Coupling Configuration Matrix Research Articles

Global exponential synchronization of complex networks with reaction diffusions and finite distributed delays coupling

This article investigates global exponential synchronization of complex networks with reaction diffusions and finite distributed delays coupling (RDDCCNs). Some restrictions on the coupling configuration matrix and time-varying delays are removed. By constructing comparison functions, utilizing analytic method and inequality techniques, a criterion for exponential synchronization of RDDCCNs is attained under adaptive intermittent control. A corresponding outcome without RDs is also acquired. At last, two simulations are proposed to illustrate validity of the obtained criteria.

Matrix projective synchronization for a class of discrete-time complex networks with commonality via controlling the crucial node

This article proposes two control strategies to realize matrix projective synchronization (MPS) for a class of discrete-time complex dynamical networks (DTCDNs). It is worth pointing out that our network model is composed of nodes with different state dimensions and its outer coupling configuration matrix can be asymmetric. In addition, since a small percentage of nodes that play more important role than others usually exist in a network, thus, one important node, which is named as crucial node in this paper, is taken into account in our network model. Besides, the commonality between each node and the crucial node is also precisely expressed. Further, based on the crucial node and the commonality and associated with the Lyapunov stability theory, two control strategies are addressed to realize the exponential matrix projective synchronization (EMPS) and asymptotic matrix projective synchronization (AMPS), respectively. It should be stressed that only the crucial node is controlled and only the state of the crucial node is applied to design the EMPS and AMPS controllers. This is promising to reduce the control cost as much as possible and to improve the feasibility of our MPS strategies. The effectiveness and feasibility of our MPS schemes are rigorously proved in theory as well as verified by two numerical examples.

Matrix projective cluster synchronization for arbitrarily coupled networks with different dimensional nodes via nonlinear control

SummaryRecently, many control mechanisms have been put forward to realize the cluster synchronization of networks. However, these results are derived by making very conservative assumptions on the network's nodes and its outer coupling configuration matrix, which greatly limit these results' potential applications. This paper focuses on weakening these restrictive assumptions as much as possible and further proposes nonlinear controllers for our general networks to achieve matrix projective cluster synchronization (MPCS). One outstanding characteristic of our introduced network model is that the state dimensions of the nodes in different clusters are different, which extends the case that the network is composed of identical dimensional nodes. Another distinctive characteristic is that no other restrictions are imposed on our network's outer coupling matrix except for being bounded, namely, both the outer coupling matrix and its block matrices may be time varying and nonzero‐row‐sum. Besides, the exponential MPCS (EMPCS) and asymptotic MPCS (AMPCS) are defined to make a better description of the cluster synchronization phenomena in this paper. It is worth noted that these two definitions extend the existing ones about cluster synchronization. Furthermore, by means of rigorous theoretical analysis, some nonlinear control strategies are proposed for our network to realize EMPCS or AMPCS. At last, a proper numerical example along with two control patterns is presented to demonstrate the validity and effectivity of our results in this paper.

Matrix projective synchronization for time-varying disturbed networks with uncertain nonlinear structures and different dimensional nodes

Abstract This paper investigates the matrix projective synchronization (MPS) problem for time-varying disturbed complex dynamical networks (CDNs) with uncertain nonlinear structures and different dimensional nodes. In order to well describe the practical networks, the unavoidable unknown nonlinear structure and the inevitable uncertain disturbance of each node are considered into our network model, which is different from the previous works. Besides, it is worth pointing out that the outer coupling configuration matrix, which represents the coupling strength and the topological structure, is not restricted by the dissipatively coupled condition in this paper. The definition of MPS is also introduced for the networks with different dimensional nodes. Moreover, several MPS schemes are respectively put forward for our network model according to the norm bound of the uncertain nonlinear structures and disturbances being unknown or not. Finally, two proper examples associated with numerical simulations are given to verify the effectiveness of our theoretical results.

Event-triggered synchronization control for complex networks with actuator saturation

This paper investigates the problem of event-triggered synchronization control for complex networks. Each node is subject to actuator saturation. Based on Lyapunov theory and algebraic graph theory, novel sufficient conditions are derived, under which the synchronization error system is asymptotically stable. Compared with the existing results, the proposed event-riggered synchronization control strategy can be applied to the network with the unknown coupling configuration matrix. A numerical example is presented to demonstrate the effectiveness of the proposed control strategy.

Modified function projective synchronization of complex dynamical networks with mixed time-varying and asymmetric coupling delays via new hybrid pinning adaptive control

This paper investigates modified function projective synchronization (MFPS) for complex dynamical networks with mixed time-varying and hybrid asymmetric coupling delays, which is composed of state coupling, time-varying delay coupling and distributed time-varying delay coupling. In contrast to previous results, the coupling configuration matrix needs not be symmetric or irreducible. The MFPS of delayed complex dynamical networks is considered via either hybrid control or hybrid pinning control with nonlinear and adaptive linear feedback control, which contains error linear term, time-varying delay error linear term and distributed time-varying delay error linear term. Based on Lyapunov stability theory, adaptive control technique, the parameter update law and the technique of dealing with some integral terms, we will show that control may be used to manipulate the scaling functions matrix such that the drive system and response networks could be synchronized up to the desired scaling function matrix. Numerical examples are given to demonstrate the effectiveness of the proposed method. The results in this article generalize and improve the corresponding results of the recent works.

