time-varying delay and coupling distributed timevarying delay. All the coupling terms are subjected to stochastic disturbances described in terms of Brownian motion, which reflects a more realistic dynamical behaviour of coupled systems in practice. Based on a simple adaptive feedback controller and stochastic stability theory, several sufficient criteria are presented to ensure the synchronization of linearly stochastically coupled complex networks with coupling mixed time-varying delays. Finally, numerical simulations illustrated by scale-free complex networks verify the effectiveness of the proposed controllers.     

Synchronization and coherence resonance in chaotic neural networks

Synchronization and coherence of chaotic Morris--Lecar (ML) neural networks have been investigated by numerical methods. The synchronization of the neurons can be enhanced by increasing the number of the shortcuts, even though all neurons are chaotic when uncoupled. Moreover, the coherence of the neurons exhibits a non-monotonic dependence on the density of shortcuts. There is an optimal number of shortcuts at which the neurons' motion is most ordered, i.e. the order parameter (the characteristic correlation time) that is introduced to measure the coherence of the neurons has a maximum. These phenomena imply that stochastic shortcuts can tame spatiotemporal chaos. The effects of the coupling strength have also been studied. The value of the optimal number of shortcuts goes down as the coupling strength increases.     

Synchronization speed of identical oscillators on community networks

Synchronizability of complex oscillators networks has attracted much research interest in recent years. In contrast, in this paper we investigate numerically the synchronization speed, rather than the synchronizability or synchronization stability, of identical oscillators on complex networks with communities. A new weighted community network model is employed here, in which the community strength could be tunable by one parameter δ. The results showed that the synchronization speed of identical oscillators on community networks could reach a maximal value when δ is around 0.1. We argue that this is induced by the competition between the community partition and the scale-free property of the networks. Moreover, we have given the corresponding analysis through the second least eigenvalue λ₂ of the Laplacian matrix of the network which supports the previous result that the synchronization speed is determined by the value of λ₂.     

Outer synchronization between two different fractional-order general complex dynamical networks

Outer synchronization between two different fractional-order general complex dynamical networks is investigated in this paper.Based on the stability theory of the fractional-order system,the sufficient criteria for outer synchronization are derived analytically by applying the nonlinear control and the bidirectional coupling methods.The proposed synchronization method is applicable to almost all kinds of coupled fractional-order general complex dynamical networks.Neither a symmetric nor irreducible coupling configuration matrix is required.In addition,no constraint is imposed on the inner-coupling matrix.Numerical examples are also provided to demonstrate the validity of the presented synchronization scheme.Numeric evidence shows that both the feedback strength k and the fractional order α can be chosen appropriately to adjust the synchronization effect effectively.     

Periodic synchronization of community networks with non-identical nodes uncertain parameters and adaptive coupling strength

In this paper, we propose a novel approach for simultaneously identifying unknown parameters and synchronizing time-delayed complex community networks with nonidentical nodes. Based on the LaSalle＇s invariance principle, a cri- teflon is established by constructing an effective control identification scheme and adjusting automatically the adaptive coupling strength. The proposed control law is applied to a complex community network which is periodically synchro- nized with different chaotic states. Numerical simulations are conducted to demonstrate the feasibility of the proposed method.     

时变时滞耦合两个不同复杂网络的自适应广义同步

针对具有时变时滞耦合的两个不同复杂网络, 研究其广义同步问题. 基于Barbalat引理, 设计非线性自适应控制器使得两个网络获得广义同步. 进一步研究了具有未知拓扑结构的两个网络的广义同步问题. 数值仿真表明了本文方法的有效性.