Mapping dynamical systems onto complex networks

The objective of this study is to design a procedure to characterize chaotic dynamical systems, in which they are mapped onto a complex network. The nodes represent the regions of space visited by the system, while the edges represent the transitions between these regions. Parameters developed to quantify the properties of complex networks, including those related to higher order neighbourhoods, are used in the analysis. The methodology is tested on the logistic map, focusing on the onset of chaos and chaotic regimes. The corresponding networks were found to have distinct features that are associated with the particular type of dynamics that generated them.     

Robustness of heterogeneous complex networks

In this paper we study the robustness of heterogeneous preferential attachment networks. The robustness of a network measures its structural tolerance to the random removal of nodes and links. We numerically analyze the influence of the affinity parameters on a set of ensemble-averaged robustness metrics. We show that the presence of heterogeneity does not fundamentally alter the smooth nature of the fragmentation process of the models. We also show that a moderate level of locality translates into slight improvements in the robustness metrics, which prompts us to conjecture an evolutionary argument for the existence of real networks with power-law scaling in their connectivity and clustering distributions.     

Epidemic outbreaks in complex heterogeneous networks

We present a detailed analytical and numerical study for the spreading of infections with acquired immunity in complex population networks. We show that the large connectivity fluctuations usually found in these networks strengthen considerably the incidence of epidemic outbreaks. Scale-free networks, which are characterized by diverging connectivity fluctuations in the limit of a very large number of nodes, exhibit the lack of an epidemic threshold and always show a finite fraction of infected individuals. This particular weakness, observed also in models without immunity, defines a new epidemiological framework characterized by a highly heterogeneous response of the system to the introduction of infected individuals with different connectivity. The understanding of epidemics in complex networks might deliver new insights in the spread of information and diseases in biological and technological networks that often appear to be characterized by complex heterogeneous architectures. Received 20 September 2001 and Received in final form 4 February 2002     

Cluster synchronization in fractional-order complex dynamical networks

Cluster synchronization of complex dynamical networks with fractional-order dynamical nodes is discussed in the Letter. By using the stability theory of fractional-order differential system and linear pinning control, a sufficient condition for the stability of the synchronization behavior in complex networks with fractional order dynamics is derived. Only the nodes in one community which have direct connections to the nodes in other communities are needed to be controlled, resulting in reduced control cost. A numerical example is presented to demonstrate the validity and feasibility of the obtained result. Numerical simulations illustrate that cluster synchronization performance for fractional-order complex dynamical networks is influenced by inner-coupling matrix, control gain, coupling strength and topological structures of the networks.     

Adaptive cluster synchronization in complex dynamical networks

Cluster synchronization is investigated in different complex dynamical networks. In this Letter, a novel adaptive strategy is proposed to make a complex dynamical network achieve cluster synchronization, where the adaptive strategy of one edge is adjusted only according to its local information. A sufficient condition about the global stability arbitrarily grouped of cluster synchronization is derived. Several numerical simulations show the effectiveness of the adaptive strategy.     

Synchronization for complex dynamical Lurie networks

This paper investigates the synchronization problem for two different complex dynamical Lurie networks, The first one is with constant coupling and the second one is with constant coupling and discrete-delay coupling. Based on contraction theory and matrix measure properties, some new delay-independent synchronization conditions depending on coupling strength and network topology are proposed. Finally, simulation results are presented to support the theoretical results.     

Adaptive-impulsive synchronization of uncertain complex dynamical networks

This Letter studies adaptive-impulsive synchronization of uncertain complex dynamical networks. Based on the stability analysis of impulsive system, several network synchronization criteria for local and global adaptive-impulsive synchronization are established. Numerical example is also given to illustrate the results.     

Topology identification of weighted complex dynamical networks

Recently, various papers investigated the geometry features, synchronization and control of complex network provided with certain topology. While, the exact topology of a network is sometimes unknown or uncertain. Using Lyapunov theory, we propose an adaptive feedback controlling method to identify the exact topology of a rather general weighted complex dynamical network model. By receiving the network nodes evolution, the topology of such kind of network with identical or different nodes, or even with switching topology can be monitored. Experiments show that the methods presented in this paper are of high accuracy with good performance.     

Synchronization in complex dynamical networks with nonsymmetric coupling

Based on the work of Nishikawa and Motter, who have extended the well-known master stability framework to include non-diagonalizable cases, we develop another extension of the master stability framework to obtain criteria for global synchronization. Several criteria for global synchronization are provided which generalize some previous results. The Jordan canonical transformation method is used in stead of the matrix diagonalization method. Especially, we show clearly that, the synchronizability of a dynamical network with nonsymmetric coupling is not always characterized by its second-largest eigenvalue, even though all the eigenvalues of the nonsymmetric coupling matrix are real. Furthermore, the effects of the asymmetry of coupling on synchronizability of networks with different structures are analyzed. Numerical simulations are also done to illustrate and verify the theoretical results on networks in which each node is a dynamical limit cycle oscillator consisting of a two-cell cellular neural network.     

