No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling
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.
... 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. ...
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. ...
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. ...
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. ...
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. ...
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] . ...
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 ...
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: ...
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. ...
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 , . . . ...
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]. ...
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. ...
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. ...
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. ...
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 sucient 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. ...
The synchronization of moving agent networks with linear coupling in a two dimensional space is investigated. Base on the Lyapunov stability theory, a criterion for the synchronization is achieved via designed decentralized controllers. And, an example of typical moving agent network, having the Rössler system at each node, has been used to demonstrate and verify the design proposed. And, the numerical simulation results show the effectiveness of proposed synchronization approaches.
... 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. ...
The exponential synchronization problem is investigated for a class of moving agent networks in a two-dimensional space and exhibits time-varying topology structure. Based on the Lyapunov stability theory, adaptive feedback controllers are developed to guarantee the exponential synchronization between each agent node. New criteria are proposed for verifying the locally and globally exponential synchronization of moving agent networks under the constraint of fast switching. In addition, a numerical example, including typical moving agent network with the Rössler system at each agent node, is provided to demonstrate the effectiveness and applicability of the proposed design approach.
... 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. ...
In this paper, we study two types of synchronization between two coupled networks with interactions, including inner synchronization inside each network and outer synchronization between two networks with the adaptive controllers. By Barbalat's lemma and linear matrix inequality (LMI), we obtain a sufficient condition for each network to be asymptotic stability in terms of LMI. When the inner synchronization happens inside each network, while outer synchronization does not appear, we design the adaptive controllers to realize the inner and outer synchronization simultaneously, and investigate the identical or nonidentical coupling and interactive matrices. We then obtain a theorem for the outer synchronization with the adaptive controllers. Finally we give some numerical examples to show the effectiveness of our obtained results. Our findings would provide insights into the dynamics of coupled networks with diverse connections.
... 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]. ...
We investigate synchronization between two discrete-time networks with mutual
couplings, including inner synchronization inside each network and outer synchronization between two networks.
We then obtain a synchronized criterion for the inner synchronization inside each network by the
method of linear matrix inequality and derive a relationship between the inner and outer synchronization.
Finally, we show numerical examples to verify our theoretical analysis and discuss the effect of coupling
strengths, node dynamics, and topological structures on the inner and outer synchronization. Compared to
the inner synchronization inside each network, the outer synchronization between two networks is difficult to
achieve.
... 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. ...
This paper exploits the outer synchronization between two dynamic networks with diversely varying node behaviours at the presence of noise. Aiming to obtain a generic technical solution for network outer synchronization, the problem is addressed for the dynamic network scenarios with non-identical topological structures, nonlinear inner coupling connections and time-delayed node characteristics. The impact of noise is fully considered in the theoretical and performance analysis which makes the derived synchronization criteria be able to be adopted in practical engineering deployment with minimized hurdles. By using a Lyapunov–Krasovskii energy function, the synchronization criteria and adaptive control solution are presented and the convergence of the system error states is also proved. Finally, the numerical analysis is carried out based on the simulation study and the result verifies the correctness and effectiveness of the suggested control solution.
This paper formulates and studies a more general model of coupled switched neural networks with impulsive time window. The main feature of impulsive time window is that impulses can exist the stochastic instants of the whole switching interval not the switching instants and a pre-specified instants. Moreover, the impulsive numbers of every subsystems is not the same. Using switching Lyapunov functions and a generalized Halany inequality, some general criteria which characterize the impulses and switching effects in aggregated form, for asymptotically synchronization and exponential synchronization of this general model are established.
Stochastic synchronization for a time-varying complex delayed dynamical networks with heterogeneous nodes is studied in this paper, using adaptive synchronization scheme for detailed analysis and discussion. Based on Lyapunov stability theory, the properties of Weiner process (Brownian motions), some controllers and adaptive laws are designed to ensure achieving stochastic synchronization of the complex delayed dynamical networks model. A sufficient condition is given to ensure that the proposed controlled networks model is globally asymptotically synchronized in mean square sense. A simulation example is presented to show the effectiveness and applicability of the proposed results.
In this paper, the finite-time outer synchronization between two complex dynamical networks with on-off coupling is investigated. By using suitable on-off controllers, sufficient conditions for finite-time outer synchronization are derived based on the Lyapunov function and the finite-time differential inequality method. The theoretical results imply that the two networks will achieve finite-time outer synchronization for fixed on-off rate if the control coupling strength is large enough. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.
