Fractional-order Complex Networks Research Articles

Cluster synchronization of fractional-order complex networks via uncertainty and disturbance estimator-based modified repetitive control

The combined problems of cluster synchronization and disturbance rejection for a family of fractional-order complex networks subject to coupling delay, unknown uncertainty and disturbances (UDs) are examined in this study. In particular, the existence of coupling delay is taken into the account with both known and unknown cases. First, a new uncertainty and disturbance estimator (UDE)-based control protocol is formulated for the concerned system to estimate and compensate for the effects of UD. Although the UDE strategy has proven to be a viable tool to deal with slowly changing UDs in control design, the presence of rapidly changing UDs or sinusoidal disturbances is not an effective tool. A well-known modified iterative control (MRC) block is built internally in a closed feedback control loop to solve this problem. After implementing UDE and MRC blocks into the feedback loop, the resulting system becomes almost UD-free. Moreover, a set of sufficient linear matrix inequality constraints are established to ensure the cluster synchronization of the resulting system. Lastly, the benefits, feasibility and robustness of the established UDE-based MRC scheme are confirmed by two illustrative examples.

Cluster synchronization in fractional-order network with nondelay and delay coupling

In this paper, cluster synchronization for fractional-order complex network with nondelay and delay coupling is investigated. Based on the stability theory of fractional-order systems and the properties of fractional derivative, both static and adaptive control schemes are adopted to design effective controllers. Sufficient condition for achieving cluster synchronization about static controllers is provided. From the condition, the needed feedback gains can be estimated by simple calculations. Further, adaptive control scheme is introduced to design unified controllers. Noticeably, in the adaptive controllers, the feedback gains need not be calculated in advance and can adjust themselves to the needed values according to updating laws. Finally, numerical simulations are given to demonstrate the correctness of the obtained results.

Intermittent control for finite-time synchronization of fractional-order complex networks

This paper is concerned with the finite-time synchronization problem for fractional-order complex dynamical networks (FCDNs) with intermittent control. Using the definition of Caputo’s fractional derivative and the properties of Beta function, the Caputo fractional-order derivative of the power function is evaluated. A general fractional-order intermittent differential inequality is obtained with fewer additional constraints. Then, the criteria are established for the finite-time convergence of FCDNs under intermittent feedback control, intermittent adaptive control and intermittent pinning control indicate that the setting time is related to order of FCDNs and initial conditions. Finally, these theoretical results are illustrated by numerical examples.

Cluster Synchronization of Fractional-Order Nonlinearly-Coupling Community Networks With Time-Varying Disturbances and Multiple Delays

This paper investigates the adaptive cluster synchronization of fractional-order complex networks with internal and coupling delays as well as time-varying disturbances via the fractional-order hybrid controllers. To be more practical, the unknown disturbances and dynamical behaviors of nodes are assumed to be nonidentical. Based on the properties of fractional calculus and the fractional-order comparison principle, sufficient conditions are derived to guarantee the cluster synchronization of two kinds of fractional-order nonlinear dynamical systems, which extends the results in existing literatures. Numerical simulations are presented to show the effectiveness of our theoretical results.

Adaptive Synchronization of Fractional-Order Output-Coupling Neural Networks via Quantized Output Control.

This article focuses on the adaptive synchronization for a class of fractional-order coupled neural networks (FCNNs) with output coupling. The model is new for output coupling item in the FCNNs that treat FCNNs with state coupling as its particular case. Novel adaptive output controllers with logarithm quantization are designed to cope with the stability of the fractional-order error systems for the first attempt, which is also an effective way to synchronize fractional-order complex networks. Based on fractional-order Lyapunov functionals and linear matrix inequalities (LMIs) method, sufficient conditions rather than algebraic conditions are built to realize the synchronization of FCNNs with output coupling. A numerical simulation is put forward to substantiate the applicability of our results.

Partial Pinning Control for the Synchronization of Fractional-Order Directed Complex Networks

This paper mainly studies the synchronization problem of fractional-order directed complex networks through partial pinning control. Unlike other papers, the network studied in this paper is neither strongly connected nor contains directed spanning trees. By utilizing the directed acyclic graph condensation and Layering theory, the network is decomposed into several strong connected components and then divided into layers. It proved that all or part of nodes in the network can achieve synchronization with the pinner’s trajectory, only when the root strong connected components in the upstream of these nodes are pinned and satisfy some sufficient conditions. In addition, according to the ControlRank algorithm, an optimized strategy is designed to solve the problem of the optimal selection of the pinning nodes to ensure the specific nodes of the network can achieve synchronization eventually. Meanwhile the amount of control energy cost will also be given in this paper. Finally, two simulation examples are given to verify the reliability and feasibility of the optimized algorithm.

