Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling

This Letter investigates the global synchronization of a general complex dynamical network with non-delayed and delayed coupling. Based on Lasalle's invariance principle, adaptive global synchronization criteria is obtained. Analytical result shows that under the designed adaptive controllers, a general complex dynamical network with non-delayed and delayed coupling can globally asymptotically synchronize to a given trajectory. What is more, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the coupling matrix is not assumed to be symmetric or irreducible. Finally, numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria.     

Global synchronization of Chua＇s chaotic delay network by using linear matrix inequality

Global synchronization of Chua‘s chaotic dynamical networks with coupling delays is investigated in this paper.Unlike other approaches, where only local results were obtained, the network is found to be not linearized in this paper.Insteat, the global synchronization is obtained by using the linear matrix inequality theory. Moreover, some quite simple linear-state-error feedback controllers for global synchronization are derived, which can be easily constructed based on the minimum eigenvalue of the coupling matrix. A simulation of Chua‘s chaotic network with global coupling delays in nodes is finally given, which is used to verify the theoretical results of the proposed global synchron izationscheme.     

Periodically intermittent controlling complex dynamical networks with time-varying delays to a desired orbit

This Letter investigates the problem of synchronization in complex dynamical networks with time-varying delays. A periodically intermittent control scheme is proposed to achieve global exponential synchronization for a general complex network with both time-varying delays dynamical nodes and time-varying delays coupling. It is shown that the sates of the general complex network with both time-varying delays dynamical nodes and time-varying delays coupling can globally exponentially synchronize with a desired orbit under the designed intermittent controllers. Moreover, a typical network consisting of the time-delayed Chua oscillator with nearest-neighbor unidirectional time-varying delays coupling is given as an example to verify the effectiveness of the proposed control methodology.     

Global synchronization in arrays of coupled networks with one single time-varying delay coupling

Global synchronization in arrays of coupled networks with one single time-varying delay coupling is investigated in this Letter. A general linear coupled network with a time-varying coupling delay is proposed and its global synchronization is further discussed. Some sufficient criteria are derived based on Lyapunov functional and linear matrix inequality (LMI). It is shown that under one single delay coupling, the synchronized state changes, which is different from the conventional synchronized solution. Moreover, the degree of the nodes and the inner delayed coupling matrix play key roles in the synchronized state. In particular, the derivative of the time-varying delay can be any given value. Finally, numerical simulations are given to illustrate the theoretical results.     

New approach to synchronization in asymmetrically coupled networks

Global exponentially synchronization in asymmetrically coupled networks is investigated in this Letter. We extend eigenvalue based method to synchronization in symmetrically coupled network to synchronization in asymmetrically coupled network. A new stability criterion of eigenvalue based is derived. In this criterion, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of sum of column of asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability result is that it can analytical be applied to the asymmetrically coupled networks and overcome the complexity on calculating eigenvalues of coupling asymmetric matrix. Therefore, this condition is very convenient to use. Moreover, a necessary condition of this synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of coupling matrix.     

Adaptive cluster synchronization in complex dynamical networks

Cluster synchronization is investigated in different complex dynamical networks. In this Letter, a novel adaptive strategy is proposed to make a complex dynamical network achieve cluster synchronization, where the adaptive strategy of one edge is adjusted only according to its local information. A sufficient condition about the global stability arbitrarily grouped of cluster synchronization is derived. Several numerical simulations show the effectiveness of the adaptive strategy.     

Synchronization in complex delayed dynamical networks with nonsymmetric coupling

A new general complex delayed dynamical network model with nonsymmetric coupling is introduced, and then we investigate its synchronization phenomena. Several synchronization criteria for delay-independent and delay-dependent synchronization are provided which generalize some previous results. The matrix Jordan canonical formalization method is used instead of the matrix diagonalization method, so in our synchronization criteria, the coupling configuration matrix of the network does not required to be diagonalizable and may have complex eigenvalues. Especially, we show clearly that the synchronizability of a delayed dynamical network is not always characterized by the second-largest eigenvalue even though all the eigenvalues of the coupling configuration matrix are real. Furthermore, the effects of time-delay on synchronizability of networks with unidirectional coupling are studied under some typical network structures. The results are illustrated by delayed networks in which each node is a two-dimensional limit cycle oscillator system consisting of a two-cell cellular neural network, numerical simulations show that these networks can realize synchronization with smaller time-delay, and will lose synchronization when the time-delay increase larger than a threshold.     