In this paper, the finite-time mixed outer synchronization (FMOS) of chaotic neural networks with time-varying delay and unknown parameters is investigated. By adjusting control strengths with the updated laws, we achieve the finite-time mixed outer synchronization between two complex networks based on the finite-time stability theory and linear matrix inequality. Furthermore, the unknown parameters estimation of the networks is identified in a finite time. Finally, numerical simulations are given to demonstrate the effectiveness of the analytical results obtained here.
The exponential synchronization problem of complex networks is investigated. The complex network model considered is a moving agent network in a two dimensional space and the coupling of between nodes is nonlinear links. In order to achieve the objective of exponential synchronizaiton, linearizing error system is presented. Base on the Lyapunov stability theory, a new criterion is proposed for exponetial synchronization of moving agnets networks under the condition of fast switching. In addition, an numerical example of the Lorenz chaotic system are analyzed, and numerical simulations results show the effectiveness of proposed exponential synchronization criterion.
In this paper, an adaptive regulation method for couplings and its physical implementation are presented to deal with the problem of output synchronization of networks. The networks are supposed to suffer from a fault described by network attenuation. For the sake of eliminating the adverse impact of network attenuation, a self-regulating network is introduced by adjusting coupling strength based on adaptive technique. By using the Lyapunov stability theory for a synchronization error system, asymptotic output synchronization of the overall networks can be established for the attenuated couplings even without any control input. Moreover, based on the adaptive regulation strategy, an approach for application of knowledge of electricity is proposed to physically realize the self-regulating networks. Finally, numerical simulations on a R ssler oscillator network are given to illustrate the effectiveness of the derived results.
In the study globally exponetial synchronization problem of complex sensor networks is investigated. The network model considered is a moving agent network in a two dimensional space and exhibits time-varying topology structure. And, each node is assigned nonlinear connections with
those of neighboring nodes. Novel adaptive controllers are designed via the Lyapunov stability theory. Especially, it is shown that the controlled networks are globally exponentially synchronized with a given convergence rate. The designed adaptive controllers for network synchronization are
rather simple. Moreover, an example of complex sensor network which the Lorenz chaotic system as the nodes are analyzed, and numerical simulations results show the effectiveness of proposed synchronization approaches.
In this paper, the synchronization between two nonlinearly coupled complex dynamical networks with uncertain parameters is investigated. Based on Lyapunov stability theory, a simple adaptive controller and the update laws for system parameters of node dynamics are obtained. By means of the presented controllers and corresponding update laws, the synchronization can realize and the uncertain parameters can also be identified. An illustrative example is presented to demonstrate the application of the theoretical results.
In the process of the optimization iteration in high-dimensional space, standard particle swarm optimization algorithm falls into local optimum easily and its convergence speed is also very slow. In order to solve these problems, a self-adaptive particle swarm optimization based on the directed-weighted complex networks is proposed by introducing the complex networks. In our algorithm, we use the small-scope network with directed links to initialize the particle swarm topological structure and introduce the evolution mechanism of directed dynamic networks, which will make the particle topological structure is evolved in scale-free networks when the in-degree is complied with the power-law distribution. Therefore, our proposed algorithm will improve the learning diversity among these particles, and avoid these particles fall into local optimum easily. The simulation experiment demonstrates that the proposed algorithm will improve the premature convergence problem and enhance the algorithm convergence speed.
This paper investigates the projective synchronization between two delayed networks of different sizes with nonidentical nodes and unknown parameters. The two complex networks are of diverse dynamical nodes, different topological structures and different sizes. Both networks contain unknown parameters and coupling delay. Based on stability theory, the bidirectional coupling control condition is proposed to ensure the realization of projective synchronization between the two delayed networks. During synchronization, the unknown parameters are successful estimated by adaptive updated laws. Furthermore, in numerical simulations, the Small-World network and Scale-Free network based on well-known Lorenz system and Chen system are constructed. The results verified the effectiveness of the proposed synchronization method.
This article investigates the outer synchronization for the two discrete‐time complex dynamic networks. First, the more universal mathematical models of the networks with unknown coupling and interaction are proposed. Then, based on the available state of networks and the estimation of summarized unknown parameters shown in the models, the adaptive state feedback controller is synthesized to make the outer synchronization error of two networks converge asymptotically to an arbitrarily small neighborhood at the origin. Finally, the results of simulation show the validity and advantages of the controller.
