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.     

Generalized projective synchronization of fractional-order complex networks with nonidentical nodes

<正>This paper investigates the synchronization problem of fractional-order complex networks with nonidentical nodes. The generalized projective synchronization criterion of fractional-order complex networks with order 0 < q < 1 is obtained based on the stability theory of the fractional-order system.The control method which combines active control with pinning control is then suggested to obtain the controllers.Furthermore,the adaptive strategy is applied to tune the control gains and coupling strength.Corresponding numerical simulations are performed to verify and illustrate the theoretical results.     

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.     

Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes

In this Letter, exponential synchronization of a complex network with nonidentical time-delayed dynamical nodes is considered. Two effective control schemes are proposed to drive the network to synchronize globally exponentially onto any smooth goal dynamics. By applying open-loop control to all nodes and adding some intermittent controllers to partial nodes, some simple criteria for exponential synchronization of such network are established. Meanwhile, a pinning scheme deciding which nodes need to be pinned and a simply approximate formula for estimating the least number of pinned nodes are also provided. By introducing impulsive effects to the open-loop controlled network, another synchronization scheme is developed for the network with nonidentical time-delayed dynamical nodes, and an estimate of the upper bound of impulsive intervals ensuring global exponential stability of the synchronization process is also given. Numerical simulations are presented finally to demonstrate the effectiveness of the theoretical results.     

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.     

Outer synchronization between two nonidentical networks with circumstance noise

This paper regards the outer synchronization between two delay-coupled complex dynamical networks with nonidentical topological structures and a noise perturbation. Considering one network as the drive network and the other one as the response network, the drive-response system achieves synchronous states through a suitably designed adaptive controller. The stochastic LaSalle invariance principle is employed to theoretically prove the almost sure synchronization between two networks. Finally, two numerical examples are examined in order to illustrate the proposed synchronization scheme.     

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.     

Function projective synchronization in drive-response dynamical network

In this Letter, the function projective synchronization in the drive-response dynamical network is investigated, where the response dynamical network is affected not only by the drive system, but also coupled via a linearly feedback scheme. Based on Lyapunov stability theory, it is shown that the function projective synchronization with desired scaling function can be realized in the drive-response dynamical network by a simple control law. Moreover it is no need for the scaling function to be differentiable, bounded and nonzero all the time. The numerical simulations are provided to verify the theoretical result.     

Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions

<正>The adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions is investigated in this paper.Based on Lyapunov stability theory and Barbalat’s lemma,generalized matrix projective lag synchronization criteria are derived by using the adaptive control method.Furthermore,each network can be undirected or directed,connected or disconnected,and nodes in either network may have identical or different dynamics.The proposed strategy is applicable to almost all kinds of complex networks.In addition,numerical simulation results are presented to illustrate the effectiveness of this method,showing that the synchronization speed is sensitively influenced by the adaptive law strength,the network size,and the network topological structure.     

Generalized and projective synchronization in modulated time-delayed systems

In this Letter, we propose a method for generalized and projective synchronization in modulated time-delayed systems using nonlinear active control. Sufficient condition for generalized synchronization is calculated analytically by Krasovskii-Lyapunov stability theory. The validity of the proposed algorithm has been confirmed by simulation results. The proposed method helps to find the explicit form of the functional relation between the synchronized variables, for which very few formulations are known at the present moment. Both usual and lag-anticipatory cases can be treated on the same footing.     

噪声环境下时滞耦合网络的广义投影滞后同步

时滞和噪声在复杂网络中普遍存在,而含有耦合时滞和噪声摄动的耦合网络同步的研究工作却极其稀少. 本文针对噪声环境下具有不同节点动力学、不同拓扑结构及不同节点数目的耦合时滞网络,提出了两个网络之间的广义投影滞后同步. 首先,构建了更加贴近现实的驱动-响应网络同步的理论框架;其次,基于随机时滞微分方程LaSalle不变性原理,严格证明了在合理的控制器作用下,驱动网络和响应网络在几乎必然渐近稳定性意义下能够取得广义投影滞后同步;最后,借助于计算机仿真,通过具体的网络模型验证了理论推理的有效性. 数值模拟结果表明,驱动网络与响应网络不但能够达到广义投影滞后同步,而且同步效果不依赖于耦合时滞和比例因子的选取,同时也揭示了更新增益和耦合时滞对同步收敛速度的显著性影响. 关键词： 复杂网络 广义投影滞后同步 随机噪声 时滞     

