Outer Synchronization Research Articles

Generalized Outer Synchronization of Stochastic Neural Networks With Time-Varying Delays

Abstract In this paper, generalized outer synchronization between two different stochastic coupled complex dynamical networks with time-varying delays has been investigated. A novel controller is given and the stochastic invariance principle is applied. A stochastic disturbance which is described in term of a Brownian motion are considered in complex dynamical networks. Moreover, some sufficient conditions are derived to ensure generalized outer synchronization of stochastic neural networks. Surprisingly, it is found that complex networks with different structure can be synchronized.

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

In this paper, mixed outer synchronization between two different complex dynamical networks without and with time-varying delayed couplings is investigated. By pinning only a fraction of the nodes in the response network, some effective mixed outer synchronization criteria are obtained based on the Lyapunov stability theory. As a result, the mixed outer synchronization between two complex networks is achieved. Then, in the view of the generalized time-varying delayed different inequalities, some sufficient conditions for mixed outer exponential synchronization are derived by imposing the impulsive controllers to the nodes of the corresponding response network. Finally, some numerical simulations are presented to demonstrate the effectiveness and feasibility of the results we obtained in this paper.

Outer synchronization of uncertain small-world networks via adaptive sliding mode control

The outer synchronization of irregular coupled complex networks is investigated with nonidentical topological structures. The switching gain is estimated by an adaptive technique, and a sliding mode controller is designed to satisfy the sliding condition. The outer synchronization between two irregular coupled complex networks with different initial conditions is implemented via the designed controllers with the corresponding parameter update laws. The chaos synchronization of two small-world networks consisting of N uncertain identical Lorenz systems is achieved to demonstrate the applications of the proposed approach.

Hybrid synchronization of coupled fractional-order complex networks

This paper investigates the hybrid synchronization problem of two coupled complex networks with fractional-order dynamical nodes. Based on the fractional-order Lyapunov stability theorem, some sufficient conditions are derived to realize the hybrid synchronization of two coupled fractional-order networks. Under suitable condition, two coupled networks can reach the hybrid synchronization, i.e., the outer synchronization between the drive and response networks, and the inner synchronization in each networks. Numerical simulations demonstrate the effectiveness and feasility of the proposed synchronization protocol.

Outer synchronization analysis of two time-varying networks with double delays based on pinning control

It is well known that time delay is universal in complex networks. However, in most existing researches outer synchronization is realized between two networks with time delay by adding controllers to all nodes which may bring great economic costs and increase the difficulties in control in practice. In this paper, in order to deal with the problem of outer synchronization between two time-varying coupling networks with node delay and coupling delay, an adaptive pinning control scheme is proposed. First, a more realistic drive-response complex network model is constructed by introducing double delays and asymmetric coupling configuration matrices. Then, we design an adaptive pinning controller which is easy to implement, and choose an effective pinning strategy to control a crucial part of the nodes in the response network. Based on LaSalle' invariance principle and the linear matrix inequality, we may rigorously prove that the outer synchronization between the proposed drive-response networks can be achieved, and meanwhile some sufficient conditions are derived by adopting an appropriate Lyapunov-Krasovskii energy function. Finally, numerical simulation experiments are employed to verify the correctness and the effectiveness of the proposed method. Results indicate that the drive-response networks with double delays can indeed achieve outer synchronization by pinning control. Moreover, the synchronization is independent of coupling delay. And the remarkable influences of coupling delays on the synchronization speed are also revealed.

Outer Synchronization of Complex Networks with Nondelayed and Time-Varying Delayed Couplings via Pinning Control or Impulsive Control

The outer synchronization problem between two complex networks with nondelayed and time-varying delayed couplings via two different control schemes, namely, pinning control and impulsive control, is considered. Firstly, by applying pinning control to a fraction of the network nodes and using a suitable Lyapunov function, we obtain some new and useful synchronization criteria, which guarantee the outer synchronization between two complex networks. Secondly, impulsive control is added to the nodes of corresponding response network. Based on the generalized inequality about time-varying delayed different equation, the sufficient conditions for outer synchronization are derived. Finally, some examples are presented to demonstrate the effectiveness and feasibility of the results obtained in this paper.

