Pinning synchronization of weighted complex networks with variable delays and adaptive coupling weights

In this paper, the synchronization for time-delayed complex networks with adaptive coupling weights is studied. A pinning strategy and a local adaptive scheme to determine coupling weights and feedback gains are proposed. It is noted that our control strategies only rely on some local information other than the global information of the whole network. Finally, the developed techniques are applied to two complex networks which are respectively synchronized to an unstable equilibrium point and a chaotic attractor.     

On pinning synchronization of general coupled networks

The synchronization in general coupled networks subjected to pinning control is investigated. Some generic stability criteria based on the Lyapunov approach are derived for such general controlled networks, which guarantee that the whole network can be pinned to a synchronization state by placing feedback control on only a small fraction of nodes. A real network of television audience flows across 28 satellite channels in China and a representative BA scale-free network composed of chaotic systems are shown, respectively, for illustration and verification. It is found that pinning stability can be improved via increasing pinning density and/or pinning strength for complete diagonal inner coupling.     

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.     

图论与复杂网络

近10年来迅猛发展起来的复杂网络理论为研究复杂性与复杂系统科学提供了一个重要支撑点,它高度概括了复杂系统的重要特征,无论是在理论还是在应用方面都具有很强的生命力,而且在各个方面都得到了很大发展.重点讨论图论在复杂网络中的应用,特别是代数图论在复杂网络同步问题中的应用.首先给出一些图的最小非零与最大特征值以及同步能力的估计,并且讨论了子图与图特征向量在同步能力估计中的作用.其次以两个简单图指出同步能力与网络结构参数的关系复杂,并给出补图与加边对同步研究的意义,然后给出图运算在复杂网络同步中的作用.最后从图论与控制理论角度展望了复杂网络领域未来可能的发展方向.      

Complex projective synchronization in drive-response networks coupled with complex-variable chaotic systems

In drive-response complex-variable systems, projective synchronization with respect to a real number, real matrix, or even real function means that drive-response systems evolve simultaneously along the same or inverse direction in a complex plane. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, this paper investigates projective synchronization in drive-response networks of coupled complex-variable chaotic systems with respect to complex numbers, called complex projective synchronization (CPS). The adaptive feedback control method is adopted first to achieve CPS in a general drive-response network. For a special class of drive-response networks, the CPS is achieved via pinning control. Furthermore, a universal pinning control scheme is proposed via the adaptive coupling strength method, several simple and useful criteria for CPS are obtained, and all results are illustrated by numerical examples.     

Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization

The primary objective of this paper is to propose a new approach for analyzing pinning stability in a complex dynamical network via impulsive control. A?simple yet generic criterion of impulsive pinning synchronization for such coupled oscillator network is derived analytically. It is shown that a single impulsive controller can always pin a given complex dynamical network to a homogeneous solution. Subsequently, the theoretic result is applied to a small-world (SW) neuronal network comprised of the Hindmarsh?CRose oscillators. It turns out that the firing activities of a single neuron can induce synchronization of the underlying neuronal networks. This conclusion is obviously in consistence with empirical evidence from the biological experiments, which plays a significant role in neural signal encoding and transduction of information processing for neuronal activity. Finally, simulations are provided to demonstrate the practical nature of the theoretical results.     

Synchronization properties of interconnected network based on the vital node

In this paper, we perform an intensive study of the synchronization properties of interconnected network and the concepts of vital node, and the simplest and equivalent network is firstly introduced. We strictly derive the eigenvalues of Laplacian matrix and the synchronizability of interconnected network and its simplest and equivalent network through utilizing the master stability function approach. Firstly, we find the synchronizability of interconnected network is identical to its simplest and equivalent network. Secondly, we identify the general factors that determine the synchronizability of interconnected network and further analyze the impact of different factors on the synchronizability. Finally, theoretical analysis and numerical simulations are carried out to indicate the validity and effectiveness of current analysis. The current results are beneficial to understand the dynamical behaviors of complex networked systems.     

Synchronization in complex dynamical networks with interval time-varying coupling delays

This paper deals with the synchronization problem of complex dynamical networks with interval time-varying coupling delays. A simple local linear feedback controller is introduced to guarantee the synchronizability of the networks. Some delay-dependent synchronization conditions for the controlled complex dynamical networks are presented by using the Lyapunov–Krasovskii functional method and the reciprocally convex combination approach. Theoretical analysis and numerical examples show that the obtained conditions have less computational complexity and less conservatism than some recently reported ones.     

Impact of node dynamics parameters on topology identification of complex dynamical networks

This paper aims at investigating the topology identification problem of complex dynamical networks with varying node dynamics parameters and fixed inner coupling matrices. In particular, by employing the unified chaotic system as node dynamics, this work further explores the influence of continuously changing node dynamics parameters on topology identification of complex dynamical networks with different coupling strengths. Results show that for sufficiently small or large coupling strengths, the performance of topology identification is not affected by the change of node parameters. Specifically, for small enough coupling strengths, the topological structure can be completely identified regardless of the change of node parameters, while for sufficiently large coupling strengths, the connectivity (presence and absence of connections) cannot be successfully identified. Furthermore, for certain coupling strengths, with the increase of node dynamics parameters, the topology identification varies from completely unidentifiable to partially or event completely identifiable. Therefore, the synchronization-based topology identification depends on node dynamics. Even for the same node dynamical model, different parameters can have a significant impact on identification results. Furthermore, for networks consisting of chaotic oscillators defining node dynamics, small coupling strengths are conducive to topology identification. A broader conclusion is that projective synchronization, rather than just complete synchronization, is an obstacle to the network topology identification. The findings in this paper will add to our understanding of conditions for identifying topologies of complex networks.     