Cluster synchronization in the adaptive complex dynamical networks via a novel approach

This Letter investigates cluster synchronization in the adaptive complex dynamical networks with nonidentical nodes by a local control method and a novel adaptive strategy for the coupling strengths of the networks. In this approach, the coupling strength of each node adjusts adaptively only based on the state information of its neighborhood. By means of the proposed scheme, the sufficient conditions for achieving cluster synchronization are derived analytically by utilizing Lyapunov stability theory. It is demonstrated that the synchronization performance is sensitively affected by the control gain, the inner-coupling matrix and the network topological structure. The numerical simulations are performed to verify the effectiveness of the theoretical results.     

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.     

Adaptive modified function projective synchronization of multiple time-delayed chaotic Rossler system

In this Letter we consider modified function projective synchronization of unidirectionally coupled multiple time-delayed Rossler chaotic systems using adaptive controls. Recently, delay differential equations have attracted much attention in the field of nonlinear dynamics. The high complexity of the multiple time-delayed systems can provide a new architecture for enhancing message security in chaos based encryption systems. Adaptive control can be used for synchronization when the parameters of the system are unknown. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the state of two chaotic systems are function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

具有随机节点结构的复杂网络同步研究

针对具有随机节点结构的复杂网络, 研究其同步问题. 基于Lyapunov稳定性理论和线性矩阵不等式技术给出了复杂网络同步稳定的充分性条件, 该充分性条件不仅与复杂网络的状态时延有关, 还与节点结构的概率分布有关. 数值仿真表明本文方法的有效性. 关键词： 复杂网络 随机节点 同步稳定 时滞     

Adaptive projective synchronization with different scaling factors in networks

We study projective synchronization with different scaling factors （PSDF） in N coupled chaotic systems networks. By using the adaptive linear control, some sufficient criteria for the PSDF in symmetrical and asymmetrical coupled networks are separately given based on the Lyapunov function method and the left eigenvalue theory. Numerical simulations for a generalized chaotic unified system are illustrated to verify the theoretical results.     

Generalized projective synchronization between two different general complex dynamical networks with delayed coupling

In this Letter, generalized projective synchronization (GPS) between two different complex dynamical networks with delayed coupling is investigated. Two complex networks are distinct if they have diverse node dynamics, or different number of nodes, or different topological structures. By using the adaptive control scheme, a sufficient synchronization criterion for this GPS is derived based on the LaSalle invariance principle. Three corollaries are also obtained. It is noticed that the synchronization speed sensitively depends on the adjustable positive constants μi. Furthermore, the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. In addition, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results.     

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.     

Adaptive synchronization between two complex networks with nonidentical topological structures

This paper addresses the theoretical analysis of synchronization between two complex networks with nonidentical topological structures. By designing effective adaptive controllers, we achieve synchronization between two complex networks. Both the cases of identical and nonidentical network topological structures are considered and several useful criteria for synchronization are given. Illustrative examples are presented to demonstrate the application of the theoretical results.     

Adaptive function projective synchronization of uncertain complex dynamical networks with disturbance

We investigate the problem of function projective synchronization (FPS) in drive-response dynamical networks with non-identical nodes. An adaptive controller is proposed for the FPS of complex dynamical networks with uncertain parameters and disturbance. Not only are the unknown parameters of the networks estimated by the adaptive laws obtained from the Lyapunov stability theory and Taylor expansions, but the unknown bounded disturbances are also simultaneously conquered by the proposed control. Finally, a numerical simulation is provided to illustrate the feasibility and effectiveness of the obtained result.