Impulsive synchronization of coupled dynamical networks with nonidentical Duffing oscillators and coupling delays

This paper aims to investigate the synchronization problem of coupled dynamical networks with nonidentical Duffing-type oscillators without or with coupling delays. Different from cluster synchronization of nonidentical dynamical networks in the previous literature, this paper focuses on the problem of complete synchronization, which is more challenging than cluster synchronization. By applying an impulsive controller, some sufficient criteria are obtained for complete synchronization of the coupled dynamical networks of nonidentical oscillators. Furthermore, numerical simulations are given to verify 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.     

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understanding synchronization phenomena in natural systems now take advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also take an overview of the new emergent features coming out from the interplay between the structure and the function of the underlying patterns of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.     

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。     

Fixed-Time Synchronization Control of Delayed Dynamical Complex Networks

Fixed-time synchronization problem for delayed dynamical complex networks is explored in this paper. Compared with some correspondingly existed results, a few new results are obtained to guarantee fixed-time synchronization of delayed dynamical networks model. Moreover, by designing adaptive controller and discontinuous feedback controller, fixed-time synchronization can be realized through regulating the main control parameter. Additionally, a new theorem for fixed-time synchronization is used to reduce the conservatism of the existing work in terms of conditions and the estimate of synchronization time. In particular, we obtain some fixed-time synchronization criteria for a type of coupled delayed neural networks. Finally, the analysis and comparison of the proposed controllers are given to demonstrate the validness of the derived results from one numerical example.     

Synchronization in uncertain complex networks

We consider the problem of synchronization in uncertain generic complex networks. For generic complex networks with unknown dynamics of nodes and unknown coupling functions including uniform and nonuniform inner couplings, some simple linear feedback controllers with updated strengths are designed using the well-known LaSalle invariance principle. The state of an uncertain generic complex network can synchronize an arbitrary assigned state of an isolated node of the network. The famous Lorenz system is stimulated as the nodes of the complex networks with different topologies. We found that the star coupled and scale-free networks with nonuniform inner couplings can be in the state of synchronization if only a fraction of nodes are controlled.     

Global synchronization for a class of dynamical complex networks

This paper is concerned with the problem of global synchronization for a class of dynamical complex networks composed of general Lur’e systems. Based on the absolute stability theory and the Kalman-Yakubovich-Popov (KYP) lemma, sufficient conditions are established to guarantee global synchronization of dynamical networks with complex topology, directed and weighted couplings. Several global synchronization criteria formulated in the form of linear matrix inequalities (LMIs) or frequency-domain inequalities are also proposed for undirected dynamical networks. In order to obtain global results, no linearization technique is involved through derivation of the synchronization criteria. Numerical examples are provided to demonstrate the effectiveness of the proposed results.     

具有双重时滞的时变耦合复杂网络的牵制外同步研究

针对同时具有节点时滞和耦合时滞的时变耦合复杂网络的外同步问题, 提出一种简单有效的自适应牵制控制方法. 首先构建一种贴近实际的驱动-响应复杂网络模型, 在模型中引入双重时滞和时变不对称外部耦合矩阵. 进一步设计易于实现的自适应牵制控制器, 对网络中的一部分关键节点进行控制. 构造适当的Lyapunov泛函, 利用 LaSalle不变集原理和线性矩阵不等式, 给出两个复杂网络实现外同步的充分条件. 最后, 仿真结果表明所提同步方法的有效性, 同时揭示耦合时滞对同步收敛速度的影响.     

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 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.      

Robust outer synchronization between two complex networks with fractional order dynamics

