Comparison of Synchronization Ability of Four Types of Regular Coupled Networks

We investigate the synchronization ability of four types of regular coupled networks. By introducing the proper error variables and Lyapunov functions, we turn the stability of synchronization manifold into that of null solution of error equations, further, into the negative definiteness of some symmetric matrices, thus we get the sufficient synchronization stability conditions. To test the valid of the results, we take the Chua's circuit as an example. Although the theoretical synchronization thresholds appear to be very conservative, they provide new insights about the influence of topology and scale of networks on synchronization, and that the theoretical results and our numerical simulations are consistent.     

Adaptive mechanism between dynamical synchronization and epidemic behavior on complex networks

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.     

非自治混沌系统的脉冲同步

研究两个非自治混沌系统之间的脉冲同步的实用稳定性,将问题转化为同步误差系统的原点的实用稳定性．利用脉冲微分方程的相关理论,给出了同步误差系统原点实用稳定的相关判据,并给出了误差范围的理论表达式．计算机仿真实验的结果验证了理论结果 关键词：     

Adaptive Synchronization of Fractional Order Complex-Variable Dynamical Networks via Pinning Control

In this paper, the synchronization of fractional order complex-variable dynamical networks is studied using an adaptive pinning control strategy based on close center degree. Some effective criteria for global synchronization of fractional order complex-variable dynamical networks are derived based on the Lyapunov stability theory. From the theoretical analysis, one concludes that under appropriate conditions, the complex-variable dynamical networks can realize the global synchronization by using the proper adaptive pinning control method. Meanwhile, we succeed in solving the problem about how much coupling strength should be applied to ensure the synchronization of the fractional order complex networks. Therefore, compared with the existing results, the synchronization method in this paper is more general and convenient. This result extends the synchronization condition of the real-variable dynamical networks to the complex-valued field, which makes our research more practical. Finally, two simulation examples show that the derived theoretical results are valid and the proposed adaptive pinning method is effective.     

Novel criteria for exponential synchronization of inner time-varying complex networks with coupling delay

This paper mainly investigates the exponential synchronization of an inner time-varying complex network with coupling delay. Firstly, the synchronization of complex networks is decoupled into the stability of the corresponding dynamical systems. Based on the Lyapunov function theory, some sufficient conditions to guarantee its stability with any given convergence rate are derived, thus the synchronization of the networks is achieved. Finally, the results are illustrated by a simple time-varying network model with a coupling delay. All involved numerical simulations verify the correctness of the theoretical analysis.     

Stability and synchronization of fractional-order complex-valued neural networks with time delay: LMI approach

In this paper, we investigate the problem of stability and synchronization of fractional-order complex-valued neural networks with time delay. By using Lyapunov–Krasovskii functional approach, some linear matrix inequality (LMI) conditions are proposed to ensure that the equilibrium point of the addressed neural networks is globally Mittag–Leffler stable. Moreover, some sufficient conditions for projective synchronization of considered fractional-order complex-valued neural networks are derived in terms of LMIs. Finally, two numerical examples are given to demonstrate the effectiveness of our theoretical results.     

Exponential synchronization of stochastic neural networks with leakage delay and reaction-diffusion terms via periodically intermittent control

We explore the issue of integrating leakage delay, stochastic noise perturbation, and reaction-diffusion effects into the study of synchronization for neural networks with time-varying delays. By using Lyapunov stability theory and stochastic analysis approaches, a periodically intermittent controller is proposed to guarantee the exponential synchronization of proposed coupled neural networks based on p-norm. Some existing results are improved and extended. The usefulness and superiority of our theoretical results are illustrated by a numerical example.     

Combination projection synchronization of fractional-order complex dynamic networks with time-varying delay couplings and disturbances

In this paper, the problem of combination projection synchronization of fractional-order complex dynamic networks with time-varying delay couplings and external interferences is studied. Firstly, the definition of combination projection synchronization of fractional-order complex dynamic networks is given, and the synchronization problem of the drive-response systems is transformed into the stability problem of the error system. In addition, time-varying delays and disturbances are taken into consideration to make the network synchronization more practical and universal. Then, based on Lyapunov stability theory and fractional inequality theory, the adaptive controller is formulated to make the drive and response systems synchronization by the scaling factors. The controller is easier to realize because there is no time-delay term in the controller. At last, the corresponding simulation examples demonstrate the effectiveness of the proposed scheme.     

