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Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling

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Abstract

This paper investigates the adaptive synchronization between two nonlinearly delay-coupled complex networks with the bidirectional actions and nonidentical topological structures. Based on LaSalle’s invariance principle, some criteria for the synchronization between two coupled complex networks are achieved via adaptive control. To validate the proposed methods, the unified chaotic system as the nodes of the networks are analyzed in detail, and numerical simulations are given to illustrate the theoretical results.Research highlights► We investigate the outer synchronization of two nonlinear coupled complex network. ► We not only consider their own network of coupling, but also take into account the bidirectional actions and time delays. ► Nonidentical topological structures are also considered.

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... Different from the inner synchronization, the outer synchronization is synchronization between two coupled networks, which has quickly gained extensive attention since Li et al. [15] first studied synchronization of two unidirectional coupled networks with identical topological structures, and derived analytically a criterion for the outer synchronization by applying the open-plus-closed-loop control method. Afterwards, the expanded works on the outer synchronization, such as including different node dynamics, nonidentical topological structures and time-varying delays, can be found in the literature [16][17][18][19][20][21][22] and references therein. ...
... Furthermore, many real complex networks cannot synchronize themselves or synchronize with the desired orbits. So far, many control schemes have been used to design effective controllers, such as adaptive control [16,19,[23][24][25] , pinning control [22,[26][27][28][29][30][31][32]54] , impulsive control [21,[33][34][35][36][37][38][39] and intermittent control [40][41][42][43] , and so on. Generally, a typical real dynamical network consists of large number of interconnected nodes, and it is usually impractical and even impossible to control all nodes to synchronize the desired trajectory. ...
... One is internal delay occurring inside the dynamical node [38,40,43,56] . The other is coupling delay caused by exchange of information between nodes [18][19][34][35][36]39,42,46,47] . According to the fact that internal delay is more complex than coupling delay, thus, the internal delay will be considered in this paper. ...
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In this paper, the problem of pinning and impulsive synchronization between two complex dynamical networks with non-derivative and derivative coupling is investigated. A hybrid controller, which contains a pinning controller and a pinning impulsive controller, is proposed simultaneously. Based on the Lyapunov stability theory and mathematical analysis technique, some novel criteria of synchronization are derived, which can guarantee that the response network asymptotically synchronizes to the drive network by combining pinning control and pinning impulsive control. Moreover, the restrictions about non-derivative coupling matrix, impulsive intervals and the number of pinned nodes are removed. Numerical examples are presented finally to illustrate the effectiveness of the theoretical results.
... Over the last few decades, considerable attentions have been denoted into the research of coupled delayed neural networks which often show stability, bifurcation or consensus [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. One of the chief reasons is that most practical systems can be modeled by complex dynamical networks, such as the Internet, the world Wide Web, metabolic systems, food webs, etc. ...
... where d l = max{hρ l , ρ l exp(λ σ (l) − ln ρ l l )τ }. Then lim t→+∞ (ϕ(t, t 0 )) = −∞ implies that the coupled DSNNs (7) with impulses time window are synchronization. ...
... In theorem 2, if t k+1 − t k ≤ M < ∞, and there exist constants λ andλ satisfying λ i < λ <λ such that ln d l +λ(t l − t l−1 ) ≤ 0, l = 1.2 . . ., then the coupled DSNNs with impulsive time window (7) are exponential synchronized. ...
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Synchronization of coupled delayed switched neural networks (DSNNs) with impulsive time window is one of the most challenging problems in the field of complex networks. In this paper, some sufficient conditions ensuring the synchronization of coupled DSNNs with all subsystems self-synchronizing are presented. Then this results are extended to the following two cases: all subsystems are desynchronizing and some of subsystems are self-synchronizing. By applying switching Lyapunov function method, some general criteria which characterize the impulse and switching effects in aggregated form are given. Moreover, from our results, one can easily observes that the impulsive time window control strategy is more general and more applicable than the fixed impulses controllers. Finally, numerical simulations are presented to further demonstrate the effectiveness of the proposed results.
... Because of the complexity of system structure and communication interaction in practical applications, there may be the bidirectional actions between networked systems. In [34], Zheng et al. investigate outer consensus between two nonlinear coupled first-order networks under nonidentical topological and bidirectional actions. Sun et al. [35] address outer and inner consensus between two coupled networks with bidirectional actions simultaneously. ...
