Chaos synchronization in general complex dynamical networks with coupling delays

On the basis of Lyapunov stability theory, chaos synchronization of a general complex dynamical network with coupling delays is investigated. Some delay-independent and delay-dependent criteria for exponential synchronization are derived via adopting the free weighting matrix approach; these are less conservative than those previously reported. As an example, the upper bound of the coupling delay for a Duffing system is obtained, and is larger than those reported previously. Finally, some simulation results obtained with different outer-coupling matrices are given to demonstrate the effectiveness of the results that we obtained, and these are compared with existing conclusions to show the advantage of our results.     

Impulsive synchronization seeking in general complex delayed dynamical networks

The present paper investigates the issues of impulsive synchronization seeking in general complex delayed dynamical networks with nonsymmetrical coupling. By establishing the extended Halanay differential inequality on impulsive delayed dynamical systems, some simple yet generic sufficient conditions for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. Compared with some existing works, the distinctive features of these sufficient conditions indicate two aspects: on the one hand, these sufficient conditions can provide an effective impulsive control scheme to synchronize an arbitrary given delayed dynamical network to a desired synchronization state even if the original given network may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as a weighted average of all the states in the network for the purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchronization seeking processes of the dynamical networks. It is shown that impulses play an important role in making the delayed dynamical networks globally exponentially synchronized. Furthermore, the results are applied to a typical nearest-neighbor unidirectional time-delay coupled networks composed of chaotic FHN neuron oscillators, and numerical simulations are given to demonstrate the effectiveness of the proposed control methodology.     

Impulsive synchronization of general continuous and discrete-time complex dynamical networks

This paper mainly investigates the impulsive synchronization of a general complex continuous and discrete-time dynamical network. Firstly, for the continuous complex networks, we give a sufficient condition to guarantee its synchronization. When the sufficient condition is not satisfied, the impulsive controllers are utilized, and some novel criteria are derived to guarantee the network synchronization in this case. What is more significant is that the similar work is extended to the discrete-time networks model. Finally, the results are, respectively, illustrated by a continuous network composed with the chaotic Chen oscillators and a discrete-time network consisting of Hénon map. All numerical simulations verify the effectiveness of the theoretical analysis.     

Local and global synchronization in general complex dynamical networks with delay coupling

Local and global synchronization of complex dynamical networks are studied in this paper. Some simple yet generic criteria ensuring delay-independent and delay-dependent synchronization are derived in terms of linear matrix inequalities (LMIs), which can be verified easily via interior-point algorithm. The assumption that the coupling configuration matrix is symmetric and irreducible, which is frequently used in other literatures, is removed. A network with a fixed delay and a special coupling scheme is given as an example to illustrate the theoretical results and the effectiveness of the proposed synchronization scheme.     

Function projective synchronization in drive-response dynamical networks with non-identical nodes

This paper investigates the problem of function projective synchronization (FPS) in drive-response dynamical networks (DRDNs) with non-identical nodes. Based on the adaptive open-plus-closed-loop (AOPCL) method, a general method of function projective synchronization is derived, which is robust to limited accuracy of data and effects of noise. Corresponding numerical simulations on the Lorenz system are performed to verify and illustrate the analytical results.     

Adaptive lag synchronization for uncertain complex dynamical network with delayed coupling

This paper proposes an adaptive control method to achieve the lag synchronization between uncertain complex dynamical network having delayed coupling and a non-identical reference node. Unknown parameters of both the network and reference node are estimated by adaptive laws obtained by Lyapunov stability theory. With the estimated parameters, the proposed method guarantees the globally asymptotical synchronization of the network in spite of unknown bounded disturbances. The effectiveness of our work is verified through a numerical example and simulation.     

Projective lag synchronization in chaotic systems

In this paper, a new projective lag synchronization is proposed, where a driven chaotic system synchronizes the past state of the driver up to a scaling factor α. An active control method is employed to design a controller to achieve the global synchronization of two identical chaotic systems. Based on Lyapunov stability theorem, a sufficient condition is then given for the asymptotical stability of the null solution of an error dynamics. The effectiveness of the proposed schemes is verified via numerical simulations.     

Observer-based lag synchronization between two different complex networks

In this paper, some new criteria for lag synchronization between two or more complex networks are proposed based on the theory of state observer. Some adaptive controllers are designed to make the drive and response systems achieve lag synchronization, no matter whether the nodes in the two systems are with the same dynamical character or the coupling configuration matrices are nonidentical. In addition, based on the output coupling, the amount of coupling variables between two connected nodes is flexible, which can save a lot of channel resources, simplify the network topology and has more significant meanings in engineering applications. At last, the effects of the lag synchronization criteria are verified through some simulation experiments.     

Synchronization in complex networks with distinct chaotic nodes

We present an approach to the chaos synchronization of complex networks with distinct nodes. The chaotic synchronization is achieved by adding a derivative coupling term in the network equation. We assume that node in networks are different and are given by the Lorenz, Rössler, Chen and Sprott chaotic systems. The derivative term is capable to induce the synchronous behavior in the network. Moreover such a coupling leads the global behavior to a chaotic attractor. We found that without derivative coupling the network is leaded only to an equilibrium point or a limit cycle. Numerical simulations are provided to illustrate the result. Complementary the network synchrony can be chaotic in presence of the derivative coupling.     