Synchronization rate of synchronized coupled systems

Synchronization phenomena, an emergent property in networks of interacting dynamical elements, are widely observed in nature, and have become the subject of intense research. Here we will investigate the synchronization rate problem in coupled limit-cycle and chaotic oscillators. Based on the mode decomposition method and Gershgörin's discs theorem, some sufficient conditions for synchronization of coupled systems are obtained, and a synchronization rate is then derived. Such a synchronization rate indicates that the error functions between state variables of underlying individual systems tend to zero in the exponential form as time tends to the infinity. Several numerical examples are also given to validate the theoretical results.     

具有非线性控制的Chua电路的混沌同步

Chua电路是一个非光滑系统.本文通过广义哈密顿系统和观测器方法,将具有非线性控制的Chua电路的混沌同步问题转化成研究具有非线性控制的光滑系统的零解稳定性;进而利用滑模控制对该光滑系统的零解稳定性进行了研究,从而使得Chua电路达到了混沌同步.最后,将上述方法应用到具体系统,数值结果也表明其正确性.     

Novel criteria of synchronization stability in complex networks with coupling delays

Complex networks are wide spread in the real world, arising in fields as disparate as sociology, physics and biology. The information spreading through a complex network is often associated with time delays due to the finite speeds of signal transmission over a distance. Hence, complex networks with coupling delays have gained increasing attention in various fields of science and engineering today. In this paper, based on the theory of asymptotic stability of linear time-delay systems, synchronization stability in complex dynamical networks with coupling delays is investigated, and we derive novel criteria of synchronization state for both delay-independent and delay-dependent stabilities. As illustrative examples, we use the networks with coupling delays and a given coupling scheme to test the theoretical results.     

Synchronization of general complex networks via adaptive control schemes

In this paper, the synchronization problem of general complex networks is investigated by using adaptive control schemes. Time-delay coupling, derivative coupling, nonlinear coupling etc. exist universally in real-world complex networks. The adaptive synchronization scheme is designed for the complex network with multiple class of coupling terms. A criterion guaranteeing synchronization of such complex networks is established by employing the Lyapunov stability theorem and adaptive control schemes. Finally, an illustrative example with numerical simulation is given to show the feasibility and efficiency of theoretical results.     

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     

Exponential stability of synchronization in asymmetrically coupled dynamical networks

Based on the original definition of the synchronization stability, a general framework is presented for investigating the exponential stability of synchronization in asymmetrically coupled networks. By choosing an appropriate Lyapunov function, we prove that the mechanism of the exponential synchronization stability is the asymmetrical coupling matrix with diffusive condition. We deduce the second largest eigenvalue of a symmetric matrix to govern the exponential stability of synchronization in asymmetrically coupled networks. Moreover, we have given the threshold value which can guarantee that the states of the asymmetrically coupled network achieve the exponential stability of synchronization.     

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 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.     

Cluster projective synchronization between community networks with nonidentical nodes

In this paper, cluster projective synchronization between community networks with nonidentical nodes is investigated. Outer synchronization between two identical or nonidentical complex networks has been extensively studied, in which all the nodes synchronized each other in a common manner. However, in real community networks, different communities in networks usually synchronize with each other in a different manner, i.e., achieving cluster projective synchronization. Based on Lyapunov stability theory, sufficient conditions for achieving cluster projective synchronization are derived through designing proper controllers. Numerical simulations are provided to verify the correctness and effectiveness of the derived theoretical results.     

Synchronization control of stochastic delayed neural networks

In this paper, synchronization control of stochastic neural networks with time-varying delays has been considered. A novel control method is given using the Lyapunov functional method and linear matrix inequality (LMI) approach. Several sufficient conditions have been derived to ensure the global asymptotical stability in mean square for the error system, and thus the drive system synchronize with the response system. Also, the estimation gains can be obtained. With these new and effective methods, synchronization can be achieved. Simulation results are given to verify the theoretical analysis in this paper.     

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.     

Synchronization in complex delayed dynamical networks with impulsive effects

The present paper is mainly concerned with the issues of synchronization dynamics of complex delayed dynamical networks with impulsive effects. A general model of complex delayed dynamical networks with impulsive effects is formulated, which can well describe practical architectures of more realistic complex networks related to impulsive effects. Based on impulsive stability theory on delayed dynamical systems, some simple but less conservative criterion are derived for global synchronization of such dynamical network. It is shown that synchronization of the networks is heavily dependent on impulsive effects of connecting configuration in the networks. Furthermore, the theoretical results are applied to a typical SF network composing of impulsive coupled chaotic delayed Hopfield neural network nodes, and are also illustrated by numerical simulations.