... In addition, two simulation examples are presented to show that all the corresponding system states can achieve consensus by applying the proposed adaptive controller. Obviously, compared with the existing work [34], when the first-order system is extended to the second-order system and mix-order system, the dynamic behavior of the system becomes more complex, and thus more factors need to be considered for designing the controller. This paper is structured as follows. ...
... and the nonlinear functions f i (i = 1, 2, 3) are chosen as the following unified chaotic systems that are similar to the example in [34] and satisfy Assumption 1. ...
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The outer consensus between two coupled heterogeneous networked systems with bidirectional actions and nonidentical topologies is addressed in this paper. Based on the adaptive theory, a control protocol is designed for two coupled heterogeneous networked systems, where each system consists of double-integrator and single-integrator nodes with different nonlinear dynamics. In the light of Lyapunov method and LaSalle’s invariance principle, we prove that the coupled heterogeneous networked systems can reach outer consensus by using the reported controller. Meanwhile, the adaptive controller is also given for the second-order nonlinear systems under identical and nonidentical topologies. The correctness of the obtained theory is demonstrated by two simulation examples.
... So far, in most of the existing works the networks with coupling time delays were considered. However, the time delays in the dynamical nodes [22][23][24][25][26][27][28], which can be more complicated, are still neglected in most of the existing works. In addition, it maybe unpractical to always use the hypothesis that all of the network nodes are the same since some real-world complex networks can be modeled by using different dynamical nodes [25]. ...
... When the dynamics of nodes in a complex network are allowed to be nonidentical, the synchronization approaches for networks consisting of identical nodes will 2 Journal of Control Science and Engineering not work anymore. Thus, it is of great importance to develop new synchronization approaches for time-delayed complex networks with nonidentical nodes [25][26][27][28][29][30]. In [26], some exponential synchronization criteria for a class of complex networks with nonidentical nodes were established via combining the local intermittent controller with the open-loop controller. ...
... In [26], some exponential synchronization criteria for a class of complex networks with nonidentical nodes were established via combining the local intermittent controller with the open-loop controller. In [27], the outer synchronization of two complex networks with nonidentical dynamical nodes and coupling time delays was investigated by employing the adaptive control technique. In [28], by using pinning control scheme, the cluster synchronization of complex dynamical networks with time-delayed coupling and dynamic nodes was studied. ...
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In this paper, we investigated the finite-time synchronization (FTS) problem for a class of time-delayed complex networks with nonidentical nodes onto any uniformly smooth state. By employing the finite-time stability theorem and designing two types of novel controllers, we obtained some simple sufficient conditions for the FTS of addressed complex networks. Furthermore, we also analyzed the effects of control variables on synchronization performance. Finally, we showed the effectiveness and feasibility of our methods by giving two numerical examples.
... However, to the best of our knowledge, it can be realized mainly by the open-plus-closed-loop method [19,20] or based on the drive-response concept [21][22][23][24][25][26][27] considering only the intranetwork coupling of network itself. Zheng et al. [28] and Wu et al. [29] further studied the outer synchronization between two complex networks considering two kinds of internetwork coupling, but nevertheless, they both still derived the synchronization criteria based on driveresponse concept and did not place the outer synchronization in the context of interdependent networks. ...
... Assumption 3. Suppose that the time-varying delays 1 ( ), Remark 4. Assumptions 2 and 3 are both general assumptions, which hold for a broad class of real-world chaotic systems, such as Lorenz system, Chua's oscillator, Chen system, and Lü system [28]. Hence, in the following sections, we always assume that both assumptions hold. ...
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This paper mainly focuses on the generalized mutual synchronization between two controlled interdependent networks. First, we propose the general model of controlled interdependent networks A and B with time-varying internetwork delays coupling. Then, by constructing Lyapunov functions and utilizing adaptive control technique, some sufficient conditions are established to ensure that the mutual synchronization errors between the state variables of networks A and B can asymptotically converge to zero. Finally, two numerical examples are given to illustrate the effectiveness of the theoretical results and to explore potential application in future smart grid. The simulation results also show how interdependent topologies and internetwork coupling delays influence the mutual synchronizability, which help to design interdependent networks with optimal mutual synchronizability.
