Synchronization analysis of networks with both delayed and non-delayed couplings via adaptive pinning control method

In this paper, simple controllers are designed to realize the synchronization of complex networks with time delays, in which the coupling configuration matrix and inner coupling matrix are not restricted to be symmetric matrix. Several adaptive synchronization criteria are obtained based on Lyapunov stability theory. These criteria relay on the coupling strength and the number of nodes pinning to the networks. For a given complex dynamical network with both delayed and non-delayed couplings, we give the minimum number of controllers under which synchronization can be achieved. One example shows the effectiveness of the proposed pinning adaptive controller.     

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.     

Pinning synchronization with low energy cost

The pinning synchronization on complex networks with low energy cost was investigated. Two basic problems were considered, in this paper. One is how to achieve synchronization on complex network with weak coupling. Another is how to reduce the energy cost in the process of achieving and keeping synchronization. On the other hand, for a good pinning strategy, the synchronizing speed should also be ensured. In this paper, a switching pinning strategy was proposed to enhance the synchronizing speed. And from the point of view of energy saving, some selected links were strengthened. Theoretical analysis and simulations on different complex networks indicate that the proposed pinning strategy is effective. Compare with conventional existing method, the superiority of our method is significant.     

Pinning synchronization of directed networks with disconnected switching topology via averaging method

This paper mainly focuses on pinning synchronization of directed networks under disconnected switching topology. Firstly, by introducing the uniformly directed spanning tree on average (UDSTA) for a time-varying digraph and designing a switching pinning controller, the asymptotic synchronization of complex networks with large number of switching topologies is investigated, in which strong connectivity and structural balance for switching topology are no longer required. Furthermore, a class of almost-period (AP) switching mode is introduced. By means of matrix decomposition, the pinning synchronization criteria of complex networks under AP switching topology are derived and the pinning nodes are selected according to maximal strongly connected root-subgraphs (MSCRSs) of average topology. Finally, a type of coupled neural networks with numerical simulations is introduced to illustrate the theoretical results.     

Cluster mixed synchronization via pinning control and adaptive coupling strength in community networks with nonidentical nodes

In this paper, complex networks with community structure and nonidentical nodes are investigated. The cluster mixed synchronization of these networks is studied by using some linear pinning control schemes. Only the nodes in one community which have direct connections to the nodes in other communities are controlled. Adaptive coupling strength method is adopted to achieve the synchronization as well. According to Lyapunov stability theory, several sufficient conditions for the network to achieve cluster mixed synchronization are derived. Numerical simulations are provided to verify the correctness and the effectiveness of the theoretical results.     

Topology identification and adaptive synchronization of uncertain complex networks with adaptive double scaling functions

This paper discusses topology identification and adaptive synchronization of uncertain complex networks with adaptive the scaling functions. In comparison with those of the existing scaling function synchronization, the scaling function can be identified by adaptive laws in this paper. Moreover, the uncertain network topological structure are identified simultaneously in the process of synchronization. Illustrative examples are presented to demonstrate the application of the theoretical results.     

Hybrid synchronization of heterogeneous chaotic systems on dynamic network

For researching the hybrid synchronization of heterogeneous chaotic systems on the complex dynamic network, there are two important issues to be discussed and analyzed. One is how to build a dynamic complex network which the connection between nodes is dynamic. Another is comparing and analyzing the synchronization characteristics of heterogeneous chaotic systems on the dynamic and static complex network. In this paper, the theoretical analysis and numerical simulation are implemented to study the synchronization on different dynamic and static complex networks. The results indicate it is feasible to realize the hybrid synchronization of heterogeneous chaotic systems on the complex dynamic network.     

ADAPTIVE PINNING SYNCHRONIZATION OF COUPLED NEURAL NETWORKS WITH MIXED DELAYS AND VECTOR-FORM STOCHASTIC PERTURBATIONS＊

In this article,we consider the global chaotic synchronization of general coupled neural networks,in which subsystems have both discrete and distributed delays.Stochastic perturbations between subsyste...     