Exponential synchronization of complex dynamical networks with Markovian jumping parameters using sampled-data and mode-dependent probabilistic time-varying delays

In this paper, the problem of exponential synchronization of complex dynamical networks with Markovian jumping parameters using sampled-data and Mode-dependent probabilistic time-varying coupling delays is investigated. The sam- pling period is assumed to be time-varying and bounded. The information of probability distribution of the time-varying delay is considered and transformed into parameter matrices of the transferred complex dynamical network model. Based on the condition, the design method of the desired sampled data controller is proposed. By constructing a new Lyapunov functional with triple integral terms, delay-distribution-dependent exponential synchronization criteria are derived in the form of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Cluster synchronization in fractional-order complex dynamical networks

Cluster synchronization of complex dynamical networks with fractional-order dynamical nodes is discussed in the Letter. By using the stability theory of fractional-order differential system and linear pinning control, a sufficient condition for the stability of the synchronization behavior in complex networks with fractional order dynamics is derived. Only the nodes in one community which have direct connections to the nodes in other communities are needed to be controlled, resulting in reduced control cost. A numerical example is presented to demonstrate the validity and feasibility of the obtained result. Numerical simulations illustrate that cluster synchronization performance for fractional-order complex dynamical networks is influenced by inner-coupling matrix, control gain, coupling strength and topological structures of the networks.     

Adaptive synchronization of weighted complex dynamical networks through pinning

This paper considers the problem of controlling weighted complex dynamical networks by applying adaptive control to a fraction of network nodes. We investigate the local and global synchronization of the controlled dynamical network through the construction of a master stability function and a Lyapunov function. Analytical results show that a certain number of nodes can be controlled by using adaptive pinning to ensure the synchronization of the entire network. We present numerical simulations to verify the effectiveness of the proposed scheme. In comparison with feedback pinning, the proposed pinning control scheme is robust when tested by noise, different weighting and coupling structures, and time delays.     

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.     

Projective synchronization in drive-response dynamical networks of partially linear systems with time-varying coupling delay

Recently, projective synchronization (PS) has been widely studied in more than one system. In this Letter, we propose a linear controller and an updated law to realize the PS in drive-response dynamical networks of partially linear systems with time-varying coupling delay, based on the Lyapunov stability theory. A sufficient condition is obtained. Moreover, numerical simulations are provided to verify the correctness and effectiveness of the scheme.     

一类动力学系统通过函数耦合实现混沌同步

非线性动力学系统的混沌同步, 一般采用单向线性耦合的控制方式, 对于函数耦合方式研究的比较少. 这就存在一个问题, 对于非线性动力学系统, 在线性耦合实现混沌同步后, 是否其他函数的耦合方式都可以实现混沌同步? 本文对于一类非线性动力学系统, 研究了其线性耦合同步与函数耦合同步的关系, 证明当线性耦合实现同步后, 函数在满足一定的条件下, 可以通过函数耦合实现系统的混沌同步. 最后对于Duffing系统采用两种函数耦合进行了仿真计算, 证明了结论的正确性.     

Pinning synchronization of the complex networks with non-delayed and delayed coupling

In this Letter, without assuming the symmetry of the coupling matrix, we investigate the global synchronization of the complex networks with non-delayed and delayed coupling based on the pinning controllers. Some sufficient conditions for the global synchronization by adding linear and adaptive feedback controllers to a part of nodes are obtained. Numerical examples are also provided to demonstrate the effectiveness of the theory.     