... On the other hand, the synchronization of complex networks is generally achieved through transferring information among interconnected nodes via couplings. Moreover, the synchronization of complex networks and adaptive control methods have been described in [10,16,19,20,22,34,[57][58][59] . In [20] , adaptive lag synchronization for uncertain CDNs with delayed coupling has been analyzed. ...
... In [20] , adaptive lag synchronization for uncertain CDNs with delayed coupling has been analyzed. The adaptive synchronization of two nonlinearly coupled CDNs with delayed coupling has been investigated in [59] . Based on Lyapunov stability theory the performance of adaptive learning control scheme for complex dynamical networks was studied in [10] and adaptive event-triggered synchronization control problem for a class of complex networks is investigated [34] . ...
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This article inspects the problem of robust synchronization for uncertain Markovian jumping complex interconnected neural networks with randomly occurring uncertainties and time delays. The uncertainties considered here occur randomly and are assumed to follow certain mutually uncorrelated Bernoulli distributed white noise sequences. The presence of sensor faults may cause degradation or even instability of the entire network. Therefore, control laws are designed with sensor faults to ensure the controlled synchronization of the complex interconnected neural networks. Three types of fault-tolerant controls are designed based on the Lyapunov stability theory and adaptive schemes which include passive and adaptive fault-tolerant control laws. By constructing a new Lyapunov-Krasovskii functional (LKF) and by using Jensen’s inequality with a free-weighting matrix approach, some new delay-dependent synchronization criteria are obtained in terms of linear matrix inequalities (LMIs). By using the Lyapunov stability theory, the existence condition for the adaptive controller that guarantees the robust mean-square synchronization of complex interconnected neural networks in terms of LMIs are derived. Finally, a numerical example is presented to demonstrate the performance of the developed approach.
... The basic idea of network synchronization is to design suitable controllers or control inputs so that the network states can track the synchronization target states asymptotically. Until now, a lot of technologies have been developed for network synchronization, for examples, master stability function method [7], adaptive method [8,9], impulsive control [10], pinning control [11,12], backstepping technology [13], sliding mode control [14], etc. ...
... and considering Eqs. (8) and (9), it can be further obtained ...
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We propose a new technology to synchronize the uncertain dynamical network with the switching topology. In this new technology, we construct the Lyapunov function of network through designing a special function to obtain the network synchronization condition, which effectively avoids the complicated calculations for solving the second largest eigenvalue of the coupling matrix of the dynamical network. At the same time, the uncertain parameters in state equations at network nodes can also be identified accurately by the designed identification laws of uncertain parameters. Our results are universal without assumption about the symmetry of the coupling matrix in network, which can be widely used to research various topologies, no matter whether they are undirected or directed, weighted or unweighted, time-invariant or switching. And there are not any limitations for the synchronization target of network.
... Since chaotic systems defy synchronization, how to design effective controllers for synchronizing coupled chaotic systems becomes an important and challenging problem. Many effective methods including pinning control [11][12][13][14], adaptive control [15][16][17][18][19][20], impulsive control [21][22][23][24][25][26], and intermittent control [27][28][29] have been adopted to design proper controllers. Inner synchronization, that is, the synchronization of all the nodes within a network, has been investigated recently. ...
... Corollary 7. Consider the complex networks (16) and (17), if Assumptions 2 and 3 hold. Use the following adaptive controllers and updated laws: ...
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This paper investigates the exponential synchronization between two nonlinearly coupled complex networks with time-varying delay dynamical nodes. Based on the Lyapunov stability theory, some criteria for the exponential synchronization are derived with adaptive control method. Moreover, the presented results here can also be applied to complex dynamical networks with single time delay case. Finally, numerical analysis and simulations for two nonlinearly coupled networks which are composed of the time-delayed Lorenz chaotic systems are given to demonstrate the effectiveness and feasibility of the proposed complex network synchronization scheme.
... In the literature, much research effort has been made to address the technical challenges from different aspects (e.g., [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]). The pioneer work was carried out in [7] and showed that the outer synchronization can be implemented between coupled unidirectional networks with identical topological structures. ...
