The synchronization of general complex dynamical network via pinning control

In this paper, the globally synchronization of the general complex network is investigated. Firstly, we discuss the synchronization problem of the linearly coupled and directed network under the pinning control, and make comparison with the previous work about the undirected network. Sufficient conditions are obtained to guarantee the realization of synchronization. Secondly, the synchronization problem of nonlinearly coupled and undirected network under the pinning control is studied, and a criteria of getting synchronization is given. Furthermore, we introduced the adaptive adjustment of the coupling strength in nonlinearly coupled network. At last, we give simulation examples to verify our theoretical results.     

Chaotic burst synchronization in heterogeneous small-world neuronal network with noise

Chaotic bursting is a fundamental behavior of neurons. In this paper, global and local burst synchronization is studied in a heterogeneous small-world neuronal network of non-identical Hindmarsh-Rose (HR) neurons with noise. It is found that the network can achieve global burst synchronization much more easily than phase synchronization and nearly complete synchronization. Moreover, local burst synchronized clusters have already formed before global burst synchronization happens. We study the effect of the shortcut-adding probability and the heterogeneity coefficient on local and global burst synchronization of the network and find that the introduction of shortcuts facilitates burst synchronization while the heterogeneity has little effect. Moreover, we study the spatiotemporal pattern of the network and find that there is an optimal coupling strength range in which the periodicity of the network is very apparent.     

Pinning adaptive synchronization of complex dynamical network with multi-links

Networks with multi-links are universal in the real world such as communication networks, transport networks, and social networks. It is important for us to investigate the control of complex dynamical network with multi-links. In this paper, both local and global stabilities of dynamical network with multi-links are analyzed by applying adaptive control theory and mathematical tools, and some new criteria are proposed to ensure the pinning synchronization. We find that the number of pinned nodes satisfies an inequality for synchronization. Additionally, we solve the problem of how much the coupling strength we need to achieve network synchronization with one pinned node in the network system with multi-links. Finally, numerical examples are used to illustrate the effectiveness of the proposed method.     

Practical bipartite synchronization via pinning control on a network of nonlinear agents with antagonistic interactions

This paper studies the synchronization phenomena in a network of heterogeneous nonlinear systems over signed graph, which can be considered as the perturbation of a network of homogeneous nonlinear systems. Assume that the signed graph is structurally balanced, the nonlinear system satisfies one-sided Lipschitz condition, and a leader pins a subset of agents. Under some proper initial conditions of the leader, we derive some conditions under which bipartite synchronization error can be kept arbitrary small by choosing a proper pining scheme. This property is called practical pinning bipartite synchronization as an alternative synchronization notion for network of heterogeneous nonlinear systems over signed graph. Finally, we present a numerical example to illustrate the effectiveness of the obtained results.     

Outer synchronization of complex networks with delay via impulse

Synchronization between the driving network and the responding network (outer synchronization) has attracted increasing attention from various fields of science and engineering. In this paper, we address outer synchronization of complex networks with delays. Both the cases of coupling delay and node delay are considered. Employing the impulsive control method which is simple, efficient, low cost, and easy to implement in practical applications, we obtain some sufficient conditions of outer synchronization. It indicates that outer synchronization can be achieved if the maximal impulsive intervals are less than a critical value. Numerical simulations are also given to demonstrate the effectiveness of the proposed impulsive control scheme.     

Effects of time-periodic intercoupling strength on burst synchronization of a clustered neuronal network

In this paper, we investigate the effect of time-periodic intercoupling strength on burst synchronization of a clustered neuronal network. We mainly focus on discussing the effects of amplitude and frequency of the time-periodic intercoupling strength on burst synchronization. We found that by tuning the frequency, burst synchrony of the clustered neuronal network could change from higher synchronized states to low synchronized states, and vice versa. While for the amplitude, we surprisingly found that with increasing of the amplitude, burst synchrony of the clustered neuronal network is not always enhanced. We know that synchronization has close relationship with cognitive activities and brain disorders. Thus, our results could give us some useful insight on the important role of time-dependent couplings in neuronal systems.     

Nonlinear synchronization on connected undirected networks

In this paper, we give sufficient conditions to have a complete synchronization of oscillators in connected undirected networks. The oscillators we are considering are not necessarily identical and the synchronization terms can be nonlinear. Many results in the literature deal with sufficient conditions insuring complete synchronization. This is a difficult problem since such conditions require to take into account the individual dynamics of the oscillators and also the graph topology. In this paper, we extend one of these results, the connection graph stability method, to nonlinear coupling functions and we also give an existence condition of trajectories of the oscillators. The sufficient conditions for synchronization presented in this paper are based on the study of a Lyapunov function and on the use of pseudometrics which enable us to link network dynamics and graph theory. These results are applied to a network of Chua’s oscillators and lead to an explicit condition insuring the complete synchronization of the oscillators.     

