Finite-time synchronization of coupled neural networks via discontinuous controllers

This paper investigates finite-time synchronization of an array of coupled neural networks via discontinuous controllers. Based on Lyapunov function method and the discontinuous version of finite-time stability theory, some sufficient criteria for finite-time synchronization are obtained. Furthermore, we propose switched control and adaptive tuning parameter strategies in order to reduce the settling time. In addition, pinning control scheme via a single controller is also studied in this paper. With the hypothesis that the coupling network topology contains a directed spanning tree and each of the strongly connected components is detail-balanced, we prove that finite-time synchronization can be achieved via pinning control. Finally, some illustrative examples are given to show the validity of the theoretical results.

Robust synchronization analysis in nonlinear stochastic cellular networks with time-varying delays, intracellular perturbations and intercellular noise

Naturally, a cellular network consisted of a large amount of interacting cells is complex. These cells have to be synchronized in order to emerge their phenomena for some biological purposes. However, the inherently stochastic intra and intercellular interactions are noisy and delayed from biochemical processes. In this study, a robust synchronization scheme is proposed for a nonlinear stochastic time-delay coupled cellular network (TdCCN) in spite of the time-varying process delay and intracellular parameter perturbations. Furthermore, a nonlinear stochastic noise filtering ability is also investigated for this synchronized TdCCN against stochastic intercellular and environmental disturbances. Since it is very difficult to solve a robust synchronization problem with the Hamilton-Jacobi inequality (HJI) matrix, a linear matrix inequality (LMI) is employed to solve this problem via the help of a global linearization method. Through this robust synchronization analysis, we can gain a more systemic insight into not only the robust synchronizability but also the noise filtering ability of TdCCN under time-varying process delays, intracellular perturbations and intercellular disturbances. The measures of robustness and noise filtering ability of a synchronized TdCCN have potential application to the designs of neuron transmitters, on-time mass production of biochemical molecules, and synthetic biology. Finally, a benchmark of robust synchronization design in Escherichia coli repressilators is given to confirm the effectiveness of the proposed methods.

Topology identification of complex dynamical networks

Recently, some researchers investigated the topology identification for complex networks via LaSalle's invariance principle. The principle cannot be directly applied to time-varying systems since the positive limit sets are generally not invariant. In this paper, we study the topology identification problem for a class of weighted complex networks with time-varying node systems. Adaptive identification laws are proposed to estimate the coupling parameters of the networks with and without communication delays. We prove that the asymptotic identification is ensured by a persistently exciting condition. Numerical simulations are given to demonstrate the effectiveness of the proposed approach.

Cluster synchronization in networks of coupled nonidentical dynamical systems

In this paper, we study cluster synchronization in networks of coupled nonidentical dynamical systems. The vertices in the same cluster have the same dynamics of uncoupled node system but the uncoupled node systems in different clusters are different. We present conditions guaranteeing cluster synchronization and investigate the relation between cluster synchronization and the unweighted graph topology. We indicate that two conditions play key roles for cluster synchronization: the common intercluster coupling condition and the intracluster communication. From the latter one, we interpret the two cluster synchronization schemes by whether the edges of communication paths lie in inter- or intracluster. By this way, we classify clusters according to whether the communications between pairs of vertices in the same cluster still hold if the set of edges inter- or intracluster edges is removed. Also, we propose adaptive feedback algorithms to adapting the weights of the underlying graph, which can synchronize any bi-directed networks satisfying the conditions of common intercluster coupling and intracluster communication. We also give several numerical examples to illustrate the theoretical results.

Globally exponential synchronization and synchronizability for general dynamical networks

The globally exponential synchronization problem for general dynamical networks is considered in this paper. One quantity will be distilled from the coupling matrix to characterize the synchronizability of the corresponding dynamical networks. The calculation of such a quantity is very convenient even for large-scale networks. The network topology is assumed to be directed and weakly connected, which implies that the coupling configuration matrix can be asymmetric, weighted, or reducible. This assumption is more consistent with the realistic network in practice than the constraint of symmetry and irreducibility. By using the Lyapunov functional method and the Kronecker product techniques, some criteria are obtained to guarantee the globally exponential synchronization of general dynamical networks. In addition, numerical examples, including small-world and scale-free networks, are given to demonstrate the theoretical results. It will be shown that our criteria are available for large-scale dynamical networks.

Coordinate transformation and matrix measure approach for synchronization of complex networks

Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.

Generalized synchronization of complex networks

We consider generalized synchronization of complex networks, which are unidirectionally coupled in the drive-response configuration. The drive network consists of linearly and diffusively coupled identical chaotic systems. By choosing suitable driving signals, we can construct the response network to generally synchronize the drive network in a predefined functional relationship. This extends both generalized synchronization of chaotic systems and synchronization inside a network. Theoretical analysis and numerical simulations fully verify our main results.