... Most recently, the criterion of generalized synchronization between two coupled networks was investigated in [13]. Also, the synchronization criteria of adaptive synchronization between two nonlinearly delay coupled complex networks with bidirectional interactions and nonidentical structures were studied in [14,15]. The hybrid synchronization problem between two networks with nondelayed and delayed coupling was addressed in [16,17] and it demonstrated that the hybrid synchronization can be realized by adding adaptive pinning controller at a number of network nodes. ...
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This paper exploits the network outer synchronization problem in a generic context for complex networks with nonlinear time-delay characteristics and nonidentical time-varying topological structures. Based on the classic Lyapunov stability theory, the synchronization criteria and adaptive control strategy are presented, respectively, by adopting an appropriate Lyapunov-Krasovskii energy function and the convergence of the system error can also be well proved. The existing results of network outer synchronization can be obtained by giving certain conditions, for example, treating the coupling matrices as time-invariant, and by applying the suggested generic synchronization criteria and control scheme. The numerical simulation experiments for networks scenarios with dynamic chaotic characteristics and time-varying topologies are carried out and the result verifies the correctness and effectiveness of the proposed control solution.
... Recently, dynamic analysis of coupled networks has always been a hot research issue, see for instance, [1][2][3][4][5][6][7]. Therein, synchronization of coupled networks has attracted many researchers' attentions, since there exist many benefits of having synchronization in many real systems, such as biological systems, chemical reactions, information science, and so on. ...
... Assume that the coefficients of drive network (1) and response network (2) satisfy the usually local Lipschitz condition and linear growth condition. Hence, for any given initial values x 0 , y 0 ∈ R m and r 0 ∈ S, the existence of unique solution to the drive-response systems (1) and (2) can be guaranteed by [45]. Denote them by y(t) = (y 1 (t) T , . . . ...
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In this paper, the issue of exponential synchronization for a class of stochastic coupled networks with Markovian switching is investigated. Based on some results in graph theory, Lyapunov stability theory and state feedback control technique, several sufficient criteria have been derived to ensure the pth moment exponential stability and almost sure exponential stability for the error network. That is, under these sufficient criteria, drive–response coupled networks can be pth moment exponentially synchronized and almost surely exponentially synchronized, respectively. Finally, stochastic Cohen–Grossberg neural networks with Markovian switching is employed to illustrate our feasible results.
... In reality, many practical systems, such as the different species development in balance [31], the groups of Drosophila clock neurons [32], and the spread of diseases, such as SARS and bird flu, between two communities [33], can be used to illustrate the outer synchronization phenomenon between two networks. Due to its theoretical and practical importance, outer synchronization between two dynamical networks has drawn much attention in recent years [34][35][36][37][38]. In [34], the outer synchronization was studied between two delay-coupled complex dynamical networks with nonidentical topological structures and a noise perturbation. ...
... In [34], the outer synchronization was studied between two delay-coupled complex dynamical networks with nonidentical topological structures and a noise perturbation. Zheng et al. [35] investigated the adaptive synchronization between two nonlinearly delay-coupled complex networks with the bidirectional actions and nonidentical topological structures. Outer synchronization between drive and response networks via adaptive impulsive pinning control was investigated in [36]. ...
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In this paper, the problem on outer synchronization is investigated for a class of mixed delayed complex networks by using the pinning control strategy. Together with some Lyapunov–Krasovskii functional and effective mathematical techniques, several conditions are derived to guarantee a class of complex networks with mixed delays to be outer synchronization. By proposing a novel functional condition which has not been proposed so far, further improved synchronization criteria are proposed. Finally, two examples are given to illustrate the effectiveness of the results.
... Many studies have pointed out that topological structure plays a significant role in the formation of network synchronization [2,3]. So far, the great majority of research activities have been focused on static networks, whose topology and coupling configuration in the network are time invariant [4][5][6][7][8][9]. However, these static connection topologies do not fit most realistic network systems; e.g., in biological, communication, social, and epidemiological networks the connection topology of the network generally changes in time. ...
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this paper studies the problem of synchronizing a moving agent network by decentralized adaptive control strategy. And, each agent node is assigned nonlinear connections with those of neighboring agents. The moving agent network model exhibits a time-varying topological structure in two dimensional spaces. Base on the Lyapunov stability theory, some criteria for the synchronization are achieved via adaptive control under the constraint of fast switching. To validate the proposed methods, the Lorenz chaotic system as the nodes of the networks are analyzed, and numerical simulations results show the effectiveness of proposed synchronization approaches.