Improving synchronous ability between complex networks

We investigate the synchronization ability between complex networks and propose a near optimal connection strategy based on one connection. Numerical simulations on scale-free, small-world and random network are presented to prove the effectiveness of the proposed strategy. Furthermore, we find that the synchronization ability of the networks can be improved more largely by enhancing inter-network coupling strength than by enhancing intra-network coupling strength. We find that there is an upper limit for the synchronization ability of the complex networks, and we analyze the corresponding reason.     

Parameter-dependent synchronization of coupled neurons in cold receptor model

In this paper, synchronization in two coupled neurons with spiking, bursting and chaos firings is investigated as the coupling strength gets increased. Synchronization state can be identified by means of the bifurcation diagram, the correlation coefficient and ISI-distance. It is illustrated that the coupled neurons can exhibit different types of synchronization state when the coupling strength increases. The different synchronization processes appear similar, but their detailed processes are different depending on the parameter values. The synchronization of neuronal network with two different network connectivity patterns is also studied. It is shown that chaotic and high period pattern are more difficult to get complete synchronization than the situation in single spike and low period pattern. It is also demonstrated that the synchronization status of multiple neurons is dependent on the network connectivity patterns. These results may be instructive to understand synchronization in neuronal systems.     

Cluster synchronization and rhythm dynamics in a complex neuronal network with chemical synapses

Cluster synchronization and rhythm dynamics are studied for a complex neuronal network with the small world structure connected by chemical synapses. Cluster synchronization is considered as that in-phase burst synchronization occurs inside each group of the network but diversity may take place among different groups. It is found that both one-cluster and multi-cluster synchronization may exist for chemically excitatory coupled neuronal networks, however, only multi-cluster synchronization can be achieved for chemically inhibitory coupled neuronal networks. The rhythm dynamics of bursting neurons can be described by a quantitative characteristic, the width factor. We also study the effects of coupling schemes, the intrinsic property of neurons and the network topology on the rhythm dynamics of the small world neuronal network. It is shown that the short bursting type is robust with respect to the coupling strength and the coupling scheme. As for the network topology, more links can only change the type of long bursting neurons, and short bursting neurons are also robust to the link numbers.     

Chimera states and synchronization behavior in multilayer memristive neural networks

To study the effect of electromagnetic induction on the spatiotemporal dynamic behavior of neural networks, in this paper, we have mainly studied both the synchronization behavior and the evolution of chimera states (CS) in coupled neural networks. To do this, a multilayer memristive neural network was constructed by selecting the Hindmarsh–Rose neurons as the network nodes, and the effect of electromagnetic induction is introduced by using the cubic flux-controlled memristive model as synapse. For simplicity, the following coupling scheme is adopted: only the coupling connections for the neurons between different layers are considered with memristive synapses, while those neurons in each layer are still bidirectional coupled with the classical electrical synapses. It is found that, by adjusting the coupled strength of electrical synapses and the parameters of memristive synapses, the coexistence behavior of coherent and incoherent states, i.e., the CS, could appear in each layer. It is interesting that the CS are also found in inter-layer memristive synapse network. Furthermore, we have discussed the synchronization behavior in this multilayer memristive neural network, one can find when the whole multilayer network is in a synchronization state, not only the spatiotemporal consistency of the CS in each layer neural networks is observed, but also the memductance of all memristive synapses is completely synchronized. Our results suggest that the electromagnetic induction may play an important role in regulating the dynamic behavior of neural networks, and the introduction of memristive synapse into the biological neural network will provide useful clues for revealing the memory behavior of the neural system in human brain.     

Observer-based synchronization in fractional-order leader–follower complex networks

This paper is devoted to synchronization behavior of complex dynamical networks with a Caputo fractional-order derivative. In particular, we propose a fractional-order leader–follower complex network where the leader is independent, has its own dynamics, and is followed by all the other nodes. Using the state observer approach, the leader and the followers are designed to connect through only a scalar coupling signal instead of the commonly used full state coupling scheme. On the basis of stability theory of the fractional-order differential system, a sufficient condition for global network synchronization is presented, which only relates to the linear part of individual nodes and can be easily solved by the pole-placement design technique. The analytic results are complemented with numerical simulations for a network whose nodes are governed by the fractional-order Chua’s circuit.     