Pinning synchronization between two general fractional complex dynamical networks with external disturbances

In this paper, the pinning synchronization between two fractional complex dynamical networks with nonlinear coupling, time delays and external disturbances is investigated. A Lyapunov-like theorem for the fractional system with time delays is obtained. A class of novel controllers is designed for the pinning synchronization of fractional complex networks with disturbances. By using this technique, fractional calculus theory and linear matrix inequalities, all nodes of the fractional complex networks reach complete synchronization. In the above framework, the coupling-configuration matrix and the innercoupling matrix are not necessarily symmetric. All involved numerical simulations verify the effectiveness of the proposed scheme.

Outer synchronization of fractional-order complex dynamical networks

The outer synchronization between two complex dynamical networks with fractional-order chaotic nodes is studied. A new fractional-order controller for outer synchronization of complex networks is presented. And some new sufficient synchronization criteria are proposed based on the LaSalle invariance principle and the Lyapunov stability theory. This method can apply to arbitrary fractional-order complex networks in which the coupling-configuration matrix and the inner-coupling matrix are not assumed to be symmetric or irreducible. It means that this method is more general and effective. Numerical simulations of three fractional-order complex networks demonstrate the universality and the effectiveness of the proposed method.

Synchronization analysis of linearly coupled delayed neural networks with discontinuous activations

In this study, without assuming the symmetry or irreducibility of the coupling configuration matrix, we investigate the synchronization of linearly coupled dynamical systems with discontinuous identical nodes and time delay. First, we design a feedback controller to realize the complete synchronization and exponential synchronization of discontinuous complex networks using a generalized Lyapunov approach. Second, we propose a set of adaptive controllers and adaptive laws to achieve complete synchronization. Third, we study the adaptive pinning synchronization of the discontinuous complex networks. The results of numerical simulations are presented to demonstrate the effectiveness of the controllers and the correctness of the theoretical results.

Delay-dependent synchronization for non-diffusively coupled time-varying complex dynamical networks

This paper investigates the delay-dependent synchronization schemes for the non-diffusively coupled time-varying complex dynamical networks. The outer coupling configuration matrix in our network model may be non-diffusive, time-varying, uncertain, asymmetric and irreducible. Different time-varying coupling delays for different nodes are also put into consideration in this paper. Besides, the nodes may have different state dimensions. Furthermore, only the common bound of the outer coupling coefficients (CBOCC) is used to design the synchronization controllers. If the CBOCC is known, our delay-dependent synchronization scheme can guarantee the network achieving exponential synchronization. And when the CBOCC is uncertain, the adaptive synchronization scheme, where only one adaptive law is needed, is proposed to guarantee the network realizing asymptotic synchronization. Simulation examples are provided to verify the effectiveness and feasibility of our theoretical results.

Synchronization of fractional order complex dynamical networks

In this letter the synchronization of complex dynamical networks with fractional order chaotic nodes is studied. A fractional order controller for synchronization of complex network is presented. Some new sufficient synchronization criteria are proposed based on the Lyapunov stability theory and the LaSalle invariance principle. These synchronization criteria can apply to an arbitrary fractional order complex network in which the coupling-configuration matrix and the inner-coupling matrix are not assumed to be symmetric or irreducible. It means that this method is more general and effective. Numerical simulations of two fractional order complex networks demonstrate the universality and the effectiveness of the proposed method.

Synchronization for time-varying complex dynamical networks with different-dimensional nodes and non-dissipative coupling

This paper investigates synchronization for time-varying complex dynamical networks with different-dimensional nodes via decentralized control. The outer coupling configuration matrix (OCCM) in our network model, which represents the coupling strength and the topological structure, can be time-varying, non-dissipatively coupled, asymmetric and uncertain. In addition, the nonlinearly coupled inner state functions are admissible in the network model. The paper mainly focuses on the synthesis of decentralized controllers in two cases for uncertain coupling coefficients in OCCM, respectively. Firstly, the decentralized nonlinear state feedback controllers are synthesised based on the known coupling coefficients common bound. The proposed controllers can guarantee exponential synchronization of the networks. Then, an adaptive mechanism with only one parameter being adjusted online is introduced to synthesize the decentralized adaptive controllers according to the unknown coupling coefficients common bound so that the network realizes asymptotic synchronization. Finally, a numerical example is given to test the effectiveness of the theoretical results.