Synchronization criteria for a generalized complex delayed dynamical network model

It has been demonstrated that most complex networks display synchronization phenomena, and this problem has attracted great attention of various fields including science and engineering. In this paper, a generalized complex dynamical network model with time-varying delays was presented. Kronecker product was adopted to investigate this model. Moreover, several synchronization criteria were derived for both delay-independent and delay-dependent asymptotical stability. Especially, it has been shown that synchronization of such dynamical network is determined by the linear matrix inequality consisting of coupling configuration matrices, inner-coupling matrices and isolated cells. At last, illustrative examples were given to validate the above-acquired.     

Observer-based synchronization in complex dynamical networks with nonsymmetric coupling

Based on a general complex dynamical network model with nonsymmetric coupling, some criteria for synchronization are proposed based on the approach of state observer design. Unlike the nonobserver-based dynamical networks, where the coupling between two connected nodes is defined by an inner coupling matrix and full state coupling is typically needed, in this paper, smaller amount of coupling variables or even only a scalar output signal of each node is needed to synchronize the network. Unlike the commonly researched complex network model, where the coupling between nodes is symmetric, here, in our network model, the coupling configuration matrix is not assumed to be symmetric and may have complex eigenvalues. The matrix Jordan canonical formalization method is used instead of the matrix diagonalization method, so in our synchronization criteria, the coupling configuration matrix is not required to be diagonalizable. Especially, the proposed step-by-step approach is simpler in computation than the existent ones, which usually rely heavily on numerical toolbox, and may be done by hand completely. An example is given to illustrate the step-by-step approach, in which each node is a two-dimensional dynamical limit cycle oscillator system consisting of a two-cell cellular neural network, and numerical simulations are also done to verify the results of design.     

Global synchronization in general complex delayed dynamical networks and its applications

The main objective of the present paper is further to investigate global synchronization of a general model of complex delayed dynamical networks. Based on stability theory on delayed dynamical systems, some simple yet less conservative criteria for both delay-independent and delay-dependent global synchronization of the networks are derived analytically. It is shown that under some conditions, if the uncoupled dynamical node is stable itself, then the network can be globally synchronized for any coupling delays as long as the coupling strength is small enough. On the other hand, if each dynamical node of the network is chaotic, then global synchronization of the networks is heavily dependent on the effects of coupling delays in addition to the connection configuration. Furthermore, the results are applied to some typical small-world (SW) and scale-free (SF) complex networks composing of coupled dynamical nodes such as the cellular neural networks (CNNs) and the chaotic FHN neuron oscillators, and numerical simulations are given to verify and also visualize the theoretical results.     

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.     

不同耦合下混沌神经元网络的同步

采用Hindmarsh-Rose神经元动力学模型, 对二维点阵上的神经元网络的同步进行了研究. 为了解不同耦合对网络同步的影响, 提出了一般反馈耦合、分层反馈耦合和分层局域平均场反馈耦合三种方案.研究表明:在耦合强度较小的近邻耦合下, 一般反馈耦合不能使网络达到完全同步, 而分层反馈耦合和分层局域平均场反馈耦合可以使网络出现局部同步和全局同步. 不同形式的耦合会导致网络出现不同的斑图, 随着耦合强度的增大, 网络从不同步到同步的过程也不相同, 一般反馈耦合和分层反馈耦合网络是突然出现全局同步, 同步之前网络出现非周期性的相干斑图; 对于分层局域平均场反馈耦合网络, 同层神经元之间先出现从簇放电同步到同步的转变, 形成靶波, 然后同步区由中心向外逐渐扩大, 最终达到网络的全局同步. 这些结果表明, 只有适当的耦合才能实现信号的无损耗的传递. 此外我们发现分层局域平均场反馈耦合可以促进网络的同步.     

Adaptive synchronization of weighted complex dynamical networks with coupling time-varying delays

This Letter considers the problem of controlling a weighted complex dynamical network with coupling time-varying delay toward an assigned evolution. Adaptive controllers have been designed for nodes of the controlled network. Analytical results show that the states of the weighted dynamical network can globally asymptotically synchronize onto a desired orbit under the designed controllers. In comparison with the common linear feedback controllers, the adaptive controllers have strong robustness against asymmetric coupling matrix, time-varying weights, delays, and noise. Numerical simulations illustrated by a nearest-neighbor coupling network verify the effectiveness of the proposed controllers.