This paper looks into the outer synchronization issue of two networks with nonlinear, time-delay and different topological characteristics. Given two general dynamic network models, a set of synchronization criteria based on the Lyapunov stability analysis are proposed and the theoretical analysis is presented. In addition, to expand the generality of the proposed control criteria, two synchronization schemes for two linear coupling networks and similar topologies are derived respectively. Unlike the existing solutions obtained from specific studies, the proposed outer synchronization scheme can be applicable for a class of complex networks with nonlinear and time-delay characteristics and the synchronization criteria are not restricted by the symmetry of the network topological structures. This greatly improves the applicability and generality of the controller in practical deployment. The numerical simulation study on the network with dynamic chaotic characteristic are carried out and the result validates and demonstrates the effectiveness of the suggested control mechanism.
The stabilization of stochastic coupled systems with time delay and time-varying coupling structure (SCSTT) via feedback control is investigated. We generalize systems with constant coupling structure to the time-varying coupling structure. Combining the graph theory with the Lyapunov method, a systematic method is provided to construct a Lyapunov function for SCSTT, and a Lyapunov-type theorem and a coefficient-type criterion are obtained to guarantee the stabilization in the sense of pth moment exponential stability. Furthermore, theoretical results are applied to analyze the stabilization of stochastic-coupled oscillators with time delay and time-varying coupling structure in order to illustrate the practicability of the results. Finally, two numerical examples are given to illustrate the effectiveness and feasibility of theoretical results.
This article discusses the synchronization problem of singular neutral complex dynamical networks (SNCDN) with distributed delay and Markovian jump parameters via pinning control. Pinning control strategies are designed to make the singular neutral complex networks synchronized. Some delay-dependent synchronization criteria are derived in the form of linear matrix inequalities based on a modified Lyapunov-Krasovskii functional approach. By applying the Lyapunov stability theory, Jensen’s inequality, Schur complement, and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the stability of the system. Finally, numerical examples are presented to illustrate the effectiveness of the obtained results.
This study examines the problem of impulsive and pinning control synchronization of Markovian jumping master-slave complex dynamical networks with hybrid coupling and additive interval time-varying delays. The linear couplings include both the discrete time-varying delay and the distributed time-varying delays. Two kinds of control schemes are utilized to synchronize the considered dynamical network system. Impulsive and pinning control strategies are designed to synchronize the master-slave complex networks. By applying Lyapunov stability theory, Jensen’s inequality, Schur complement and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the asymptotic stability of this system. We provided numerical examples to illustrate the feasibility and effectiveness of the results obtained.
Some existing papers focused on finite-time parameter identification and synchronization, but provided incomplete theoretical analyses. Such works incorporated conflicting constraints for parameter identification, therefore, the practical significance could not be fully demonstrated. To overcome such limitations, the underlying paper presents new results of parameter identification and synchronization for uncertain complex dynamical networks with impulsive effect and stochastic perturbation based on finite-time stability theory. Novel results of parameter identification and synchronization control criteria are obtained in a finite time by utilizing Lyapunov function and linear matrix inequality respectively. Finally, numerical examples are presented to illustrate the effectiveness of our theoretical results.
We consider the outer synchronization between drive-response systems on networks with time-varying delays, where we focus on the case when the underlying networks are not strongly connected. A hierarchical method is proposed to characterize large-scale networks without strong connectedness. The hierarchical algorithm can be implemented by some programs to overcome the difficulty resulting from the scale of networks. This method allows us to obtain two different kinds of sufficient outer synchronization criteria without the assumption of being strongly connected, by combining the theory of asymptotically autonomous systems with Lyapunov method and Kirchhoff's Matrix Tree Theorem in graph theory. The theory improves some existing results obtained by graph theory. As illustrations, the theoretic results are applied to delayed coupled oscillators and a numerical example is also given.
We investigate a finite-time synchronization problem of hybrid-coupled delayed dynamical network via pinning control. According to linear feedback principle and finite-time control theory, the finite-time synchronization can be achieved by pinning control with suitable continuous finite-time controller. Some sufficient conditions are given for finite-time synchronization of undirected and directed complex network by applying finite-time stability lemma. Numerical simulations are finally presented to demonstrate the effectiveness of the theoretical results.
The stability of coupled systems with time-varying coupling structure (CSTCS) is considered in this paper. The graph-theoretic method on a digraph with constant weight has been successfully generalized into a digraph with time-varying weight. In addition, we construct a global Lyapunov function for CSTCS. By using the graph theory and the Lyapunov method, a Lyapunov-type theorem and some sufficient criteria are obtained. Furthermore, the theoretical conclusions on CSTCS can successfully be applied to the predator-prey model with time-varying dispersal. Finally, a numerical example of CSTCS is given to illustrate the effectiveness and feasibility of our results.