Decomposition approach to exponential synchronisation for a class of non‐linear singularly perturbed complex networks

This study is concerned with the globally exponential synchronisation problem for a class of non-linear singularly perturbed complex networks where all nodes possess the same structure and properties. The addressed global synchronisation problem is converted into the one for two lower-order sub-networks, namely, a non-linear slow sub-network and a linear fast sub-network, which are obtained by using the classical singular perturbation decomposition method. The network topology is directed and weighted, which means that the coupling configuration matrix is allowed to be asymmetric. By using the Lyapunov functional method and the Kronecker product technique, sufficient conditions are obtained under which the synchronisation is achieved, respectively, for the two sub-networks and the original complex network. These conditions can be easily verified by using the semi-definite programming method. A numerical example is finally simulated to validate the theoretical results and the effectiveness of the proposed synchronisation scheme.

Finite-time stabilization and synchronization of complex dynamical networks with nonidentical nodes of different dimensions

A class of complex dynamical networks, in which the nodes have different state dimensions, is investigated in this paper. Since the networks constructed by nonidentical nodes with different state dimensions may exhibit different dynamical behavior, the appropriate control strategies are proposed for the stabilization and synchronization of such complex networks. By employing suitable controllers, sufficient conditions for finite-time stabilization and synchronization are derived based on the finite-time stability theory. It is noticed that the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. Finally, numerical examples are presented to show the effectiveness of the proposed control methods.

Adaptive cluster synchronization of directed complex networks with time delays.

This paper studied the cluster synchronization of directed complex networks with time delays. It is different from undirected networks, the coupling configuration matrix of directed networks cannot be assumed as symmetric or irreducible. In order to achieve cluster synchronization, this paper uses an adaptive controller on each node and an adaptive feedback strategy on the nodes which in-degree is zero. Numerical example is provided to show the effectiveness of main theory. This method is also effective when the number of clusters is unknown. Thus, it can be used in the community recognizing of directed complex networks.

Adaptive pinning cluster synchronization of fractional-order complex dynamical networks

The problem about cluster synchronization of fractional-order CDNs is studied via a pinning adaptive approach in this paper. Based on the stability theory of fractional differential equations, some sufficient criteria for local and global cluster synchronization of fractional-order CDNs are derived. In this paper, the coupling configuration matrix can be asymmetric as well as reducible and the inner coupling matrix can also be asymmetric. Moreover, the number of pinning nodes in each cluster can be evaluated. Especially, when the coupling strength is large enough and the coupling configuration matrix is symmetric, cluster synchronization can be achieved via pinning a single node in each cluster. Finally, some typical examples are given to illustrate the correctness and effectiveness of our results, a surprising finding is that the synchronization performance will become better as the fractional order decreases in this simulation.

Modified Projective Outer Synchronization between Two Fractional Order Complex Networks via Adaptive Control

The modified projective outer synchronization between two different fractional order complex dynamical networks with different node dynamics and different topological structures is investigated in this paper. Based on the stability theorem of fractional order systems, the adaptive controllers and sufficient criteria for achieving the synchronization are given by applying the nonlinear control. The proposed method is also applicable to modified projective outer synchronization between two fractional order complex networks with identical topological structures. In addition, there are fewer restrictions on the coupling configuration matrix and inner-coupling matrix. The corresponding numerical results coincide with theoretical analysis, which also show that feedback strength and fractional order can effectively affect the synchronization performance.

Synchronisation analysis for coupled networks with multiple delays

This paper concerns the synchronisation problems for coupled networks with multiple time-varying delays. By linearising complicated system to some time-delayed subsystems, some novel criteria are given to ensure the robust synchronisation for the complex systems with multiple delays. Moreover, the presented results are also applied to the system with single time delay, and the coupling configuration matrix is not necessarily symmetric. Since the results are expressed in the form of linear matrix inequality , the obtained criteria can be easily checked in practice. Finally, numerical examples are provided to show the effectiveness of the obtained results.

Bounded synchronisation of a time-varying dynamical network with nonidentical nodes

This paper investigates the global bounded synchronisation problem of complex dynamical networks of coupled nonidentical nodes with general time-varying topology through the Lyapunov function and graph theory. Several sufficient conditions in form of scalar inequalities are established so that the global bounded synchronisation of the general dynamical network can be evaluated by the stability of a linear time-varying system and the boundedness of a nonlinear function, both associated with the dynamics of nonidentical nodes. These analytical results are simple yet generic, without assuming the symmetry coupling configuration matrix or calculating their eigenvalues. They can be used to explore synchronisation issues of various complex networks. Numerical simulations show their effectiveness.

Synchronization stability in complex interconnected neural networks with nonsymmetric coupling

Some global stability criteria for arrays of linearly coupled delayed neural networks with nonsymmetric coupling are established on the basis of linear matrix inequality (LMI) method, in which the coupling configuration matrix may be arbitrary matrix with appropriate dimensions. Meanwhile, the relations of coupling matrix to stability and synchronization are discussed in detail, and some analysis methods to the synchronization stability of complex networks are also argued. The difficulty arising from the nonsymmetry of the coupling matrix has been overcome in this work. Numerical simulation is also given to verify the effectiveness of the theoretical results.