... For the past few years, progressive control methods, such as pinning control [6][7][8][9]15], adaptive control [23], impulsive control [24,25], intermittent control [6,9,25,26], and so on, have been proposed to accomplish the synchronization of a complex network. In particular, pinning control is one of the valid schemes in the cluster synchronization control of a complex network, which is economy, simplicity, and practicality. ...
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Abstract Throughout this article, the conundrum on finite-time cluster synchronization is investigated for time-varying delayed complex dynamical networks using a kind of new hybrid control scheme. In the light of Lyapunov stability theorem and finite-time control theory, the finite-time cluster synchronization criteria can be achieved. Besides, we introduce the distinction between cluster synchronization and complete synchronization, which is how to select the controlling nodes. We discuss the differences among cluster synchronization with time-varying delays, complete synchronization with time-varying delays, cluster synchronization with a single delay, and cluster synchronization without delay, which is on constructing Lyapunov functional and designing the finite-time hybrid controllers. Finally, numerical simulations are presented to demonstrate the availability of the theoretical consequences.
... Time delays greatly influence behaviors of dynamical systems. Many literatures are focused on synchronization and control of complex networks with coupling delay among different nodes [18, 19]. Noise is another important factor affecting behaviors of dynamical systems, as it is inevitable due to environmental disturbance and uncertainties. ...
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In the real world, many complex systems are represented not by single networks but rather by sets of interdependent ones. In these specific networks, nodes in one network mutually interact with nodes in other networks. This paper focuses on a simple representative case of two-layer networks (the so-called duplex networks) with unidirectional inter-layer couplings. That is, each node in one network depends on a counterpart in the other network. Accordingly, the former network is called the response layer and the latter network is the drive layer. Specifically, synchronization between each node in the drive layer and its counterpart in the response layer (counterpart synchronization (CS)) in these kinds of duplex networks with delayed nodes and noise perturbation is investigated. Based on the LaSalle-type invariance principle, a control technique is proposed and a sucient condition is developed for realizing CS of duplex networks. Furthermore, two corollaries are derived as special cases. In addition, node dynamics within each layer can be varied and topologies of the two layers are not necessarily identical. Therefore, the proposed synchronization method can be applied to a wide range of multiplex networks. Numerical examples are provided to illustrate the feasibility and effectiveness of the results.
... The synchronization properties of a complex network are mainly determined by its topological structures connections between nodes. In the current study of complex networks, most of the existing works on synchronization consider a static networks, that is, the topological structures of which do not change as time evolves [4][5][6][7][8][9][10]. However, numerous real-world networks such as biological, communication, social, and epidemiological networks generally evolve with time-varying topological structures. ...
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... The synchronization properties of a complex network are mainly determined by its topological structures connections between nodes. In the current study of complex networks, most of the existing works on synchronization consider static networks, whose topological structures do not change as time evolves [4][5][6][7][8][9][10]. The Master-stability function (MSF) approach [11] allows us to determine the stability of a linearly coupled dynamical network with a constant coupling (or Laplacian) matrix. ...
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... Another example is that two unmanned vehicles are assigned to accomplish independent tasks, such as cooperative searches and attacks [22], which focuses on the inner synchronization inside each network. Presently a lot of studies applied the control methods (e.g., the adaptive, impulsive and pinning) [17,[26][27][28][29] to realize the outer synchronization, while the inner and outer synchronization were not studied simultaneously. If the outer synchronization is achieved by the appropriate interactions, then the controllers are not necessary. ...
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... In [15], Wu et al. studied the generalized outer synchronization between two networks with different dimensions of node dynamics. In addition, there are many works on the outer synchronization, that is, introducing the noise, time delay, fractional order node dynamics, and unknown parameters [16][17][18][19][20][21]. ...