Robust synchronization for stochastic delayed complex networks with switching topology and unmodeled dynamics via adaptive control approach

In this paper, the robust synchronization problem for a class of stochastic delayed complex networks with switching topology and unmodeled dynamics is investigated by using the adaptive control scheme. Different from some existing uncertain complex network models, the uncertain terms in the complex networks are only needed to be bounded. Based on the stability theory of stochastic delayed differential equation, a novel adaptive controller which ensures that all the nodes in the complex networks robustly synchronize with the target node is derived. Finally, a numerical example is provided to show the effectiveness of the proposed method.     

Robust synchronization of singular complex switched networks with parametric uncertainties and unknown coupling topologies via impulsive control

This paper presents a general model of singular complex switched networks, in which the nodes can be singular dynamic systems and switching behaviors act on both nodes and edges. The parametric uncertainties and unknown coupling topologies are also considered in this model. Two robust synchronization schemes are discussed respectively. In one scheme, the network is synchronized to a homogeneous orbit and in the other one the network is synchronized to a weighted average of all the nodes. Based on the Lyapunov stability theory, different robust synchronization conditions for the two schemes are obtained for this singular complex switched network model via impulsive control. The similarities and differences between these synchronization conditions for the two schemes are discussed. In addition, three useful robust results for the special cases of the singular complex switched networks are presented. Two systematic-design procedures are presented for the two schemes, and three numerical examples are provided for illustrations.     

Cluster synchronization for directed community networks via pinning partial schemes

In this paper, we focus on driving a class of directed networks to achieve cluster synchronization by pinning schemes. The desired cluster synchronization states are no longer decoupled orbits but a set of un-decoupled trajectories. Each community is considered as a whole and the synchronization criteria are derived based on the information of communities. Several pinning schemes including feedback control and adaptive strategy are proposed to select controlled communities by analyzing the information of each community such as indegrees and outdegrees. In all, this paper answers several challenging problems in pinning control of directed community networks: (1) What communities should be chosen as controlled candidates? (2) How many communities are needed to be controlled? (3) How large should the control gains be used in a given community network to achieve cluster synchronization? Finally, an example with numerical simulations is given to demonstrate the effectiveness of the theoretical results.     

Outer synchronization between two hybrid‐coupled delayed dynamical networks via aperiodically adaptive intermittent pinning control

This article is concerned with the problem of pinning outer synchronization between two complex delayed dynamical networks via adaptive intermittent control. At first, a general model of hybrid‐coupled dynamical network with time‐varying internal delay and time‐varying coupling delay is given. Then, an aperiodically adaptive intermittent pinning‐control strategy is introduced to drive two such delayed dynamical networks to achieve outer synchronization. Some sufficient conditions to guarantee global outer‐synchronization are derived by constructing a novel piecewise Lyapunov function and utilizing stability analytical method. Moreover, a simple pinned‐node selection scheme determining what kinds of nodes should be pinned first is provided. It is noted that the adaptive pinning control type is aperiodically intermittent, where both control period and control width are non‐fixed. Finally, a numerical example is given to illustrate the validity of the theoretical results. © 2016 Wiley Periodicals, Inc. Complexity 21: 593–605, 2016     

Finite-time synchronization and identification of complex delayed networks with Markovian jumping parameters and stochastic perturbations

In this paper, the finite-time synchronization and identification for the uncertain system parameters and topological structure of complex delayed networks with Markovian jumping parameters and stochastic perturbations is studied. On the strength of finite time stability theorem and appropriate stochastic Lyapunov–Krasovskii functional under the Itô’s formula, some sufficient conditions are obtained to assurance that the complex delayed networks with Markovian switching dynamic behavior can be identified the uncertain parameters and topological structure matrix in finite time under stochastic perturbations. In addition, three numerical simulations of different situation and dimension are presented to illustrate the effectiveness and feasibility of the theoretical results.     