Synchronization of multi-links impulsive fractional-order complex networks via feedback control based on discrete-time state observations

This paper addresses the issue of global Mittag-Leffler synchronization of multi-links impulsive fractional-order complex networks (MIFCNs). A class of feedback control based on discrete-time state observations is first adopted for synchronization of MIFCNs. By combining Lyapunov method with graph-theoretic approach, a synchronization criterion for MIFCNs is derived. In particular, this synchronization criterion is related to the order of fractional derivatives, the parameters of control and the topological structure of networks. Furthermore, as an application of our theoretical results, synchronization of multi-links impulsive fractional-order BAM neural networks is studied in detail and a synchronization criterion is also provided. Finally, a numerical example is presented to illustrate the validity and feasibility of theoretical results.

Finite-time synchronization of delayed fractional-order heterogeneous complex networks

This paper is devoted to exploring the finite-time (FET) synchronization problem of time-varying delay fractional-order (FO) coupled heterogeneous complex networks (TFCHCNs) with external interference via a discontinuous feedback controller. Firstly, we propose a novel Lemma which is useful for discussing the FET stability and synchronization problem of FO systems. Secondly, based on the proposed Lemma, a discontinuous feedback controller is designed to guarantee the FET synchronization of TFCHCNs with external interference. Moreover, the upper bound of settling-time function is obtained. Finally, two simulation examples are provided to verify the practicability of our findings.

Non-fragile robust finite-time synchronization for fractional-order discontinuous complex networks with multi-weights and uncertain couplings under asynchronous switching

This paper focus on the global robust non-fragile finite-time synchronization issue for fractional-order complex networks with multi-weights and uncertain couplings under asynchronous switching topology, where nonlinear dynamic nodes are discontinuous and non-decreasing. Firstly, the non-fragile controller with switching gains is designed. Secondly, under the fractional Filippov differential inclusion framework and the network topologies with no-delayed and delayed couplings, by adopting multiple Lur’e Postnikov-type Lyapunov functional, nonsmooth analysis method and the average dwell time technique, the global robust non-fragile finite-time synchronization conditions are achieved in terms of linear matrix inequalities (LMIs), respectively. Finally, the validity of the theoretical results developed in this paper is verified by two numerical simulations.

Complex projection synchronization of fractional order uncertain complex-valued networks with time-varying coupling

In this paper, the complex projection synchronization problem of fractional complex-valued dynamic networks is investigated. Considering the time-varying coupling and unknown parameters of the fractional order complex network, several decentralized adaptive strategies are designed to adjust the coupling strength and controller feedback gain in order to investigate the complex projection synchronization problem of the system. Moreover, based on the designed identification law, the uncertain parameters in the network can be estimated. Using adaptive law which balances the time-varying coupling strength and the feedback gain of the controller, some sufficient conditions are obtained for the complex projection synchronization of complex networks. Finally, numerical simulation examples are provided to illustrate the efficiency of the complex projection synchronization strategies of the fractional order complex dynamic networks.

Projective synchronization via adaptive pinning control for fractional-order complex network with time-varying coupling strength

This paper presents an adaptive projective pinning control method for fractional-order complex network. First, based on theories of complex network and fractional calculus, some preliminaries of mathematics are given. Then, an analysis is conducted on the adaptive projective pinning control theory for fractional-order complex network. Based on the projective synchronization control method and the combined adaptive pinning feedback control method, suitable projection synchronization scale factor, adaptive feedback controller and the node selection algorithm are designed to illustrate the synchronization for fractional-order hyperchaotic complex network. Simulation results show that all nodes are stabilized to equilibrium point. Theoretical analysis and simulation results demonstrate that the designed adaptive projective pinning controllers are efficient.

Global synchronization in finite time for fractional-order coupling complex dynamical networks with discontinuous dynamic nodes

The global Mittag–Leffler synchronization and global synchronization problem in finite time for fractional-order complex networks which with discontinuous nodes is studied in this paper. When the coupling matrix is time-varying and unknown, under the designed adaptive law with respect to the coupling matrix, by utilizing nonsmooth analysis method and Lyapunov functional approach as well as Laplace transform technique, the conditions of global Mittag–Leffler synchronization are achieved in terms of linear matrix inequalities (LMIs). When the coupling matrix is invariable and known, under the designed feedback controller, and by constructing a Lur’e Postnikov-type Lyapunov functional, the conditions of global synchronization in finite time are established, which are in the form of LMIs, and the upper bound of the settling time is explicitly evaluated. Finally, two examples are provided to illustrate the validity of the proposed design method and theoretical results.

Global synchronization between two fractional-order complex networks with non-delayed and delayed coupling via hybrid impulsive control

This paper focuses on the global synchronization problem for two fractional-order complex networks with non-delayed and delayed coupling (FCNNDC) by using hybrid impulsive control strategy. First, a fractional-order comparison principle is established firstly. Based on our proposed comparison principle and combining algebraic graph theory and laplace transform as well as mathematical induction, some novel synchronization criteria revealing the interplay among the impulsive gain, impulsive interval, networks’ topology structure and fractional order, are derived to guarantee the global synchronization of FCNNDC. Finally, some numerical results are presented to illustrate the effectiveness of the theoretical results.