In this paper, we study lag synchronization between two coupled networks and apply two types of control schemes, including the open-plus-closed-loop (OPCL) and adaptive controls. We then design the corresponding control algorithms according to the OPCL and adaptive feedback schemes. With the designed controllers, we obtain two theorems on the lag synchronization based on Lyapunov stability theory and Barbalat's lemma. Finally we provide numerical examples to show the effectiveness of the obtained controllers and see that the adaptive control is stronger than the OPCL control when realizing the lag synchronization between two coupled networks with different coupling structures.
In this article, a synchronization problem for master-slave Markovian switching complex dynamical networks with time-varying delays in nonlinear function via sliding mode control is investigated. On the basis of the appropriate Lyapunov-Krasovskii functional, introducing some free weighting matrices, new synchronization criteria are derived in terms of linear matrix inequalities (LMIs). Then, an integral sliding surface is designed to guarantee synchronization of master-slave Markovian switching complex dynamical networks, and the suitable controller is synthesized to ensure that the trajectory of the closed-loop error system can be driven onto the prescribed sliding mode surface. By using Dynkin's formula, we established the stochastic stablity of master-slave system. Finally, numerical example is provided to demonstrate the effectiveness of the obtained theoretical results.
This study examines the problem of exponential stability of complex dynamical networks with impulse control and semi-Markovian switching parameters. By utilizing a supplementary variable technique and a plant transformation, the semi-Markovian switching complex dynamical networks can be equivalently expressed as its associated Markovian switching complex dynamical networks. By applying the Lyapunov stability theory, Jensen’s inequality, Dynkins formula, Schur complement and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the exponential stability of the equilibrium point. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the results obtained.
This paper addresses the adaptive asymptotic synchronization problem of a class of leader-follower large scale networked systems against communicated signal attenuation and actuator bias faults. Adaptive mechanisms are constructed to estimate each unknown faulty factor of communications and actuators on-line. Based on the estimations, an adaptive compensation controller is designed to automatically remove the unexpected impacts of communication and actuator faults. Through the adaptive adjustment parameters and Lyapunov functions, the achievement of asymptotic synchronization of the leader-follower large-scale networked system with the improper actuator and faulty transmitted signals is obtained. Finally, the synchronization results are validated by a multiple vehicle large-scale device. © 2016 Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag Berlin Heidelberg
In this paper, the problem of linear generalized outer synchronization between two complex dynamical networks with the time-varying coupling delay, different connection topologies and diverse dynamical nodes, is investigated. Based on LaSalle’s invariance principle, some criteria for the linear generalized outer synchronization between two complex dynamical networks are achieved via the adaptive control method. Moreover, the presented results here can also be applied to a great many complex dynamical networks. Some numerical simulations are provided to show the effectiveness and feasibility of the proposed synchronization method.
In this paper, we study hybrid synchronization and parameter identification of uncertain interacted networks and investigate inner synchronization and outer synchronization between these two networks. Based on the linear matrix inequality (LMI), we obtain the synchronous conditions for the inner synchronization in the form of LMIs. When the node dynamical equation has an unknown parameter vector, we design the appropriate controllers to realize the outer synchronization and simultaneously identify the unknown parameter. Finally, we provide two numerical examples to show the efficiency of the obtained theoretical results.
In this article, a general complex dynamical network which contains multiple delays and uncertainties is introduced, which contains time-varying coupling delays, time-varying node delay, and uncertainties of both the inner- and outer-coupling matrices. A robust adaptive synchronization scheme for these general complex networks with multiple delays and uncertainties is established and raised by employing the robust adaptive control principle and the Lyapunov stability theory. We choose some suitable adaptive synchronization controllers to ensure the robust synchronization of this dynamical network. The numerical simulations of the time-delay Lorenz chaotic system as local dynamical node are provided to observe and verify the viability and productivity of the theoretical research in this paper. Compared to the achievement of previous research, the research in this paper seems quite comprehensive and universal.