Robust synchronization of fractional-order complex dynamical networks with parametric uncertainties

The study investigates robust synchronization of fractional-order complex dynamical networks with parametric uncertainties. Based on the properties of the kronecker product and the stability of the fractional-order system, the robust synchronization criteria are derived by applying the nonlinear control. These criteria are in the form of linear matrix inequalities which can be readily solved by applying the LMI toolbox. The coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix needs not to be symmetric, diagonal or positive definite. Two numerical examples are provided to demonstrate the validity of the presented synchronization scheme.     

Finite-time synchronization control of complex dynamical networks with time delay

In this paper, the finite-time synchronization between two complex networks with non-delayed and delayed coupling is proposed by using the impulsive control and the periodically intermittent control. Some novel and useful finite-time synchronization criteria are derived based on finite-time stability theory. Especially, the traditional synchronization criteria are improved by using the impulsive control and the periodically intermittent control in the convergence time, the results of this paper are important. Finally, numerical examples are given to verify the effectiveness and correctness of the synchronization criteria.     

Robust global synchronization of complex networks with neutral-type delayed nodes

In this paper, the problems of robust global exponential synchronization for a class of complex networks with time-varying delayed couplings are considered. Each node in the network is composed of a class of delayed neural networks described by a nonlinear delay differential equation of neutral-type. Since model errors commonly exist in practical applications, the parameter uncertainties are involved in the considered model. Sufficient conditions that ensure the complex networks to be robustly globally exponentially synchronized are obtained by using the Lyapunov functional method and some properties of Kronecker product. An illustrative example is presented to show the effectiveness of the proposed approach.     

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes

Generalized function projective (lag, anticipated and complete) synchronization between two different complex networks with nonidentical nodes is investigated in this paper. Based on Barbalat’s lemma, some sufficient synchronization criteria are derived by applying the nonlinear feedback control. Although previous work studied function projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In our work, the dynamics of the nodes of the complex networks are any chaotic systems without the limitation of the partial linearity. In addition, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. The proposed strategy is applicable to almost all kinds of complex networks. Numerical simulations further verify the effectiveness and feasibility of the proposed synchronization method. Numeric evidence shows that the synchronization rate is sensitively influenced by the feedback strength, the time delay, the network size and the network topological structure.     

Adaptive synchronization of drive-response fractional-order complex dynamical networks with uncertain parameters

This paper investigates the adaptive synchronization in the drive-response fractional-order dynamical networks with uncertain parameters. By means of both the stability theory of fractional-order differential system and the adaptive control technique, a novel adaptive synchronization controller is developed with a more general and simpler analytical expression, which does not contain the parameters of the complex network, and effective adaptive laws of parameters. Furthermore, the very strong and conservative uniformly Lipschitz condition on the node dynamics of complex network is released. To demonstrate the validity of the proposed method, the examples for the synchronization of systems with the chaotic and hyper-chaotic node dynamics are presented.     

Generalized outer synchronization between non-dissipatively coupled complex networks with different-dimensional nodes

This paper investigates the generalized outer synchronization (GOS) between two non-dissipatively coupled complex dynamical networks (CDNs) with different time-varying coupling delays. Our drive-response networks also possess nonlinear inner coupling functions and time-varying outer coupling configuration matrices. Besides, in our network models, the nodes in the same network are nonidentical and the nodes in different networks have different state dimensions. Asymptotic generalized outer synchronization (AGOS) and exponential generalized outer synchronization (EGOS) are defined for our CDNs. Our main objective in this paper is to design AGOS and EGOS controllers for our drive-response networks via the open-plus-closed-loop control technique. Distinguished from most existing literatures, it is the partial intrinsic dynamics of each node in response network that is restricted by the QUAD condition, which is easy to be satisfied. Representative simulation examples are given to verify the effectiveness and feasibility of our GOS theoretical results in this paper.     

Pinning impulsive synchronization of complex dynamical networks with various time-varying delay sizes

This paper studies the pinning impulsive synchronization problem for a class of complex dynamical networks with time-varying delay. By applying the Lyapunov stability theory and mathematical analysis technique, sufficient verifiable criterion for the synchronization of delayed complex dynamical networks with small delay is derived analytically. It is shown that synchronization can be achieved by only impulsively controlling a small fraction of network nodes. Moreover, a novel sufficient condition is constructed to relax the restrictions on the size of time-delay and guarantee the synchronization of concerned networks with large delay. Two numerical examples are presented to illustrate the effectiveness of the obtained results.     

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

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.     

Exponential synchronization of the complex dynamical networks with a coupling delay and impulsive effects

Based on the stability analysis of the impulsive functional differential equation, the exponential synchronization of the complex dynamical network with a coupling delay and impulses is investigated in the paper. The criteria for the exponential synchronization are derived by the geometrical decomposition of network states and linear matrix inequality method. Two examples are given to show the effectiveness of the proposed criteria.     

Adaptive synchronization of dynamical networks via states of several nodes as target orbit

In this paper, based on the invariance principle of differential equation, a simple adaptive control method is proposed to synchronize the dynamical networks with the general coupling functions. Comparing with other adaptive control methods, the weighted average of a few nodes’ states is used as target orbit to design controller. To show the effectiveness of proposed method, some numerical simulations are performed.     

Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation

In this paper, the generalized outer synchronization between two different delay-coupled complex dynamical networks with noise perturbation is investigated. With a nonlinear control scheme, the sufficient condition for almost sure generalized outer synchronization is developed based on the LaSalle-type invariance principle for stochastic differential equations. Numerical examples are examined to illustrate the effectiveness of the analytical results. The theoretic result is also applied to investigate the outer synchronization between two delay-coupled Hindmarsh–Rose neuronal networks with noise perturbation.