Investigating the properties of self-organization and synchronization in electronic systems

Nonlinear cooperative behavior appears naturally in many systems, such as cardiac cell oscillations; cellular calcium oscillations; oscillatory chemical reactions, and fireflies. Such systems have been studied in detail due to their inherent properties of robustness, adaptability, scalability, and emergence. In this paper, such nonlinear cooperative behaviors are considered within the domain of electronic system design. We investigate these desirable properties in a system composed of electronic oscillators. The paper presents a series of circuit simulation results showing that self-organizing principles, which can be emulated in an electronic circuit, enable the systems to show a phase transition to synchronization, in a manner similar to those of natural systems. Circuit simulation results presented here show that the circuits are robust to the unreliable performance of the electronic oscillators and tolerant to their run-time faults. These are important findings for future engineering applications in which the system's elements are likely to be unreliable and faulty, such as in molecular- and nanoelectronic systems.

Cluster synchronization in community networks with nonidentical nodes

In this paper dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community are considered. The cluster synchronization of these networks with or without time delay is studied by using some feedback control schemes. Several sufficient conditions for achieving cluster synchronization are obtained analytically and are further verified numerically by some examples with chaotic or nonchaotic nodes. In addition, an essential relation between synchronization dynamics and local dynamics is found by detailed analysis of dynamical networks without delay through the stage detection of cluster synchronization.

Pinning control of fractional-order weighted complex networks

In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The well-studied integer-order complex networks are the special cases of the fractional-order ones. The network model considered can represent both directed and undirected weighted networks. First, based on the eigenvalue analysis and fractional-order stability theory, some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated. A surprising finding is that the fractional-order complex networks can stabilize itself by reducing the fractional-order q without pinning any node. Second, numerical algorithms for fractional-order complex networks are introduced in detail. Finally, numerical simulations in scale-free complex networks are provided to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter beta, the larger overall coupling strength c, the more capacity that the pinning scheme may possess to enhance the control performance of fractional-order complex networks.

Generalized outer synchronization between complex dynamical networks

In this paper, the problem of generalized outer synchronization between two completely different complex dynamical networks is investigated. With a nonlinear control scheme, a sufficient criterion for this generalized outer synchronization is derived based on Barbalat's lemma. Two corollaries are also obtained, which contains the situations studied in two lately published papers as special cases. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results.

Pinning synchronization of delayed dynamical networks via periodically intermittent control

This paper investigates the synchronization problem for a class of complex delayed dynamical networks by pinning periodically intermittent control. Based on a general model of complex delayed dynamical networks, using the Lyapunov stability theory and periodically intermittent control method, some simple criteria are derived for the synchronization of such dynamical networks. Furthermore, a Barabasi-Albert network consisting of coupled delayed Chua oscillators is finally given as an example to verify the effectiveness of the theoretical results.

On the relationship between pinning control effectiveness and graph topology in complex networks of dynamical systems

This paper concerns pinning control in complex networks of dynamical systems, where an external forcing signal is applied to the network in order to align the state of all the systems to the forcing signal. By considering the control signal as the state of a virtual dynamical system, this problem can be studied as a synchronization problem. The main focus of this paper is to study how the effectiveness of pinning control depends on the underlying graph. In particular, we look at the relationship between pinning control effectiveness and the complex network asymptotically as the number of vertices in the network increases. We show that for vertex balanced graphs, if the number of systems receiving pinning control does not grow as fast as the total number of systems, then the strength of the control needed to effect pinning control will be unbounded as the number of vertices grows. Furthermore, in order to achieve pinning control in systems coupled via locally connected graphs, as the number of systems grows, both the pinning control and the coupling among all systems need to increase. Finally, we give evidence to show that applying pinning control to minimize the distances between all systems to the pinned systems can lead to a more effective pinning control.

Synchronization of complex networks through local adaptive coupling

Two local adaptive strategies for the synchronization of complex networks are discussed in this paper. One, termed as vertex-based, uses local adaptive coupling gains at each node in the network. The other, named edge-based, associates to each edge in the network an adaptive coupling gain, determined solely on the basis of local information. The global asymptotic stability of the synchronous evolution is proven for both strategies using appropriate Lyapunov-based techniques. The effectiveness of the adaptive methodologies presented in the paper is shown via two representative examples: adaptive consensus and the adaptive synchronization of a network on N coupled Chua's circuits.

Impulsive control of stochastic systems with applications in chaos control, chaos synchronization, and neural networks

Real systems are often subject to both noise perturbations and impulsive effects. In this paper, we study the stability and stabilization of systems with both noise perturbations and impulsive effects. In other words, we generalize the impulsive control theory from the deterministic case to the stochastic case. The method is based on extending the comparison method to the stochastic case. The method presented in this paper is general and easy to apply. Theoretical results on both stability in the pth mean and stability with disturbance attenuation are derived. To show the effectiveness of the basic theory, we apply it to the impulsive control and synchronization of chaotic systems with noise perturbations, and to the stability of impulsive stochastic neural networks. Several numerical examples are also presented to verify the theoretical results.