Synchronization between two coupled complex networks with fractional-order dynamics, hereafter referred to as outer synchronization, is investigated in this work. In particular, we consider two systems consisting of interconnected nodes. The state variables of each node evolve with time according to a set of (possibly nonlinear and chaotic) fractional-order differential equations. One of the networks plays the role of a master system and drives the second network by way of an open-plus-closed-loop (OPCL) scheme. Starting from a simple analysis of the synchronization error and a basic lemma on the eigenvalues of matrices resulting from Kronecker products, we establish various sets of conditions for outer synchronization, i.e., for ensuring that the errors between the state variables of the master and response systems can asymptotically vanish with time. Then, we address the problem of robust outer synchronization, i.e., how to guarantee that the states of the nodes converge to common values when the parameters of the master and response networks are not identical, but present some perturbations. Assuming that these perturbations are bounded, we also find conditions for outer synchronization, this time given in terms of sets of linear matrix inequalities (LMIs). Most of the analytical results in this paper are valid both for fractional-order and integer-order dynamics. The assumptions on the inner (coupling) structure of the networks are mild, involving, at most, symmetry and diffusivity. The analytical results are complemented with numerical examples. In particular, we show examples of generalized and robust outer synchronization for networks whose nodes are governed by fractional-order Lorenz dynamics.     

Cost and effect of pinning control for network synchronization

The problem of pinning control for the synchronization of complex dynamical networks is discussed in this paper. A cost function of the controlled network is defined by the feedback gain and the coupling strength of the network. An interesting result is that a lower cost is achieved by using the control scheme of pinning nodes with smaller degrees. Some strict mathematical analyses are presented for achieving a lower cost in the synchronization of different star-shaped networks. Numerical simulations on some non-regular complex networks generated by the Barabási--Albert model and various star-shaped networks are performed for verification and illustration.     

Robustness of optimal synchronization in real networks

Experimental studies can provide powerful insights into the physics of complex networks. Here, we report experimental results on the influence of connection topology on synchronization in fiber-optic networks of chaotic optoelectronic oscillators. We find that the recently predicted nonmonotonic, cusplike synchronization landscape manifests itself in the rate of convergence to the synchronous state. We also observe that networks with the same number of nodes, same number of links, and identical eigenvalues of the coupling matrix can exhibit fundamentally different approaches to synchronization. This previously unnoticed difference is determined by the degeneracy of associated eigenvectors in the presence of noise and mismatches encountered in real-world conditions.     

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.     

Stochastic synchronization for time-varying complex dynamical networks

This paper studies the stochastic synchronization problem for time-varying complex dynamical networks. This model is totally different from some existing network models. Based on the Lyapunov stability theory, inequality techniques, and the properties of the Weiner process, some controllers and adaptive laws are designed to ensure achieving stochastic synchronization of a complex dynamical network model. A sufficient synchronization condition is given to ensure that the proposed network model is mean-square stable. Theoretical analysis and numerical simulation fully verify the main results.     

Finite-time complex projective synchronization of fractional-order complex-valued uncertain multi-link network and its image encryption application

Multi-link networks are universal in the real world such as relationship networks, transportation networks, and communication networks. It is significant to investigate the synchronization of the network with multi-link. In this paper, considering the complex network with uncertain parameters, new adaptive controller and update laws are proposed to ensure that complex-valued multilink network realizes finite-time complex projective synchronization (FTCPS). In addition, based on fractional-order Lyapunov functional method and finite-time stability theory, the criteria of FTCPS are derived and synchronization time is given which is associated with fractional order and control parameters. Meanwhile, numerical example is given to verify the validity of proposed finite-time complex projection strategy and analyze the relationship between synchronization time and fractional order and control parameters. Finally, the network is applied to image encryption, and the security analysis is carried out to verify the correctness of this method.     

Impulsive synchronization of two coupled complex networks with time-delayed dynamical nodes

In this paper, we investigate the impulsive synchronization between two coupled complex networks with time-delayed dynamical nodes. Based on the Lyapunov stability, the linear feedback control and the impulsive control theories, the linear feedback and the impulsive controllers are designed separately. By using the generalized Barbalat's lemma, the global asymptotic impulsive synchronization of the drive—response complex networks is derived and some corresponding sufficient conditions are also obtained. Numerical examples are presented to verify the effectiveness and the correctness of the synchronization criteria.     

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 for complex dynamical Lurie networks

This paper investigates the synchronization problem for two different complex dynamical Lurie networks, The first one is with constant coupling and the second one is with constant coupling and discrete-delay coupling. Based on contraction theory and matrix measure properties, some new delay-independent synchronization conditions depending on coupling strength and network topology are proposed. Finally, simulation results are presented to support the theoretical results.