Global Synchronization of General Delayed Dynamical Networks

Global synchronization of general delayed dynamical networks with linear coupling are investigated. A sufficient condition for the global synchronization is obtained by using the linear matrix inequality and introducing a reference state. This condition is simply given based on the maximum nonzero eigenvalue of the network coupling matrix. Moreover, we show how to construct the coupling matrix to guarantee global synchronization of network, which is very convenient to use. A two-dimension system with delay as a dynamical node in network with global coupling is finally presented to verify the theoretical results of the proposed global synchronization scheme.     

不同耦合下混沌神经元网络的同步

采用Hindmarsh-Rose神经元动力学模型, 对二维点阵上的神经元网络的同步进行了研究. 为了解不同耦合对网络同步的影响, 提出了一般反馈耦合、分层反馈耦合和分层局域平均场反馈耦合三种方案.研究表明:在耦合强度较小的近邻耦合下, 一般反馈耦合不能使网络达到完全同步, 而分层反馈耦合和分层局域平均场反馈耦合可以使网络出现局部同步和全局同步. 不同形式的耦合会导致网络出现不同的斑图, 随着耦合强度的增大, 网络从不同步到同步的过程也不相同, 一般反馈耦合和分层反馈耦合网络是突然出现全局同步, 同步之前网络出现非周期性的相干斑图; 对于分层局域平均场反馈耦合网络, 同层神经元之间先出现从簇放电同步到同步的转变, 形成靶波, 然后同步区由中心向外逐渐扩大, 最终达到网络的全局同步. 这些结果表明, 只有适当的耦合才能实现信号的无损耗的传递. 此外我们发现分层局域平均场反馈耦合可以促进网络的同步.     

Synchronization of Rössler oscillators on scale-free topologies

We study the synchronization of Rössler oscillators as prototypes of chaotic systems on scale-free complex networks. As it turns out, the underlying topology crucially affects the global synchronization properties. In particular, we show that the existence of loops facilitates the synchronizability of the system, whereas Rössler oscillators do not synchronize on tree-like topologies beyond a certain size. Moreover, it is not the mere number of loops that counts for synchronization but also the type of loops. By considering Cayley trees modified by additional loops in different ways, we find out that also the distribution of shortest path lengths between two oscillators plays an important role for the global synchronization.     

Adaptive mechanism between dynamical synchronization and epidemic behavior on complex networks

Many realistic epidemic networks display statistically synchronous behavior which we will refer to as epidemic synchronization. However, to the best of our knowledge, there has been no theoretical study of epidemic synchronization. In fact, in many cases, synchronization and epidemic behavior can arise simultaneously and interplay adaptively. In this paper, we first construct mathematical models of epidemic synchronization, based on traditional dynamical models on complex networks, by applying the adaptive mechanisms observed in real networks. Then, we study the relationship between the epidemic rate and synchronization stability of these models and, in particular, obtain the conditions of local and global stability for epidemic synchronization. Finally, we perform numerical analysis to verify our theoretical results. This work is the first to draw a theoretical bridge between epidemic transmission and synchronization dynamics and will be beneficial to the study of control and the analysis of the epidemics on complex networks.     

Adaptive Synchronization of Fractional Order Complex-Variable Dynamical Networks via Pinning Control

In this paper, the synchronization of fractional order complex-variable dynamical networks is studied using an adaptive pinning control strategy based on close center degree. Some effective criteria for global synchronization of fractional order complex-variable dynamical networks are derived based on the Lyapunov stability theory. From the theoretical analysis, one concludes that under appropriate conditions, the complex-variable dynamical networks can realize the global synchronization by using the proper adaptive pinning control method. Meanwhile, we succeed in solving the problem about how much coupling strength should be applied to ensure the synchronization of the fractional order complex networks. Therefore, compared with the existing results, the synchronization method in this paper is more general and convenient. This result extends the synchronization condition of the real-variable dynamical networks to the complex-valued field, which makes our research more practical. Finally, two simulation examples show that the derived theoretical results are valid and the proposed adaptive pinning method is effective.     