Out Lag Synchronization of Fractional Order Delayed Complex Networks with Coupling Delay via Pinning Control

This paper discusses the out lag synchronization of fractional order complex networks (FOCN) including both internal delay and coupling delay and with the employment of pinning control scheme. Using comparison theorem and constructing the auxiliary function, several synchronization criterions by linear feedback pinning control are presented. The model and the obtained results in this work are more general than the previous works. Correctness and effectiveness of the theoretical results are validated through numerical simulations.

Synchronization for a Class of Fractional-order Linear Complex Networks via Impulsive Control

Up to now, the research topic about fractional-order complex networks is mainly focused on the synchronization. In this paper, synchronization for a class of fractional-order linear complex networks is realized via impulsive control. The general expression of solution for a fractional-order impulsive error system is deduced by utilizing iteration algorithm. Some inequality conditions are established to guarantee that the largest Lyapunov exponents of each node are negative, which means that the corresponding error system is asymptotic stable and synchronization is realized. It is the first time to achieve the synchronization of fractional-order systems based on the largest Lyapunov exponent. Finally, examples are present to illustrate the validity and effectiveness of proposed conclusions. Numerical simulations also indicate that the fractional-order parameter has a great influence on the largest Lyapunov exponent, although it is not reflected in the theoretical analysis.

Finite-time synchronization of fractional-order complex networks via hybrid feedback control

Abstract This paper addresses the problem with respect to finite-time (FT) synchronization for fractional-order (FO) complex networks. Based on hybrid feedback control technique, Lyapunov approach, and some novel analysis techniques of fractional calculation, some sufficient conditions are obtained to guarantee FT synchronization, and the estimation bound of the setting time for synchronization is given. It is meaningful to find that the setting time is dependent on fractional order, controller parameters and initial value of networks. At last, numerical simulations are presented to illustrate the effectiveness and applicability of our proposed control method and verify our theoretical results.

Adaptive Synchronization of Fractional-Order Nonlinearly Coupled Complex Networks With Time Delay and External Disturbances

This paper considers both local and global synchronizations of fractional-order nonlinearly-coupled complex networks with time delay and unknown external disturbances. Here, neither delayed or non-delayed configuration matrices are necessarily irreducible or symmetric. Combined with the fractional order compare theorem and fractional order stability theory, some novel sufficient conditions are obtained that guarantee the realization of local and global asymptotic synchronization via adaptive control and pinning control. The network model and conclusion in this paper are more practical and general than those in the existing literature. The experimental results show the feasibility of our theoretical analysis.

Synchronization in a fractional-order dynamic network with uncertain parameters using an adaptive control strategy

This paper studies synchronization of all nodes in a fractional-order complex dynamic network. An adaptive control strategy for synchronizing a dynamic network is proposed. Based on the Lyapunov stability theory, this paper shows that tracking errors of all nodes in a fractional-order complex network converge to zero. This simple yet practical scheme can be used in many networks such as small-world networks and scale-free networks. Unlike the existing methods which assume the coupling configuration among the nodes of the network with diffusivity, symmetry, balance, or irreducibility, in this case, these assumptions are unnecessary, and the proposed adaptive strategy is more feasible. Two examples are presented to illustrate effectiveness of the proposed method.

Synchronization of fractional-order linear complex networks with directed coupling topology

The synchronization of fractional-order complex networks with general linear dynamics under directed connected topology is investigated. The synchronization problem is converted to an equivalent simultaneous stability problem of corresponding independent subsystems by use of a pseudo-state transformation technique and real Jordan canonical form of matrix. Sufficient conditions in terms of linear matrix inequalities for synchronization are established according to stability theory of fractional-order differential equations. In a certain range of fractional order, the effects of the fractional order on synchronization is clearly revealed. Conclusions obtained in this paper generalize the existing results. Three numerical examples are provided to illustrate the validity of proposed conclusions.

Adaptive Synchronization of Fractional Order Complex-Variable Dynamical Networks via Pinning Control**Supported by National Natural Science Foundation of China under Grant No. 61201227, National Natural Science Foundation of China Guangdong Joint Fund under Grant No. U1201255, the Natural Science Foundation of Anhui Province under Grant No. 1208085MF93, 211 Innovation Team of Anhui University under Grant Nos. KJTD007A and KJTD001B, and also

In this paper, the synchronization of fractional order complex-variable dynamical networks is studied using an adaptive pinning control strategy based on close center degree. Some effective criteria for global synchronization of fractional order complex-variable dynamical networks are derived based on the Lyapunov stability theory. From the theoretical analysis, one concludes that under appropriate conditions, the complex-variable dynamical networks can realize the global synchronization by using the proper adaptive pinning control method. Meanwhile, we succeed in solving the problem about how much coupling strength should be applied to ensure the synchronization of the fractional order complex networks. Therefore, compared with the existing results, the synchronization method in this paper is more general and convenient. This result extends the synchronization condition of the real-variable dynamical networks to the complex-valued field, which makes our research more practical. Finally, two simulation examples show that the derived theoretical results are valid and the proposed adaptive pinning method is effective.