Outer Synchronization of Simple Firefly Discrete Models in Coupled Networks

Synchronization is one of the most important emerging collective behaviors in nature, which results from the interaction in groups of organisms. In this paper, network synchronization of discrete-time dynamical systems is studied. In particular, network synchronization with fireflies oscillators like nodes is achieved by using complex systems theory. Different cases of interest on network synchronization are studied, including for a large number of fireflies oscillators; we consider synchronization in small-world networks and outer synchronization among different coupled networks topologies; for all presented cases, we provide appropriate ranges of values for coupling strength and extensive numerical simulations are included. In addition, for illustrative purposes, we show the effectiveness of network synchronization by means of experimental implementation of coupled nine electronics fireflies in different topologies.

Outer synchronization of nearest-neighbor and small-world chaotic networks

In this work, outer synchronization of coupled networks with nonidentical topological structures is reported. In particular, outer synchronization in nearest-neighbor and small-world networks with chaotic nodes is achieved by using complex systems theory. By means of extensive numerical simulations, we show the advantages in outer synchronization of small-world networks against nearest-neighbor. Different cases of interest are studied for a large number of Chua's oscillators like chaotic nodes, including synchronization without chaotic master network, and synchronization of chaotic networks with master-slave configuration.

Adaptive outer synchronization of delay-coupled nonidentical complex networks in the presence of intrinsic time delay and circumstance noise

This paper studies the outer synchronization of time delay complex networks of nonidentical topological structures in the presence of intrinsic time delay, time-varying coupling delay, and perturbation by noise. Considering one network as the drive network and the other as the response network, the drive-response networks achieve synchronization through a suitably designed adaptive controller. The stochastic LaSalle invariance principle is employed to theoretically prove the almost certain synchronization between two complex networks. The proposed controller is applicable to complex networks that have either intrinsic time delay or coupling delay or both. Two examples are examined to illustrate the proposed synchronization scheme and demonstrate its effectiveness.

Recovering Structures of Complex Dynamical Networks Based on Generalized <newline/>Outer Synchronization

The topological structures of complex networks play a crucial role in determining their evolutionary mechanisms and functional behaviors, and may have significant consequences for many real-world applications. Many researchers focused on the geometric features, collective behaviors and control of complex networks provided with precisely known structures. However, the exact topology of a network is usually unknown or uncertain in practical situations. Therefore, accurate and timely topology identification is of great necessity and importance. This paper presents a novel scheme for topology identification. Specifically, an auxiliary complex network with a very general form is constructed and some adaptive controllers are designed to recover the topology of the considered network upon generalized outer synchronization. Different from previous schemes, the network constructed can be composed of any kind of nodes. If the network with an unknown topology has very complicated node dynamics or a high node dimension, one can construct a response network consisting of nodes with much simpler dynamics, which is very practical for circuit design. The effectiveness of the theoretical findings has been illustrated by three numerical examples. This work provides a convenient approach to recovering network topology, which can facilitate the selection of practical circuits and reduce application costs.

Star-Like Network Synchronization of a New Four-Wing Chaotic System

In this paper, an inner synchronization and outer synchronization of star-like complex network formed by a new four-wing chaotic system as node dynamics is proposed. First, a suitable linear variable separation approach is taken to the nonlinear function of the system. Then, based on the stability theory of the linear system and the nonlinear asymmetric coupling method, a simple synchronization scheme of complex network is presented, which can realize not only the inner synchronization within the same sub-network but also the outer synchronization between different sub-networks only by a switch control scheme on the central node. Besides, we discuss the influence of the coupling intensity factors on the synchronization of complex network and give the corresponding stable range. Finally, two numerical simulations are shown to verify the feasibility and effectiveness of the proposed scheme.

Generation of lag outer synchronization of complex networks with noise coupling

In this paper, we investigate the lag outer synchronization problem of complex dynamical networks with noise coupling. Based on the stability theory of stochastic differential equations, we obtain the sufficient conditions for the lag outer synchronization. The theoretical results show that the lag outer synchronization could be achieved only via white noise-based coupling. Furthermore, we show that noise can also be used to enhance the lag outer synchronization between two networks when two networks have both conventional linear coupling and white noise-based coupling. Finally, numerical simulations are presented to demonstrate the effectiveness of the theoretical results.