Pinning synchronization of the complex networks with non-delayed and delayed coupling

In this Letter, without assuming the symmetry of the coupling matrix, we investigate the global synchronization of the complex networks with non-delayed and delayed coupling based on the pinning controllers. Some sufficient conditions for the global synchronization by adding linear and adaptive feedback controllers to a part of nodes are obtained. Numerical examples are also provided to demonstrate the effectiveness of the theory.     

Adaptive exponential synchronization of delayed chaotic networks

This paper deals with the global exponential synchronization of a class of delayed chaotic networks. Under some simple conditions, the global synchronization of a network about its all variables is derived by only considering the global synchronization of its partial variables. Furthermore, based on the Halanay inequality technique, some delay-independent criteria are obtained to ensure the adaptive exponential synchronization of the model. And the simpler, less conservative and more efficient results are easy to be verified in engineering applications. Finally, an illustrative example is given to demonstrate the effectiveness of the presented synchronization scheme.     

Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix

In this paper, the global synchronization for an array of nonlinearly coupled identical chaotic systems is investigated. A distinctive feature of this work is to address synchronization issues for nonlinearly coupled complex networks with an asymmetrical coupling matrix. By projecting the nonlinear coupling function onto a linear one and assuming the difference between them as a disturbing function, we give some criteria for the global synchronization in virtual of the left eigenvector corresponding to the zero eigenvalue of the coupling matrix. Numerical examples are also provided to demonstrate the effectiveness of the theory.     

Global synchronization of Chua＇s chaotic delay network by using linear matrix inequality

Global synchronization of Chua‘s chaotic dynamical networks with coupling delays is investigated in this paper.Unlike other approaches, where only local results were obtained, the network is found to be not linearized in this paper.Insteat, the global synchronization is obtained by using the linear matrix inequality theory. Moreover, some quite simple linear-state-error feedback controllers for global synchronization are derived, which can be easily constructed based on the minimum eigenvalue of the coupling matrix. A simulation of Chua‘s chaotic network with global coupling delays in nodes is finally given, which is used to verify the theoretical results of the proposed global synchron izationscheme.     

Global synchronization for a class of dynamical complex networks

This paper is concerned with the problem of global synchronization for a class of dynamical complex networks composed of general Lur’e systems. Based on the absolute stability theory and the Kalman-Yakubovich-Popov (KYP) lemma, sufficient conditions are established to guarantee global synchronization of dynamical networks with complex topology, directed and weighted couplings. Several global synchronization criteria formulated in the form of linear matrix inequalities (LMIs) or frequency-domain inequalities are also proposed for undirected dynamical networks. In order to obtain global results, no linearization technique is involved through derivation of the synchronization criteria. Numerical examples are provided to demonstrate the effectiveness of the proposed results.     

Impulsive synchronization of two coupled complex networks with time-delayed dynamical nodes

In this paper, we investigate the impulsive synchronization between two coupled complex networks with time-delayed dynamical nodes. Based on the Lyapunov stability, the linear feedback control and the impulsive control theories, the linear feedback and the impulsive controllers are designed separately. By using the generalized Barbalat's lemma, the global asymptotic impulsive synchronization of the drive—response complex networks is derived and some corresponding sufficient conditions are also obtained. Numerical examples are presented to verify the effectiveness and the correctness of the synchronization criteria.     

Partial synchronization in linearly and symmetrically coupled ordinary differential systems

In this paper, we discuss partial synchronization of linearly and symmetrically coupled ordinary differential equations (LSCODEs). The synchronization phenomena are investigated via invariant synchronization manifolds. On the basis of geometrical analysis of the synchronization manifold, several criteria for the global attractivity of the invariant synchronization manifold are obtained. Combining these criteria with some numerical examples, we investigate how topological structure affects partial synchronization, and give a valuable discussion about the possibility of partial synchronization with increasing coupling strength.     