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... However, most findings and solutions of network synchronization have focused on the inner synchronization, which considers the collective dynamic behaviours of the nodes inside a network (Dai, Chen, Xie, & Jia, 2018;Lee, Park, Ji, Kwon, & Lee, 2012;Sakthivel, Sathishkumar, Kaviarasan, & Marshal Anthoni, 2017;Wang, Li, Yang, & Fei, 2012;Xiang & Zhu, 2011;Zhou, Feng, & Chen, 2011). Unlike inner synchronization, the synchronization problem between two or multiple networks, known as outer synchronization, has been investigated in recent years to look into the interactive behaviours between networks (Fang, Yang, & Yan, 2014;Li, Lü, Yang, Zhou, & Hong, 2018;Liu & Wang, 2017;Wu, Li, Wu, & Kurths, 2012;Yang, Zhang, & Chen, 2012;Zheng, Wang, Dong, & Bi, 2012). Though in (Fang et al., 2014), the outer synchronization problem with nonlinear time-delay characteristics and nonidentical time-varying topological structures has been explored, it should be highlighted that the existing research effort has merely considered the scenario that all the network nodes have the same dynamic characteristics. ...
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When a transmission delay occurs in the interconnection of linearly coupled systems described by ordinary differential equations (LCODEs), both synchronization and the final synchronized state will vary. In this paper, mathematical analysis is presented on the synchronization phenomena of LCODEs with a single coupling delay. Criteria are derived for both local and global synchronization. It is known that in addition to the dynamical behaviors of the underlying uncoupled system and the coupling configuration, the coupling strength and the coupling delay also play key roles on the stability of synchronization. Both theoretical and numerical analysis indicate that under some conditions, if the coupling strength is large enough, the coupled system can be completely synchronized for any coupling delay. On the other hand, in some cases, the coupled system can be synchronized if the coupling delay is small enough.
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In this paper, synchronization between two discrete-time networks, called "outer synchronization" for brevity, is theoretically and numerically studied. First, a sufficient criterion for this outer synchronization between two coupled discrete-time networks which have the same connection topologies is derived analytically. Numerical examples are also given and they are in line with the theoretical analysis. Additionally, numerical investigations of two coupled networks which have different connection topologies are analyzed as well. The involved numerical results show that these coupled networks with different connection matrices can reach synchronization.
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In this paper, global synchronization of linearly coupled neural network (NN) systems with time-varying coupling is investigated. The dynamical behavior of the uncoupled system at each node is general, which can be chaotic or others; the coupling configuration is time varying, i.e., the coupling matrix is not a constant matrix. Based on Lyapunov function method and the specific property of Householder transform, some criteria for the global synchronization are obtained. By these criteria, one can verify whether the coupled system with time-varying coupling is globally synchronized, which is important and useful for both understanding and interpreting synchronization phenomena and designing coupling configuration. Finally, two simulations are given to demonstrate the effectiveness of the theoretical results.
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This paper investigates synchronization dynamics of a general model of complex delayed networks as well as the effects of time delays. Some simple yet generic criteria ensuring delay-independent and delay-dependent synchronization are derived, which are less conservative than those reported so far in the literature. Moreover, a scale "nu" denoted by a function of the smallest and the second largest eigenvalues of coupling matrix is presented to analyze the effects of time delays on synchronization of the networks. Furthermore, various kinds of coupling schemes, including small-world networks and scale-free networks, are studied. It is shown that, if the coupling delays are less than a positive threshold, then the network will be synchronized. On the other hand, with the increase of the coupling delays, the synchronizability of the network will be restrained and even eventually desynchronized. The results are illustrated by a prototype composed of the chaotic Duffing oscillators. Numerical simulations are also given to verify theoretical results
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Recently, it has been demonstrated that many large complex networks display a scale-free feature, that is, their connectivity distributions are in the power-law form. In this paper, we investigate the synchronization phenomenon in scale-free dynamical networks. We show that if the coupling strength of a scale-free dynamical network is greater than a positive threshold, then the network will synchronize no matter how large it is. We show that the synchronizability of a scale-free dynamical network is robust against random removal of nodes, but is fragile to specific removal of the most highly connected nodes
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Synchronization between two networks with different topology structures and different dynamical behaviours is studied. These two different networks are driving and responding networks, respectively. Under the preconditions that the driving network gets synchronization, we give the conditions for the responding network to be synchronized to the same dynamics as the driving network with the help of the open-plus-closed-loop method. Then a example is given to verify the validity of the theoretical results.