In this paper, to deal with the problem of modified function projective synchronization (MFPS) between two different networks with delayed couplings and delayed nodes of different dimensions, a hybrid control scheme combining adaptive control and nonlinear control is proposed. First, a more realistic drive-response complex network model is constructed by introducing double delays. Then, we design hybrid feedback controllers to synchronize up the drive and response networks of different dimensions to a scaling function matrix. Based on Lyapunov stability theory and the linear matrix inequality (LMI), we rigorously prove that the MFPS between the proposed drive-response networks can be achieved and meanwhile some sufficient conditions are derived by adopting an appropriate Lyapunov–Krasovskii energy function. Noteably, many existing synchronization settings can be regarded as special cases of the present synchronization framework. Numerical simulation experiments are employed to verify the correctness and the effectiveness of the proposed method.
This paper proposes an approach to identify the topological structure and unknown system parameters of a weighted complex dynamical network with delay and noise perturbation. Based on the Barbalat-type invariance principle for stochastic differential equations, an effective adaptive feedback technique with an updated law is developed to realize generalized outer synchronization. The unknown topological structure and parameters are identified simultaneously through the established technique. The weight configuration matrix was found to be unnecessarily symmetric. Numerical examples are examined to illustrate the effectiveness of the analytical results.
In this paper, a general framework is presented for analyzing the synchronization stability of Linearly Coupled Ordinary Differential Equations
(LCODEs). The uncoupled dynamical behavior at each node is general, and can be chaotic or otherwise; the coupling configuration is also general,
with the coupling matrix not assumed to be symmetric or irreducible. On the basis of geometrical analysis of the synchronization manifold, a new
approach is proposed for investigating the stability of the synchronization manifold of coupled oscillators. In this way, criteria are obtained for
both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling
matrix play key roles in the stability analysis of the synchronization manifold. Furthermore, the roles of the uncoupled dynamical behavior on
each node and the coupling configuration in the synchronization process are also studied.
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.
It has been demonstrated that most complex networks display synchronization phenomena, and this problem has attracted great attention of various fields including science and engineering. In this paper, a generalized complex dynamical network model with time-varying delays was presented. Kronecker product was adopted to investigate this model. Moreover, several synchronization criteria were derived for both delay-independent and delay-dependent asymptotical stability. Especially, it has been shown that synchronization of such dynamical network is determined by the linear matrix inequality consisting of coupling configuration matrices, inner-coupling matrices and isolated cells. At last, illustrative examples were given to validate the above-acquired.
In this paper, a general framework is presented for analyzing the synchronization stability of Linearly Coupled Ordinary Differential Equations (LCODEs). The uncoupled dynamical behavior at each node is general, and can be chaotic or otherwise; the coupling configuration is also general, with the coupling matrix not assumed to be symmetric or irreducible. On the basis of geometrical analysis of the synchronization manifold, a new approach is proposed for investigating the stability of the synchronization manifold of coupled oscillators. In this way, criteria are obtained for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles in the stability analysis of the synchronization manifold. Furthermore, the roles of the uncoupled dynamical behavior on each node and the coupling configuration in the synchronization process are also studied.
We investigate synchronization in a network of continuous-time dynamical systems with small-world connections. The small-world network is obtained by randomly adding a small fraction of connection in an originally nearest-neighbor coupled network. We show that, for any given coupling strength and a sufficiently large number of cells, the small-world dynamical network will synchronize, even if the original nearest-neighbor coupled network cannot achieve synchronization under the same condition.
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.
Systems as diverse as genetic networks or the world wide web are best
described as networks with complex topology. A common property of many large
networks is that the vertex connectivities follow a scale-free power-law
distribution. This feature is found to be a consequence of the two generic
mechanisms that networks expand continuously by the addition of new vertices,
and new vertices attach preferentially to already well connected sites. A model
based on these two ingredients reproduces the observed stationary scale-free
distributions, indicating that the development of large networks is governed by
robust self-organizing phenomena that go beyond the particulars of the
individual systems.
We study synchronization for two unidirectionally coupled networks. This is a substantial generalization of several recent papers investigating synchronization inside a network. We derive analytically a criterion for the synchronization of two networks which have the same (inside) topological connectivity. Then numerical examples are given which fit the theoretical analysis. In addition, numerical calculations for two networks with different topological connections are presented and interesting synchronization and desynchronization alternately appear with increasing value of the coupling strength.
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.
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Full textFull text is available as a scanned copy of the original print version. Get a printable copy (PDF file) of the complete article (364K), or click on a page image below to browse page by page.
363
364
365
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
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/
On the evolution of random graphs, Publ math inst hung acad sci 5
- P Erdos
- A Renyi
Emergence of scaling in random networks
- A L Barbaasi
- R Albert
Barbaasi AL, Albert R. Emergence of scaling in random networks. Science 1999;286:509-12.