Synchronization mechanism of clapping rhythms in mutual interacting individuals

In recent years, clapping synchronization between individuals has been widely studied as one of the typical synchronization phenomena. In this paper, we aim to reveal the synchronization mechanism of clapping interactions by observing two individuals’ clapping rhythms in a series of experiments. We find that the two synchronizing clapping rhythm series exhibit long-range cross-correlations(LRCCs);that is, the interaction of clapping rhythms can be seen as a strong-anticipation process. Previous studies have demonstrated that the interactions in local timescales or global matching in statistical structures of fluctuation in long timescales can be sources of the strong-anticipation process. However, the origin of the strong anticipation process often appears elusive in many complex systems. Here, we find that the clapping synchronization process may result from the local interaction between two clapping individuals and may result from the more global coordination between two clapping individuals. We introduce two stochastic models for mutually interacting clapping individuals that generate the LRCCs and prove theoretically that the generation of clapping synchronization process needs to consider both local interaction and global matching. This study provides a statistical framework for studying the internal synchronization mechanism of other complex systems. Our theoretical model can also be applied to study the dynamics of other complex systems with the LRCCs, including finance, transportation, and climate.     

Synchronization is enhanced in weighted complex networks

The propensity for synchronization of complex networks with directed and weighted links is considered. We show that a weighting procedure based upon the global structure of network pathways enhances complete synchronization of identical dynamical units in scale-free networks. Furthermore, we numerically show that very similar conditions hold also for phase synchronization of nonidentical chaotic oscillators.     

Comment on “Synchronization in dynamic networks with nonsymmetrical time-delay coupling based on linear feedback controllers” [Physica A 387 (2008) 2111-2119]

In this paper, we point out two problems in synchronization of dynamic networks based on linear feedback controllers presented by Wu and Jiao [J. Wu, L. Jiao, Synchronization in dynamic networks with nonsymmetrical time-delay coupling based on linear feedback controllers, Physica A 387 (2008) 2111-2119]. Furthermore, an amendatory version on global exponential synchronization is given and proved in this paper.     

Concentric circular arrays of Josephson junctions as dressed models of granular superconductors

In this paper networks that optimize a combined measure of local and global synchronizability are evolved. It is shown that for low coupling improvements in the local synchronizability dominate network evolution. This leads to an expressed grouping of elements with similar native frequency into cliques, allowing for an early onset of synchronization, but rendering full synchronization hard to achieve. In contrast, for large coupling the network evolution is governed by improvements towards full synchronization, preventing any expressed community structure. Such networks exhibit strong coupling between dissimilar oscillators. Albeit a rapid transition to full synchronization is achieved, the onset of synchronization is delayed in comparison to the first type of networks. The paper illustrates that an early onset of synchronization (which relates to clustering) and global synchronization are conflicting demands on network topology.     

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-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.     

Paths to synchronization on complex networks

The understanding of emergent collective phenomena in natural and social systems has driven the interest of scientists from different disciplines during decades. Among these phenomena, the synchronization of a set of interacting individuals or units has been intensively studied because of its ubiquity in the natural world. In this Letter, we show how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks, driving the process towards a certain global synchronization degree following different paths. The dependence of the dynamics on the coupling strength and on the topology is unveiled. This study provides a new perspective and tools to understand this emerging phenomena.     

Chaotic synchronization of coupled ergodic maps

With few exceptions, studies of chaotic synchronization have focused on dissipative chaos. Though less well known, chaotic systems that lack dissipation may also synchronize. Motivated by an application in communication systems, we couple a family of ergodic maps on the N-torus and study the global stability of the synchronous state. While most trajectories synchronize at some time, there is a measure zero set that never synchronizes. We give explicit examples of these asynchronous orbits in dimensions two and four. On more typical trajectories, the synchronization error reaches arbitrarily small values and, in practice, converges. In dimension two we derive bounds on the average synchronization time for trajectories resulting from randomly chosen initial conditions. Numerical experiments suggest similar bounds exist in higher dimensions as well. Adding noise to the coupling signal destroys the invariance of the synchronous state and causes typical trajectories to desynchronize. We propose a modification of the standard coupling scheme that corrects this problem resulting in robust synchronization in the presence of noise.