Analyzing projective synchronization on different scaling factors in a drive-response coupling dynamical network with time-varying delays

In this paper, projective synchronization of drive-response coupled dynamical network with delayed system nodes and coupling time-varying delay is investigated via impulsive control, where the scaling factors are different from each other. Different controllers are designed to achieve the projective synchronization: only impulsive control is used when the scaling factors need extra limitation, while an extra controller, that is, a simple linear feedback controller, is added when the scaling factors don??t need extra limitation. Based on the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization of such coupled network are established, and an estimate of the upper bound of impulsive intervals ensuring global exponential synchronization of drive-response coupled dynamical network is also given. Numerical examples on the time-delay Lorenz chaotic systems are presented finally to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Parameter identification and synchronization of spatiotemporal chaos in uncertain complex network

Synchronization and parameter identification of a unidirectional star-network constructed by discrete spatiotemporal chaos systems with unknown parameters are studied. The synchronization principle of the network and design method of parameter recognition law are introduced. The function to be determined in the parameter recognition law and the range of adjusting parameter are obtained based on Lyapunov stability theory. Not only global synchronization of the network is realized, but also the unknown parameters in spatiotemporal chaos systems at the nodes of the network are identified. Discrete laser spatiotemporal chaos model is taken as each node of the network, and simulation results show the effectiveness of the synchronization principle and parameter recognition law.     

Effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network

In this paper, effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network is investigated. On the basis of matrix theory, the threshold values of the coupling strength and the weight parameter for cluster synchronization have been obtained by utilizing the attractiveness criteria of the invariant synchronization manifold. It shows that cluster synchronization bifurcation comes forth, which concept is first put forward. That is to say, via changing the weight parameter and the coupling strength, the purpose of controlling the number of clusters is achieved, which provides a new idea for control the number of clusters in a network. Numerical simulations are given to demonstrate the theoretical results. In addition, the theoretical results and the numerical simulations also show that full synchronization may not be realized even if the network is globally coupled when there are some negative couplings.     

Exponential synchronization for second-order nonlinear systems in complex dynamical networks with time-varying inner coupling via distributed event-triggered transmission strategy

This paper focuses on the exponential synchronization problem of complex dynamical networks (CDNs) with time-varying inner coupling via distributed event-triggered transmission strategy. Information update is driven by properly defined event, which depends on the measurement error. Each node is described as a second-order nonlinear dynamic system and only exchanges information with its neighboring nodes over a directed network. Suppose that the network communication topology contains a directed spanning tree. A sufficient condition for achieving exponential synchronization of second-order nonlinear systems in CDNs with time-varying inner coupling is derived. Detailed theoretical analysis on exponential synchronization is performed by the virtues of algebraic graph theory, distributed event-triggered transmission strategy, matrix inequality and the special Lyapunov stability analysis method. Moreover, the Zeno behavior is excluded as well by the strictly positive sampling intervals based on the upper right-hand Dini derivative. It is noted that the amount of communication among network nodes and network congestion have been significantly reduced so as to avoid the waste of network resources. Finally, a simulation example is given to show the effectiveness of the proposed exponential synchronization criteria.     

Local and global exponential output synchronization of complex delayed dynamical networks

This paper proposes a new complex dynamical network model with output coupling. This model is totally different from some existing complex network models. It is well known that the node state is difficult to be observed or measured, even the node state cannot be observed or measured at all. Moreover, sometimes only part states are needed to make synchronization to come true. For these phenomena, the output synchronization is investigated in this paper. Several criteria on local and global exponential output synchronization are derived for the proposed network model. Finally, numerical simulations are given to illustrate the effectiveness of the derived results.     

Mixed outer synchronization of dynamical networks with nonidentical nodes and output coupling

Outer synchronization between the drive network and the response network has attracted much more attention in various fields of science and engineering. In this paper, mixed outer synchronization between two complex dynamical networks with nonidentical nodes and output coupling is investigated via impulsive hybrid control, that is, an adaptive feedback controller with impulsive control effects. Moreover, both the cases of complex networks without and with coupling delay are considered. According to the stability analysis of the impulsive functional differential equation, several sufficient conditions for the networks to achieve mixed outer synchronization are derived. Numerical examples are presented finally to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

混沌网络的聚类同步方法

由于通讯技术和生物工程等科学的发展,许多的聚类同步现象陆续被发 现和研究. 然而,由于网络的复杂性,有关聚类同步的理论结果极少见报道. 从上述问题出发,对于大小为N=mn的网络, 通过构造带权重的相互作用,得 到了n个聚类的聚类同步,其中每个聚类的振子数目都等于m. 最后, 以耦合Lorenz系统为例,验证了以上结果的正确性.     

Projective synchronization of the small-world delayed network with uncertainty

Spatiotemporal chaotic systems are taken as nodes to constitute the small-world delayed network with uncertainty, and projective synchronization of the network is researched. The control input of the network and the adaptive law of adjustment parameters are designed based on Lyapunov theorem. Concretely, the Burgers systems with spatiotemporal chaotic behavior in physics domain are taken as nodes to constitute the complex network, and the Fisher–Kolmogorov system is taken as the target system. The simulation results show that the synchronization performance of the network is very stable.