Synchronization of stochastically hybrid coupled neural networks with coupling discrete and distributed time-varying delays

A general model of linearly stochastically coupled identical connected neural networks with hybrid coupling is proposed, which is composed of constant coupling, coupling discrete time-varying delay and coupling distributed timevarying delay. All the coupling terms are subjected to stochastic disturbances described in terms of Brownian motion, which reflects a more realistic dynamical behaviour of coupled systems in practice. Based on a simple adaptive feedback controller and stochastic stability theory, several sufficient criteria are presented to ensure the synchronization of linearly stochastically coupled complex networks with coupling mixed time-varying delays. Finally, numerical simulations illustrated by scale-free complex networks verify the effectiveness of the proposed controllers.     

Robust adaptive synchronization of uncertain and delayed dynamical complex networks with faulty network

<正>This paper presents a new robust adaptive synchronization method for a class of uncertain dynamical complex networks with network failures and coupling time-varying delays.Adaptive schemes are proposed to adjust controller parameters for the faulty network compensations,as well as to estimate the upper and lower bounds of delayed state errors and perturbations to compensate the effects of delay and perturbation on-line without assuming symmetry or irreducibility of networks.It is shown that,through Lyapunov stability theory,distributed adaptive controllers constructed by the adaptive schemes are successful in ensuring the achievement of asymptotic synchronization of networks in the present of faulty and delayed networks,and perturbation inputs.A Chua's circuit network example is finally given to show the effectiveness of the proposed synchronization criteria.     

Adaptive Synchronization between Two Different Complex Networks with Time-Varying Delay Coupling

A new general network model for two complex networks with time-varying delay coupling is presented. Then we investigate its synchronization phenomena. The two complex networks of the model differ in dynamic nodes, the number of nodes and the coupling connections. By using adaptive controllers, a synchronization criterion is derived. Numerical examples are given to demonstrate the effectiveness of the obtained synchronization criterion. This study may widen the application range of synchronization, such as in chaotic secure communication.     

Generalized Projective Synchronization between Two Complex Networks with Time-Varying Coupling Delay

Generalized projective synchronization （GPS） between two complex networks with time-varying coupling delay is investigated. Based on the Lyapunov stability theory, a nonlinear controller and adaptive updated laws are designed. Feasibility of the proposed scheme is proven in theory. Moreover, two numerical examples are presented, using the energy resource system and Lü＇s system [Physica A 382 （2007） 672] as the nodes of the networks. GPS between two energy resource complex networks with time-varying coupling delay is achieved. This study can widen the application range of the generalized synchronization methods and will be instructive for the demand supply of energy resource in some regions of China.     

具有双重时滞的时变耦合复杂网络的牵制外同步研究

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

Robust adaptive synchronization of general dynamical networks with multiple delays and uncertainties

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.     

Synchronization of networks

We study the synchronization of coupled dynamical systems on networks. The dynamics is governed by a local nonlinear oscillator for each node of the network and interactions connecting different nodes via the links of the network. We consider existence and stability conditions for both single- and multi-cluster synchronization. For networks with time-varying topology we compare the synchronization properties of these networks with the corresponding time-average network. We find that if the different coupling matrices corresponding to the time-varying networks commute with each other then the stability of the synchronized state for both the time-varying and the time-average topologies are approximately the same. On the other hand, for non-commuting coupling matrices the stability of the synchronized state for the time-varying topology is in general better than the time-average topology.      

基于自适应-脉冲控制方法实现时变耦合驱动-响应复杂网络的投影同步

最近,对时变延迟网络的脉冲稳定性的研究大量出现,但通过自适应-脉冲控制方法获得的时变延迟网络同步准则却很少.本文中,运用自适应-脉冲控制方法,设计自适应反馈控制器、自适应律和线性脉冲控制器,研究时变耦合部分线性系统驱动-响应复杂网络的投影同步.获得时变耦合网络的自适应-脉冲投影同步准则.并且不需要网络的耦合构造矩阵是不可约的.另外,运用数值模拟证实方案的有效性和可行性.     

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

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

Synchronization in a class of weighted complex networks with coupling delays

It is commonly accepted that realistic networks can display not only a complex topological structure, but also a heterogeneous distribution of connection weights. In addition, time delay is inevitable because the information spreading through a complex network is characterized by the finite speeds of signal transmission over a distance. Weighted complex networks with coupling delays have been gaining increasing attention in various fields of science and engineering. Some of the topics of most concern in the field of weighted complex networks are finding how the synchronizability depends on various parameters of the network including the coupling strength, weight distribution and delay. On the basis of the theory of asymptotic stability of linear time-delay systems with complex coefficients, the synchronization stability of weighted complex dynamical networks with coupling delays is investigated, and simple criteria are obtained for both delay-independent and delay-dependent stabilities of the synchronization state. Finally, an example is given as an illustration testing the theoretical results.     

