Dynamic modeling and aperiodically intermittent strategy for adaptive finite-time synchronization control of the multi-weighted complex transportation networks with multiple delays

The idea of network splitting according to time delay and weight is introduced. Based on the cyber physical systems(CPS), a class of multi-weighted complex transportation networks with multiple delays is modeled. The finite-time synchronization of the proposed complex transportation networks model is studied systematically. On the basis of the theory of stability, the technique of adaptive control, aperiodically intermittent control and finite-time control, the aperiodically intermittent adaptive finite-time synchronization controller is designed. The controller designed in this paper is beneficial for understanding the synchronization in multi-weighted complex transportation networks with multiple delays. In addition,the conditions for the existence of finite time synchronization have been discussed in detail. And the specific value of the settling finite time for synchronization is obtained. Moreover, the outer coupling configuration matrices are not required to be irreducible or symmetric. Finally, simulation results of the finite-time synchronization problem are given to illustrate the correctness of the results obtained.     

Synchronization of a network coupled with complex-variable chaotic systems

In this paper, synchronization of a network coupled with complex-variable chaotic systems is investigated. Adaptive feedback control and intermittent control schemes are adopted for achieving adaptive synchronization and exponential synchronization, respectively. Several synchronization criteria are established. In these schemes, the outer coupling matrix is not necessarily assumed to be symmetric or irreducible. Further, for a class of networks with an irreducible and balanced outer coupling matrix, a pinning control scheme is adopted for achieving synchronization. Numerical simulations are demonstrated to verify the effectiveness of the theoretical results.     

Impulsive synchronization of complex networks with non-delayed and delayed coupling

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.     

Finite-Time Generalized Outer Synchronization Between Two Different Complex Networks

This paper investigates the finite-time generalized outer synchronization between two complex dynamical networks with different dynamical behaviors. The two networks can be undirected or directed, and they may also contain isolated nodes and clusters. By using suitable controllers, sufficient conditions for finite-time generalized outer synchronization are derived based on the finite-time stability theory. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. The effect of control parameters on the synchronization time is also numerically demonstrated.     

Finite-Time Generalized Outer Synchronization Between Two Different Complex Networks

This paper investigates the finite-time generalized outer synchronization between two complex dynamical networks with different dynamical behaviors. The two networks can be undirected or directed, and they may also contain isolated nodes and clusters. By using suitable controllers, sufficient conditions for finite-time generalized outer synchronization are derived based on the finite-time stability theory. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. The effect of control parameters on the synchronization time is also numerically demonstrated.     

Outer synchronization between two different fractional-order general complex dynamical networks

Outer synchronization between two different fractional-order general complex dynamical networks is investigated in this paper.Based on the stability theory of the fractional-order system,the sufficient criteria for outer synchronization are derived analytically by applying the nonlinear control and the bidirectional coupling methods.The proposed synchronization method is applicable to almost all kinds of coupled fractional-order general complex dynamical networks.Neither a symmetric nor irreducible coupling configuration matrix is required.In addition,no constraint is imposed on the inner-coupling matrix.Numerical examples are also provided to demonstrate the validity of the presented synchronization scheme.Numeric evidence shows that both the feedback strength k and the fractional order α can be chosen appropriately to adjust the synchronization effect effectively.     

Generalized projective synchronization between two different general complex dynamical networks with delayed coupling

In this Letter, generalized projective synchronization (GPS) between two different complex dynamical networks with delayed coupling is investigated. Two complex networks are distinct if they have diverse node dynamics, or different number of nodes, or different topological structures. By using the adaptive control scheme, a sufficient synchronization criterion for this GPS is derived based on the LaSalle invariance principle. Three corollaries are also obtained. It is noticed that the synchronization speed sensitively depends on the adjustable positive constants μi. Furthermore, the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. In addition, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results.     