Combinatorial synchronization of complex multiple networks with unknown parameters

In this paper, a combinatorial inner synchronization within a sub-network, which consists of four-wing chaotic system with unknown parameters and external disturbances as node dynamics, and a combinatorial outer synchronization between different sub-networks are investigated. Based on the Lyapunov stability theory, LaSalle’s invariance principle, cluster analysis, and pinning control technique, some sufficient conditions, which can ensure not only the combinatorial inner synchronization of the nodes with identical node dynamics in a sub-network, but also the combinatorial outer synchronization of the sink nodes with identical or nonidentical ones between different sub-networks by a suitable switch control scheme, are obtained. By using the pinning control, only the sink node within a sub-network which has direct connections to the sink nodes in other sub-networks needs to be controlled. Finally, some numerical simulations are presented to demonstrate the feasibility and validity of the obtained results by taking the star-like topological structure as an example.

Synchronization of Boolean Networks with Different Update Schemes

In this paper, the synchronizations of Boolean networks with different update schemes (synchronized Boolean networks and asynchronous Boolean networks) are investigated. All nodes in Boolean network are represented in terms of semi-tensor product. First, we give the concept of inner synchronization and observe that all nodes in a Boolean network are synchronized with each other. Second, we investigate the outer synchronization between a driving Boolean network and a corresponding response Boolean network. We provide not only the concept of traditional complete synchronization, but also the anti-synchronization and get the anti-synchronization in simulation. Third, we extend the outer synchronization to asynchronous Boolean network and get the complete synchronization between an asynchronous Boolean network and a response Boolean network. Consequently, theorems for synchronization of Boolean networks and asynchronous Boolean networks are derived. Examples are provided to show the correctness of our theorems.

Second-order terminal sliding mode control for networks synchronization

This paper proposes a second-order terminal sliding mode controller for outer synchronization of a class of complex networks with disturbances. This control scheme adopts a hierarchical control structure to reduce the numbers of controllers. The multivariable error systems were decomposed into two subsystems, a controllable nonlinear subsystem and a linear subsystem. The proposed terminal sliding mode controller was only designed for the nonlinear subsystem. The linear subsystem is set to appropriate sliding mode parameters to implement synchronization. This approach possesses the features of less controllers, faster convergence and higher tracking precision. The performance of the control strategy is evaluated through the control of the complex networks consisting of \(N\) identical Rossler systems. Simulation results demonstrate the effectiveness of the proposed control method.

Inner and outer synchronization between two coupled networks with interactions

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.

Time-controllable combinatorial inner synchronization and outer synchronization of anti-star networks and its application in secure communication

In this paper, a time-controllable combinatorial inner synchronization and outer synchronization of anti-star networks, each of which consists of four-wing hyper-chaotic system as node dynamics, is investigated. Based on the adaptive technique and the stability of Lyapunov function, some sufficient conditions, which can ensure the realization of not only combinatorial inner synchronization within an anti-star network with unknown parameters and external perturbations in the computable time, but also combinatorial outer synchronization between different sub-networks with external perturbations in the computable time, are obtained. Moreover, a simple secure communication scheme, which is based on the adaptive combinatorial outer synchronization between different sub-networks under the influence of stochastic noise and time-delay, is presented. Numerical simulation results show the feasibility and validity of the proposed method.

Adaptive Impulsive Outer Synchronization Between Drive-Response Dynamical Networks

In this paper, outer synchronization between drive-response dynamical networks is investigated. Impulsive control combining with adaptive strategy is adopted to design controllers for achieving the goal. Based on the Lyapunov function method and mathematical analysis technique, a synchronization criterion with respect to the impulsive gains and intervals is analytically derived. From the criterion, the impulsive gains can adjust themselves to proper values when the impulsive intervals and some constants are fixed, and vice versa. Finally, two numerical examples are provided to verify the effectiveness of the derived result.

Outer synchronization of networks with different node dynamics

A new type of outer synchronization between two distinct networks, composed of different classes of oscillators is investigated with the help of open plus closed loop approach, proposed earlier by Jackson and Grosu. It is further assumed that all the members of the network differ in their parameter values. Asymptotic stability of the zero solution of the error equation is proved analytically. Numerical simulation reveals that the same type of members of the two networks gets synchronized.

Generalized outer synchronization between two uncertain dynamical networks

This paper investigates generalized outer synchronization between two uncertain dynamical networks with a novel feature that the couplings of each network are unknown functions. With nonlinear control schemes, two sufficient criteria for generalized outer synchronization with or without time delay are obtained by Lyapunov stability theory and Barbalat’s lemma. Our results are valid for many studies of the couplings inside each network being linear or nonlinear. Finally, numerical simulations are given to verify the effectiveness of the control schemes.