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In this paper, we study the global synchronization of nonlinearly coupled complex delayed dynamical networks with both directed and undirected graphs. Via Lyapunov–Krasovskii stability theory and the network topology, we investigate the global synchronization of such networks. Under the assumption that coupling coefficients are known, a family of delay-independent decentralized nonlinear feedback controllers are designed to globally synchronize the networks. When coupling coefficients are unavailable, an adaptive mechanism is introduced to synthesize a family of delay-independent decentralized adaptive controllers which guarantee the global synchronization of the uncertain networks. Two numerical examples of directed and undirected delayed dynamical network are given, respectively, using the Lorenz system as the nodes of the networks, which demonstrate the effectiveness of proposed results.
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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.
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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.
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Recently, it has been demonstrated that many large-scale complex dynamical networks display a collective synchronization motion. Here, we introduce a time-varying complex dynamical network model and further investigate its synchronization phenomenon. Based on this new complex network model, two network chaos synchronization theorems are proved. We show that the chaos synchronization of a time-varying complex network is determined by means of the inner coupled link matrix, the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix, rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network. Especially, we do not assume that the coupled configuration matrix is symmetric and its off-diagonal elements are nonnegative, which in a way generalizes the related results existing in the literature.
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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.
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This Letter investigates the impulsive synchronization between two complex networks with non-delayed and delayed coupling. Based on the stability analysis of impulsive differential equation, the criteria for the synchronization is derived, and a linear impulsive controller and the simple updated laws are designed. Particularly, the weight configuration matrix is not necessarily symmetric or irreducible, and the inner coupling matrix need not be symmetric. Numerical examples are presented to verify the effectiveness and correctness of the synchronization criteria.
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This Letter investigates the global exponential synchronization in arrays of coupled identical delayed neural networks (DNNs) with constant and delayed coupling. By referring to Lyapunov functional method and Kronecker product technique, some sufficient conditions are derived for global synchronization of such systems. These new synchronization criteria offer some adjustable matrix parameters, which is of important significance in the design and applications of such coupled DNNs, and the results improve and extend the earlier works. Finally, an example is given to illustrate the theoretical results.
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Recently, it has been demonstrated that many large-scale complex dynamical networks display a collective synchronization motion. In this paper, synchronization in nonlinearly coupled dynamical networks is studied. By using the invariance principle of differential equations, some simple linear feedback controllers with dynamical updated strengths are constructed to make the dynamical network synchronize with an isolate node. The feedback strength can be automatically enhanced to make the dynamical network collectively synchronized. The structure of the network can be random, regular, small-world, or scale-free. A numerical example is given to demonstrate the validity of the proposed method, in which the famous Lorenz system is chosen as the node of the network.
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In this paper, the problem of generalized outer synchronization between two completely different complex dynamical networks is investigated. With a nonlinear control scheme, a sufficient criterion for this generalized outer synchronization is derived based on Barbalat's lemma. Two corollaries are also obtained, which contains the situations studied in two lately published papers as special cases. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results.
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Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
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In this article, a new method, which constructs a coupling scheme with cooperative and competitive weight-couplings, is used to stabilize arbitrarily selected cluster synchronization patterns with several clusters for connected chaotic networks. By the coupling scheme, a sufficient condition about the global stability of the selected cluster synchronization patterns is derived. That is to say, when the sufficient condition is satisfied, arbitrarily selected cluster synchronization patterns in connected chaotic networks can be achieved via an appropriate coupled scheme. The effectiveness of the method is illustrated by an example.
We study synchronization and desynchronization of a complex network of chaotic dynamical systems, in both continuous-time and discrete-time cases. With proved synchronization conditions, we illustrate network synchronization and desynchronization processes by a prototype composing of Henon maps in a scale-free network. We show that synchronization and desynchronization of such a complex dynamical network can be determined by the network topology and the maximum Lyapunov exponent of the individual chaotic nodes.
This paper gives sufficient conditions for an array of linearly coupled systems to synchronize. A typical result states that the array will synchronize if the nonzero eigenvalues of the coupling matrix have real parts that are negative enough. In particular, we show that the intuitive idea that strong enough mutual diffusive coupling will synchronize an array of identical cells is true in general. Sufficient conditions for synchronization for several coupling configurations will be considered. For coupling that leaves the array decoupled at the synchronized state, the cells each follow their natural uncoupled dynamics at the synchronized state. We illustrate this with an array of chaotic oscillators. Extensions of these results to general coupling are discussed
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Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now 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 overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern 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. Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com/
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