Adaptive pinning synchronization in fractional-order complex dynamical networks

Synchronization of general complex dynamical networks with fractional-order dynamical nodes is addressed in this paper. Based on the stability theory of fractional-order differential systems and adaptive pinning control, some sufficient local asymptotical synchronization criteria and global asymptotical ones are derived respectively, which succeed in solving the problem about how many nodes are need to be controlled and how much coupling strength should be applied to ensure the synchronization of the entire fractional-order networks. The obtained results are more general and effective than those reported. Moreover, the coupling-configuration matrices and the inner-coupling matrices are not assumed to be symmetric and irreducible. Finally, a numerical simulation is presented to demonstrate the validity and feasibility of the proposed synchronization criteria.     

Stochastic synchronization for time-varying complex dynamical networks

This paper studies the stochastic synchronization problem for time-varying complex dynamical networks. This model is totally different from some existing network models. Based on the Lyapunov stability theory, inequality techniques, and the properties of the Weiner process, some controllers and adaptive laws are designed to ensure achieving stochastic synchronization of a complex dynamical network model. A sufficient synchronization condition is given to ensure that the proposed network model is mean-square stable. Theoretical analysis and numerical simulation fully verify the main results.     

Synchronization of complex dynamical networks with time delays

In the present paper, two kinds of dynamical complex networks are considered. The first is that elements of every node have different time delays but all nodes in such networks have the same time-delay vector. The second is that different nodes have different time-delay vectors, and the elements of each node also have different time delays. Corresponding synchronization theorems are established. Numerical examples show the efficiency of the derived theorems.     

Impulsive generalized function synchronization of complex dynamical networks

This Letter investigates generalized function synchronization of continuous and discrete complex networks by impulsive control. By constructing the reasonable corresponding impulsively controlled response networks, some criteria and corollaries are derived for the generalized function synchronization between the impulsively controlled complex networks, continuous and discrete networks are both included. Furthermore, the generalized linear synchronization and nonlinear synchronization are respectively illustrated by several examples. All the numerical simulations demonstrate the correctness of the theoretical results.     

Robust impulsive synchronization of complex delayed dynamical networks

This Letter investigates robust impulsive synchronization of complex delayed dynamical networks with nonsymmetrical coupling from the view of dynamics and control. Based on impulsive control theory on delayed dynamical systems, some simple yet generic criteria for robust impulsive synchronization are established. It is shown that these criteria can provide a novel and effective control approach to synchronize an arbitrary given delayed dynamical network to a desired synchronization state. Comparing with existing results, the advantage of the control scheme is that synchronization state can be selected as a weighted average of all the states in the network for the purpose of practical control strategy. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed control methodology.     

Synchronization of complex dynamical networks with nonidentical nodes

This Letter investigates the synchronization problem of a complex network with nonidentical nodes, and proposes two effective control schemes to synchronize the network onto any smooth goal dynamics. By applying open-loop control to all nodes and placing adaptive feedback injections on a small fraction of network nodes, a low-dimensional sufficient condition is derived to guarantee the global synchronization of the complex network with nonidentical nodes. By introducing impulsive effects to the open-loop controlled network, another synchronization scheme is developed for the network composed of nonidentical nodes, and an upper bound of impulsive intervals is estimated to ensure the global stability of the synchronization process. Numerical simulations are given to verify the theoretical results.     

Observer-based synchronization in complex dynamical networks with nonsymmetric coupling

Based on a general complex dynamical network model with nonsymmetric coupling, some criteria for synchronization are proposed based on the approach of state observer design. Unlike the nonobserver-based dynamical networks, where the coupling between two connected nodes is defined by an inner coupling matrix and full state coupling is typically needed, in this paper, smaller amount of coupling variables or even only a scalar output signal of each node is needed to synchronize the network. Unlike the commonly researched complex network model, where the coupling between nodes is symmetric, here, in our network model, the coupling configuration matrix is not assumed to be symmetric and may have complex eigenvalues. The matrix Jordan canonical formalization method is used instead of the matrix diagonalization method, so in our synchronization criteria, the coupling configuration matrix is not required to be diagonalizable. Especially, the proposed step-by-step approach is simpler in computation than the existent ones, which usually rely heavily on numerical toolbox, and may be done by hand completely. An example is given to illustrate the step-by-step approach, in which each node is a two-dimensional dynamical limit cycle oscillator system consisting of a two-cell cellular neural network, and numerical simulations are also done to verify the results of design.     

Hierarchical synchronization in complex networks with heterogeneous degrees

We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function.     

Generalized synchronization in complex dynamical networks via adaptive couplings

This paper investigates generalized synchronization of three typical classes of complex dynamical networks: scale-free networks, small-world networks, and interpolating networks. The proposed synchronization strategy is to adjust adaptively a node’s coupling strength based on the node’s local generalized synchronization information. By taking the auxiliary-system approach and using the Lyapunov function method, we prove that for any given initial coupling strengths, the generalized synchronization can take place in complex networks consisting of nonidentical dynamical systems. It is demonstrated that the coupling strengths are affected by topologies of the networks. Furthermore, it is found that there are hierarchical features in the processes of generalized synchronization in scale-free networks because of their highly heterogeneous distributions of connection degree. Finally, we discuss in detail how a network’s degree of heterogeneity affects its generalization synchronization behavior.