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.     

Adaptive synchronization of T-S fuzzy complex networks with time-varying delays via the pinning control method

In this paper, the synchronization of Takagi–Sugeno (T-S) fuzzy complex networks with time-varying delays and adaptive coupling weights is studied. Using the pinning control and adaptive feedback strategy, a new general class of complex networks with fuzzy logic is proposed and its synchronization is investigated in terms of linear matrix inequalities (LMIs). The adaptive update law of coupling weight is only related to the dynamical behaviors of directly connected nodes. Based on the Lyapunov stability theory, it is proven that the synchronization of the addressed network can be achieved under those control strategies. Finally, two numerical examples are given to verify the effectiveness of our theoretical results.     

Finite-time synchronization of complex networks with nonidentical discontinuous nodes

In this paper, we study the finite-time synchronization problem for linearly coupled complex networks with discontinuous nonidentical nodes. Firstly, new conditions for general discontinuous chaotic systems is proposed, which is easy to be verified. Secondly, a set of new controllers are designed such that the considered model can be finite-timely synchronized onto any target node with discontinuous functions. Based on a finite-time stability theorem for equations with discontinuous right-hand and inequality techniques, several sufficient conditions are obtained to ensure the synchronization goal. Results of this paper are general, and they extend and improve existing results on both continuous and discontinuous complex networks. Finally, numerical example, in which a BA scale-free network with discontinuous Sprott and Chua circuits is finite-timely synchronized onto discontinuous Chen system, is given to show the effectiveness of our new results.     

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.     

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.     

Evoking complex neuronal networks by stimulating a single neuron

The responses of electrically coupled neuronal network to external stimulus injected on a single neuron are investigated. Stimulating the largest-degree neuron in the network, it is found that as the intensity of the stimulus increases, the network will be transiting from the resting to firing states and then restoring to the resting state, thereby showing a bounded firing region in the parameter space. Furthermore, it is found that as the coupling strength among the neurons decreases, the firing region is gradually expanded and, at the weak couplings, it could be separated into several disconnected subregions. By a simplified network model, we conduct a detailed analysis on the bifurcation diagram of the network dynamics in the two-dimensional parameter space spanned by stimulating intensity and coupling strength, and, by introducing a new coefficient named effective stimulus, explore the underlying mechanisms for the modified firing region. It is revealed that the coupling strength and stimulating intensity are equally important in evoking the network, but with different mechanisms. Specifically, the effective stimuli are shifted up globally by increasing the stimulating intensity, while are drawn closer by increasing the coupling strength. The dynamical responses of small-world and random complex networks to external stimulus are also investigated, which confirm the generality of the observed phenomena. The findings shed new lights on the collective behaviors of complex neuronal networks and might help our understandings on the recent experimental results.     

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.     

Cluster synchronization of nonlinearly-coupled complex networks with nonidentical nodes and asymmetrical coupling matrix

In this paper, we investigate the cluster synchronization problem for networks with nonlinearly coupled nonidentical dynamical systems and asymmetrical coupling matrix by using pinning control. We derive sufficient conditions for cluster synchronization for any initial values through a feedback scheme and propose an adaptive feedback algorithm that adjusts the coupling strength. Some numerical examples are then given to illustrate the theoretical results.     

Online estimation of unknown delays and parameters in uncertain time delayed dynamical complex networks via adaptive observer

In this paper, based on the observer concept and adaptive control approach, we present a general and systematic research on estimating uncertain information in complex networks, which includes the unknown delays, node parameters and network topologies. Specifically, two novel approaches for simultaneous estimation of unknown coupling delays and node parameters, as well as the node delays and topology weights, are proposed and proved in this paper. Several typical examples are presented to verify the effectiveness of these approaches. The numerical results also show that the delay estimation approach is applicable for online monitoring of dynamical complex networks with time-varying delays.     

Pinning control of spatially and temporally complex dynamical networks with time-varying delays

In the present paper, two types of complex delayed dynamical networks with spatially and temporally varying state variables are proposed. The first is that all nodes in the network have the same time-varying delay. The second is that different nodes have different time-varying delays. We respectively investigate the stabilization problem of these two types of complex network models by pinning a small fraction of nodes with negative feedback controllers. With the help of Lyapunov functionals and some inequality techniques, several asymptotic stability and exponential stability conditions are established. Finally, numerical simulations are presented to illustrate the effectiveness of the results obtained here.     

Topology identification of complex networks from noisy time series using ROC curve analysis

As it is known, there is various unknown information in most real world networks, such as uncertain topological structure and node dynamics. Thus how to identify network topology from dynamical behaviors is an important inverse problem for physics, biology, engineering, and other science disciplines. Recently, with the help of noise, a method to predict network topology has been proposed from the dynamical correlation matrix, which is based on a general, one-to-one correspondence between the correlation matrix and the connection matrix. However, the success rate of this prediction method depends on the threshold, which is related to the coupling strength and noise intensity. Different coupling strength and noise intensity result in different success rate of prediction. To deal with this problem, we select a desirable threshold to improve the success rate of prediction by using Receiver Operating Characteristic (ROC) curve analysis. By the technique of ROC curve analysis, we find that the accuracy and efficiency of topology identification is mainly determined by coupling strengths. The success rate of estimation will be reduced if the coupling is too weak or too strong. The presence of noise facilitates topology identification, but the noise intensity is not always crucial to the effectiveness of topology identification.