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.     

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     

Mixed outer synchronization of coupled complex networks with time-varying coupling delay

In this paper, the problem of outer synchronization between two complex networks with the same topological structure and time-varying coupling delay is investigated. In particular, we introduce a new type of outer synchronization behavior, i.e., mixed outer synchronization (MOS), in which different state variables of the corresponding nodes can evolve into complete synchronization, antisynchronization, and even amplitude death simultaneously for an appropriate choice of the scaling matrix. A novel nonfragile linear state feedback controller is designed to realize the MOS between two networks and proved analytically by using Lyapunov-Krasovskii stability theory. Finally, numerical simulations are provided to demonstrate the feasibility and efficacy of our proposed control approach.     

Synchronization in Complex Networks with Multiple Connections

In this paper a class of networks with multiple connections are discussed. The multiple connections include two different types of links between nodes in complex networks. For this new model, we give a simple generating procedure. Furthermore, we investigate dynamical synchronization behavior in a delayed two-layer network, giving corresponding theoretical analysis and numerical examples.     

Robust projective outer synchronization of coupled uncertain fractional-order complex networks

In this work, we propose a novel projective outer synchronization (POS) between unidirectionally coupled uncertain fractional-order complex networks through scalar transmitted signals. Based on the state observer theory, a control law is designed and some criteria are given in terms of linear matrix inequalities which guarantee global robust POS between such networks. Interestingly, in the POS regime, we show that different choices of scaling factor give rise to different outer synchrony, with various special cases including complete outer synchrony, anti-outer synchrony and even a state of amplitude death. Furthermore, it is demonstrated that although stability of POS is irrelevant to the inner-coupling strength, it will affect the convergence speed of POS. In particular, stronger inner synchronization can induce faster POS. The effectiveness of our method is revealed by numerical simulations on fractional-order complex networks with small-world communication topology.     

耦合含时滞的相互依存网络的局部自适应异质同步

针对由两个子网络构成的耦合含时滞的相互依存网络,研究其局部自适应异质同步问题.时滞同时存在于两个子网络的内部耦合项和子网络间的一对一相互依赖耦合项中,且网络的耦合关系满足非线性特性和光滑性.基于李雅普诺夫稳定性理论、线性矩阵不等式方法和自适应控制技术,通过对子网络设置合适的控制器,提出了使得相互依存网络的子网络分别同步到异质孤立系统的充分条件.针对小世界网络和无标度网络构成的相互依存网络进行数值模拟,验证了提出理论的正确性和有效性.     

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 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.     

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.     

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.     

Generalized projective synchronization of two coupled complex networks of different sizes

We investigate a new generalized projective synchronization between two complex dynamical networks of different sizes. To the best of our knowledge, most of the current studies on projective synchronization have dealt with coupled networks of the same size. By generalized projective synchronization, we mean that the states of the nodes in each network can realize complete synchronization, and the states of a pair of nodes from both networks can achieve projective synchronization. Using the stability theory of the dynamical system, several sufficient conditions for guaranteeing the existence of the generalized projective synchronization under feedback control and adaptive control are obtained. As an example, we use Chua's circuits to demonstrate the effectiveness of our proposed approach.     

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 in networks with random interactions: theory and applications

Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to systems with time delayed coupling and then nonlinear systems where the synchronized dynamics can be periodic or chaotic. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.     

Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks

We study projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, where projective-anticipating and projective-lag synchronization can be achieved only on two coupled chaotic systems. In this paper, we realize projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In this paper, the dynamics of the nodes of the complex networks are time-delayed chaotic systems without the limitation of the partial linearity. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its existence and sufficient stability conditions. The validity of the proposed method is demonstrated and verified by examining specific examples using Ikeda and Mackey-Glass systems on Erdos-Renyi networks.     

An Improved Heterogeneous Mean-Field Theory for the Ising Model on Complex Networks

Heterogeneous mean-field theory is commonly used methodology to study dynamical processes on complex networks,such as epidemic spreading and phase transitions in spin models.In this paper,we propose an improved heterogeneous mean-field theory for studying the Ising model on complex networks.Our method shows a more accurate prediction in the critical temperature of the Ising model than the previous heterogeneous mean-field theory.The theoretical results are validated by extensive Monte Carlo simulations in various types of networks.     

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.     

复杂网络上动力系统同步的研究进展

本文简要介绍复杂网络的基本概念并详细总结了近年来复杂网络上动力学系统的同步的研究进展，主要内容有复杂网络同步的稳定性分析，复杂网络上动力学系统同步的特点，网络的几何特征量对同步稳定性的影响，以及提高网络同步能力的方法等。最后文章提出了这一领域的几个有待解决的问题及可能的发展方向。     

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.     

Cluster synchronization in networks of distinct groups of maps

In this paper, we study cluster synchronization in general bi-directed networks of nonidentical clusters, where all nodes in the same cluster share an identical map. Based on the transverse stability analysis, we present sufficient conditions for local cluster synchronization of networks. The conditions are composed of two factors: the common inter-cluster coupling, which ensures the existence of an invariant cluster synchronization manifold, and communication between each pair of nodes in the same cluster, which is necessary for chaos synchronization. Consequently, we propose a quantity to measure the cluster synchronizability for a network with respect to the given clusters via a function of the eigenvalues of the Laplacian corresponding to the generalized eigenspace transverse to the cluster synchronization manifold. Then, we discuss the clustering synchronous dynamics and cluster synchronizability for four artificial network models: (i) p-nearest-neighborhood graph; (ii) random clustering graph; (iii) bipartite random graph; (iv) degree-preferred growing clustering network. From these network models, we are to reveal how the intra-cluster and inter-cluster links affect the cluster synchronizability. By numerical examples, we find that for the first model, the cluster synchronizability regularly enhances with the increase of p, yet for the other three models, when the ratio of intra-cluster links and the inter-cluster links reaches certain quantity, the clustering synchronizability reaches maximal.