Observer-based synchronization in complex dynamical networks with nonsymmetric coupling

Based on a general complex dynamical network model with nonsymmetric coupling, some criteria for synchronization are proposed based on the approach of state observer design. Unlike the nonobserver-based dynamical networks, where the coupling between two connected nodes is defined by an inner coupling matrix and full state coupling is typically needed, in this paper, smaller amount of coupling variables or even only a scalar output signal of each node is needed to synchronize the network. Unlike the commonly researched complex network model, where the coupling between nodes is symmetric, here, in our network model, the coupling configuration matrix is not assumed to be symmetric and may have complex eigenvalues. The matrix Jordan canonical formalization method is used instead of the matrix diagonalization method, so in our synchronization criteria, the coupling configuration matrix is not required to be diagonalizable. Especially, the proposed step-by-step approach is simpler in computation than the existent ones, which usually rely heavily on numerical toolbox, and may be done by hand completely. An example is given to illustrate the step-by-step approach, in which each node is a two-dimensional dynamical limit cycle oscillator system consisting of a two-cell cellular neural network, and numerical simulations are also done to verify the results of design.     

Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling

This Letter investigates the global synchronization of a general complex dynamical network with non-delayed and delayed coupling. Based on Lasalle's invariance principle, adaptive global synchronization criteria is obtained. Analytical result shows that under the designed adaptive controllers, a general complex dynamical network with non-delayed and delayed coupling can globally asymptotically synchronize to a given trajectory. What is more, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the coupling matrix is not assumed to be symmetric or irreducible. Finally, numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria.     

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.     

Stochastic synchronization of coupled neural networks with intermittent control

In this Letter, we study the exponential stochastic synchronization problem for coupled neural networks with stochastic noise perturbations. Based on Lyapunov stability theory, inequality techniques, the properties of Weiner process, and adding different intermittent controllers, several sufficient conditions are obtained to ensure exponential stochastic synchronization of coupled neural networks with or without coupling delays under stochastic perturbations. These stochastic synchronization criteria are expressed in terms of several lower-dimensional linear matrix inequalities (LMIs) and can be easily verified. Moreover, the results of this Letter are applicable to both directed and undirected weighted networks. A numerical example and its simulations are offered to show the effectiveness of our new results.     

Finite-time stochastic combination synchronization of three different chaotic systems and its application in secure communication

In this paper, the finite-time stochastic combination synchronization of three different chaotic systems is investigated. Based on the adaptive technique and the properties of Weiner process, a novel sufficient condition is obtained to ensure combination synchronization under stochastic perturbations. Moreover, a secure communication scheme based on the adaptive combination synchronization of three different systems, i.e., the Lorenz system, Chen system, and Lu? system, with uncertainties, unknown parameters, and stochastic perturbation is presented. The simulation results show the feasibility of the proposed method.     

Fixed-Time Outer Synchronization of Complex Networks with Noise Coupling

In this paper,the fixed-time outer synchronization of complex networks with noise coupling is investigated.Based on the theory of fixed-time stability and matrix inequalities,sufficient conditions for fixed-time outer synchronization are established and the estimation of the upper bound of the setting time is obtained.The result shows that the setting time can be adjusted to a desired value regardless of the initial states.Numerical simulations are performed to verify the effectiveness of the theoretical results.The effects of control parameters and the density of controlled nodes on the converging time are studied.     

Finite-time synchronization of Markovian jumping complex dynamical networks and hybrid couplings

This paper of finite-time synchronization for Markovian jumping complex dynamical frameworks with hybrid couplings is studied. A state feedback control is planned for finite-time synchronization of complex frameworks is presented. Sufficient synchronization criteria are proposed in light of the Lyapunov stability theory. A sensible Lyapunov-Krasovskii functional (LKF) is worked with Kronecker products. The desired state feedback controller can be refined by comprehending a plan of linear matrix inequalities (LMIs). Numerical simulation of complex frameworks demonstrates the comprehensiveness and the ampleness of the proposed method.     