Exponential stability of synchronization in asymmetrically coupled dynamical networks

Based on the original definition of the synchronization stability, a general framework is presented for investigating the exponential stability of synchronization in asymmetrically coupled networks. By choosing an appropriate Lyapunov function, we prove that the mechanism of the exponential synchronization stability is the asymmetrical coupling matrix with diffusive condition. We deduce the second largest eigenvalue of a symmetric matrix to govern the exponential stability of synchronization in asymmetrically coupled networks. Moreover, we have given the threshold value which can guarantee that the states of the asymmetrically coupled network achieve the exponential stability of synchronization.

New eigenvalue based approach to synchronization in asymmetrically coupled networks

Locally and globally exponential stability of synchronization in asymmetrically nonlinear coupled networks and linear coupled networks are investigated in this paper, respectively. Some new synchronization stability criteria based on eigenvalues are derived. In these criteria, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of the sum of the column of the asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability results is that they can be analytically applied to the asymmetrically coupled networks and can overcome the complexity of calculating eigenvalues of the coupling asymmetric matrix. Therefore, these conditions are very convenient to use. Moreover, a necessary condition of globally exponential synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of the coupling matrix.

Global synchronization in lattices of coupled chaotic systems

Based on the concept of matrix measures, we study global stability of synchronization in networks. Our results apply to quite general connectivity topology. In addition, a rigorous lower bound on the coupling strength for global synchronization of all oscillators is also obtained. Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized.

Adaptive complete synchronization of chaotic dynamical network with unknown and mismatched parameters

In this paper, an adaptive controller is designed to synchronize the chaotic dynamical network with unknown and mismatched parameters. Based on the invariance principle of differential equations, some generic sufficient conditions for asymptotic synchronization are obtained. In order to demonstrate the effectiveness of the proposed method, an example is provided and numerical simulations are performed. The numerical results show that our control scheme is very effective and robust against the weak noise.

Synchronization in time-varying networks: a matrix measure approach

Synchronization in complex networks has attracted lots of interest in various fields. We consider synchronization in time-varying networks, in which the weights of links are time varying. We propose a useful approach--i.e., the matrix measure approach--to derive some analytically sufficient conditions for synchronization in time-varying networks. These conditions are less conservative than many existing synchronization conditions. Theoretical analysis and numerical simulations of different networks verify our main results.

Adaptive dynamical networks via neighborhood information: synchronization and pinning control

In this paper, we introduce a model of an adaptive dynamical network by integrating the complex network model and adaptive technique. In this model, the adaptive updating laws for each vertex in the network depend only on the state information of its neighborhood, besides itself and external controllers. This suggests that an adaptive technique be added to a complex network without breaking its intrinsic existing network topology. The core of adaptive dynamical networks is to design suitable adaptive updating laws to attain certain aims. Here, we propose two series of adaptive laws to synchronize and pin a complex network, respectively. Based on the Lyapunov function method, we can prove that under several mild conditions, with the adaptive technique, a connected network topology is sufficient to synchronize or stabilize any chaotic dynamics of the uncoupled system. This implies that these adaptive updating laws actually enhance synchronizability and stabilizability, respectively. We find out that even though these adaptive methods can succeed for all networks with connectivity, the underlying network topology can affect the convergent rate and the terminal average coupling and pinning strength. In addition, this influence can be measured by the smallest nonzero eigenvalue of the corresponding Laplacian. Moreover, we provide a detailed study of the influence of the prior parameters in this adaptive laws and present several numerical examples to verify our theoretical results and further discussion.

Better synchronizability predicted by crossed double cycle

In this Brief Report, we propose a network model named crossed double cycles, which are completely symmetrical and can be considered as the extensions of nearest-neighboring lattices. The synchronizability, measured by eigenratio R, can be sharply enhanced by adjusting the only parameter, the crossed length m. The eigenratio R is shown very sensitive to the average distance L, and the smaller average distance will lead to better synchronizability. Furthermore, we find that, in a wide interval, the eigenratio R approximately obeys a power-law form as R approximately L(1.5).

Synchronizing weighted complex networks

Real networks often consist of local units, which interact with each other via asymmetric and heterogeneous connections. In this work, we explore the constructive role played by such a directed and weighted wiring for the synchronization of networks of coupled dynamical systems. The stability condition for the synchronous state is obtained from the spectrum of the respective coupling matrices. In particular, we consider a coupling scheme in which the relative importance of a link depends on the number of shortest paths through it. We illustrate our findings for networks with different topologies: scale free, small world, and random wirings.