Robust decentralized adaptive synchronization of general complex networks with coupling delayed and uncertainties

The problem of robust decentralized adaptive synchronization of general complex networks with coupling delayed and uncertainties is investigated in this article. It is only assumed that the upper normal bound of uncertain inner and outer coupling matrices is positive but its concrete structure is not also required to be known. The time‐varying coupling delay is a any nonnegative continuous and bounded function and not require its derivative to be less than one, that is, general time‐varying coupling delays and uncertainties. For such a class of uncertain complex networks, a new synchronization scheme is presented by a class of continuous memoryless robust decentralized adaptive synchronization controllers. It is also shown that the synchronization error dynamics of uncertain complex networks can be guaranteed as uniformly exponentially convergent toward a ball that can be as small as desired. Finally, numerical simulations are provided to demonstrate the effectiveness and robustness of proposed complex networks synchronization schemes. © 2013 Wiley Periodicals, Inc. Complexity 19: 10–26, 2014     

Lag synchronization of complex networks via pinning control

This paper mainly investigates the lag synchronization of complex networks via pinning control. Without assuming the symmetry and irreducibility of the coupling matrix, sufficient conditions of lag synchronization are obtained by adding controllers to a part of nodes. Particularly, the following two questions are solved: (1) How many controllers are needed to pin a coupled complex network to a homogeneous solution? (2) How should we distribute these controllers? Finally, a simple example is provided to demonstrate the effectiveness of the theory.     

Adaptive pinning synchronization of a class of nonlinearly coupled complex networks

This paper is concerned with the pinning control of the robust synchronization of a class of nonlinearly coupled complex networks through adaptive techniques. The effect of perturbed couplings is addressed by adaptive compensation and adjustment methods with controllers and coupling strength designs, respectively. For the pinned nodes, a controller gain function is proposed to compensate the nonlinearities based on adaptive estimations of controller parameters on-line; while for the un-pinned nodes, adaptive adjustment laws are addressed to adjust unknown coupling factors to restrain the unexpected action of the nonlinearly coupled networks. On the basis of Lyapunov stability theory, adaptive pinning controllers and coupling strength adjusters are constructed to ensure that the synchronization errors of the networks can be reduced as small as desired in the presence of the nonlinear couplings. A numerical simulation is provided to illustrate the effectiveness of the theoretical results.     

Finite-time structure identification and synchronization of drive-response systems with uncertain parameter

This paper proposes an approach of finite-time synchronization to identify the topological structure and unknown parameters simultaneously for under general complex dynamical networks. Based on the finite-time stability theory, an effective control input and a feedback control with an updated law are designed to realize finite-time synchronization between two complex networks. The unknown network topological structure and system parameters of uncertain general complex dynamical networks are identified simultaneously. Since finite-time topology identification means the suboptimum in identified time, the results of this paper are important. Several useful criteria for finite-time synchronization are given. Finally, two examples simulations for supporting the theoretical results are also provided.     

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.     

On pinning control of some typical discrete-time dynamical networks

Recently, the pinning control of complex dynamical networks to their homogeneous states has been studied by many researchers, most of the dynamical networks are continuous-time ones, i.e., their dynamical behavior can be described by ODEs. An interesting result is that, for a continuous-time network, its desired (homogeneous) state can be achieved by pinning some nodes with relatively large degrees (also called the specifically pinning scheme [Wang XF, Chen GR. Pinning control of scale-free dynamical networks. Physica A 2002;310:521–31]). Is this specifically pinning scheme also effective for the discrete-time dynamical networks? In this paper, we demonstrate that the pinning control for a discrete-time dynamical network is difficult, and sometimes it is impossible to achieve the desired state just by controlling the nodes with larger degrees. In order to control the discrete-time dynamical networks successfully, we may need to control all the nodes. Finally, we also consider how to extend the interval for the feedback gain d for successful control.     

Projective lag synchronization of the general complex dynamical networks with distinct nodes

In this paper, projective lag synchronization of the general complex dynamical networks with different nodes is investigated. Combining Barbalat’s lemma with adaptive control technique, the adaptive feedback controllers are constructed to achieve projective lag synchronization between the dynamical network with diverse nodes and arbitrary desired trajectory. The presented synchronization method can be applied to any complex networks. It is discovered that the update gains, the time delay, the network size and the network topology have influence on the synchronization effect. Furthermore, projective lag synchronization of the dynamical networks can still be efficiently realized in presence of noise and parameter perturbations. Corresponding numerical simulations are performed to validate the effectiveness and robustness of the proposed synchronization scheme.