Exponential stability of synchronization in asymmetrically coupled dynamical networks

Based on the original definition of the synchronization stability, a general framework is presented for investigating the exponential stability of synchronization in asymmetrically coupled networks. By choosing an appropriate Lyapunov function, we prove that the mechanism of the exponential synchronization stability is the asymmetrical coupling matrix with diffusive condition. We deduce the second largest eigenvalue of a symmetric matrix to govern the exponential stability of synchronization in asymmetrically coupled networks. Moreover, we have given the threshold value which can guarantee that the states of the asymmetrically coupled network achieve the exponential stability of synchronization.     

Synchronization of a large number of continuous one-dimensional stochastic elements with time-delayed mean-field coupling

We study synchronization as a means of control of collective behavior of an ensemble of coupled stochastic units in which oscillations are induced merely by external noise. For a large number of one-dimensional continuous stochastic elements coupled non-homogeneously through the mean field with delay we developed an approach to find a boundary of synchronization domain and the frequency of the mean-field oscillations on it. Namely, the exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker-Planck equation. Here the synchronization threshold is found by solving this BVP numerically. Approximate analytics is obtained by expanding the solution of the linearized Fokker-Planck equation into a series of eigenfunctions of the stationary Fokker-Planck operator. Bistable systems with a polynomial and piece-wise linear potential are considered as examples. Multistability and hysteresis in the mean-field behavior are observed in the stochastic network at finite noise intensities. In the limit of small noise intensities the critical coupling strength is shown to remain finite, provided that the delay in the coupling function is not infinitely small. Delay in the coupling term can be used as a control parameter that manipulates the location of the synchronization threshold.     

Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay

Many existing papers investigated the geometric features, control and synchronization of complex dynamical networks provided with certain topology. However, the exact topology of a network is sometimes unknown or uncertain. Based on LaSalle’s invariance principle, we propose an adaptive feedback technique to identify the exact topology of a weighted general complex dynamical network model with time-varying coupling delay. By receiving the network nodes evolution, the topology of such a kind of network with identical or different nodes, or even with varying topology can be monitored. In comparison with previous methods, time delay is taken into account in this simple, analytical and systematic synchronization-based technique. Particularly, the weight configuration matrix is not necessarily symmetric or irreducible, and the inner-coupling matrix need not be symmetric. Illustrative simulations are provided to verify the correctness and effectiveness of the proposed scheme.     

Synchronization in complex delayed dynamical networks with nonsymmetric coupling

A new general complex delayed dynamical network model with nonsymmetric coupling is introduced, and then we investigate its synchronization phenomena. Several synchronization criteria for delay-independent and delay-dependent synchronization are provided which generalize some previous results. The matrix Jordan canonical formalization method is used instead of the matrix diagonalization method, so in our synchronization criteria, the coupling configuration matrix of the network does not required to be diagonalizable and may have complex eigenvalues. Especially, we show clearly that the synchronizability of a delayed dynamical network is not always characterized by the second-largest eigenvalue even though all the eigenvalues of the coupling configuration matrix are real. Furthermore, the effects of time-delay on synchronizability of networks with unidirectional coupling are studied under some typical network structures. The results are illustrated by delayed networks in which each node is a two-dimensional limit cycle oscillator system consisting of a two-cell cellular neural network, numerical simulations show that these networks can realize synchronization with smaller time-delay, and will lose synchronization when the time-delay increase larger than a threshold.     

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.     

Bidirectional communication using delay coupled chaotic directly modulated semiconductor lasers

Chaotic synchronization of two directly modulated semiconductor lasers with negative delayed optoelectronic feedback is investigated and this scheme is found to be useful for efficient bidirectional communication between the lasers. A symmetric bidirectional coupling is identified as a suitable method for isochronal synchronization of such lasers. The optimum values of coupling and feedback strength that can provide maximum quality of synchronization are identified. This method is successfully employed for encoding/decoding both analog and digital messages. The importance of a symmetric coupling is demonstrated by studying the variation of decoding efficiency with respect to asymmetric coupling.