Global synchronization in arrays of coupled networks with one single time-varying delay coupling

Global synchronization in arrays of coupled networks with one single time-varying delay coupling is investigated in this Letter. A general linear coupled network with a time-varying coupling delay is proposed and its global synchronization is further discussed. Some sufficient criteria are derived based on Lyapunov functional and linear matrix inequality (LMI). It is shown that under one single delay coupling, the synchronized state changes, which is different from the conventional synchronized solution. Moreover, the degree of the nodes and the inner delayed coupling matrix play key roles in the synchronized state. In particular, the derivative of the time-varying delay can be any given value. Finally, numerical simulations are given to illustrate the theoretical results.     

Periodically intermittent controlling complex dynamical networks with time-varying delays to a desired orbit

This Letter investigates the problem of synchronization in complex dynamical networks with time-varying delays. A periodically intermittent control scheme is proposed to achieve global exponential synchronization for a general complex network with both time-varying delays dynamical nodes and time-varying delays coupling. It is shown that the sates of the general complex network with both time-varying delays dynamical nodes and time-varying delays coupling can globally exponentially synchronize with a desired orbit under the designed intermittent controllers. Moreover, a typical network consisting of the time-delayed Chua oscillator with nearest-neighbor unidirectional time-varying delays coupling is given as an example to verify the effectiveness of the proposed control methodology.     

Exponential synchronization of complex dynamical networks with Markovian jumping parameters using sampled-data and mode-dependent probabilistic time-varying delays

In this paper, the problem of exponential synchronization of complex dynamical networks with Markovian jumping parameters using sampled-data and Mode-dependent probabilistic time-varying coupling delays is investigated. The sam- pling period is assumed to be time-varying and bounded. The information of probability distribution of the time-varying delay is considered and transformed into parameter matrices of the transferred complex dynamical network model. Based on the condition, the design method of the desired sampled data controller is proposed. By constructing a new Lyapunov functional with triple integral terms, delay-distribution-dependent exponential synchronization criteria are derived in the form of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Pinning synchronization of time-varying delay coupled complex networks with time-varying delayed dynamical nodes

This paper deals with the pinning synchronization of nonlinearly coupled complex networks with time-varying coupling delays and time-varying delays in the dynamical nodes.We control a part of the nodes of the complex networks by using adaptive feedback controllers and adjusting the time-varying coupling strengths.Based on the Lyapunov-Krasovskii stability theory for functional differential equations and a linear matrix inequality(LMI),some sufficient conditions for the synchronization are derived.A numerical simulation example is also provided to verify the correctness and the effectiveness of the proposed scheme.     

Adaptive projective synchronization in complex networks with time-varying coupling delay

In this Letter, adaptive projective synchronization (PS) between two complex networks with time-varying coupling delay is investigated by the adaptive control method, and this method has been applied to identify the exact topology of a weighted general complex network. To validate the proposed method, the Lü and Qi systems as the nodes of the networks are detailed analysis, and some numerical results show the effectiveness of the present method.     

Synchronizing distant nodes: a universal classification of networks

Stability of synchronization in delay-coupled networks of identical units generally depends in a complicated way on the coupling topology. We show that for large coupling delays synchronizability relates in a simple way to the spectral properties of the network topology. The master stability function used to determine the stability of synchronous solutions has a universal structure in the limit of large delay: It is rotationally symmetric around the origin and increases monotonically with the radius in the complex plane. This allows a universal classification of networks with respect to their synchronization properties and solves the problem of complete synchronization in networks with strongly delayed coupling.     

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

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

Synchronization criterion for Lur'e type complex dynamical networks with time-varying delay

In this Letter, the synchronization problem for a class of complex dynamical networks in which every identical node is a Lur'e system with time-varying delay is considered. A delay-dependent synchronization criterion is derived for the synchronization of complex dynamical network that represented by Lur'e system with sector restricted nonlinearities. The derived criterion is a sufficient condition for absolute stability of error dynamics between the each nodes and the isolated node. Using a convex representation of the nonlinearity for error dynamics, the stability condition based on the discretized Lyapunov-Krasovskii functional is obtained via LMI formulation. The proposed delay-dependent synchronization criterion is less conservative than the existing ones. The effectiveness of our work is verified through numerical examples.     

Adaptive Synchronization of Fractional-Order Complex-Valued Uncertainty Dynamical Network with Coupling Delay

Compared with real-valued complex networks, complex-valued dynamic networks have a wider application space. In addition, considering the existence of time delay and uncertainty in the actual system, the synchronization problem of fractional-order complex-valued dynamic networks with uncertain parameter and coupled delay is studied in this paper. In particular, the uncertain parameter is correlated with time delay. By using fractional derivative inequalities and linear delay fractional order equations, the synchronization of uncertainty complex networks with coupling delay is realized. Sufficient conditions for global asymptotic synchronization are obtained. The obtained synchronization results are applicable to most complex network systems with or without delay. Finally, numerical simulations verify the effectiveness of the obtained results.

     

Synchronization criteria for a generalized complex delayed dynamical network model

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.