噪声环境下时滞耦合网络的广义投影滞后同步

时滞和噪声在复杂网络中普遍存在,而含有耦合时滞和噪声摄动的耦合网络同步的研究工作却极其稀少. 本文针对噪声环境下具有不同节点动力学、不同拓扑结构及不同节点数目的耦合时滞网络,提出了两个网络之间的广义投影滞后同步. 首先,构建了更加贴近现实的驱动-响应网络同步的理论框架;其次,基于随机时滞微分方程LaSalle不变性原理,严格证明了在合理的控制器作用下,驱动网络和响应网络在几乎必然渐近稳定性意义下能够取得广义投影滞后同步;最后,借助于计算机仿真,通过具体的网络模型验证了理论推理的有效性. 数值模拟结果表明,驱动网络与响应网络不但能够达到广义投影滞后同步,而且同步效果不依赖于耦合时滞和比例因子的选取,同时也揭示了更新增益和耦合时滞对同步收敛速度的显著性影响. 关键词： 复杂网络 广义投影滞后同步 随机噪声 时滞     

Robust outer synchronization between two complex networks with fractional order dynamics

Synchronization between two coupled complex networks with fractional-order dynamics, hereafter referred to as outer synchronization, is investigated in this work. In particular, we consider two systems consisting of interconnected nodes. The state variables of each node evolve with time according to a set of (possibly nonlinear and chaotic) fractional-order differential equations. One of the networks plays the role of a master system and drives the second network by way of an open-plus-closed-loop (OPCL) scheme. Starting from a simple analysis of the synchronization error and a basic lemma on the eigenvalues of matrices resulting from Kronecker products, we establish various sets of conditions for outer synchronization, i.e., for ensuring that the errors between the state variables of the master and response systems can asymptotically vanish with time. Then, we address the problem of robust outer synchronization, i.e., how to guarantee that the states of the nodes converge to common values when the parameters of the master and response networks are not identical, but present some perturbations. Assuming that these perturbations are bounded, we also find conditions for outer synchronization, this time given in terms of sets of linear matrix inequalities (LMIs). Most of the analytical results in this paper are valid both for fractional-order and integer-order dynamics. The assumptions on the inner (coupling) structure of the networks are mild, involving, at most, symmetry and diffusivity. The analytical results are complemented with numerical examples. In particular, we show examples of generalized and robust outer synchronization for networks whose nodes are governed by fractional-order Lorenz dynamics.     

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.     

Erratum to: Rheotaxis performance increases with group size in a coupled phase model with sensory noise

This special topics issue is a collection of contributions on the recent developments of control and synchronization in time delayed systems and space time chaos. The various articles report interesting results on time delayed complex networks; fractional order delayed models; dynamics of spatio-temporal patterns; stochastic models etc. Experimental analysis on synchronization, dynamics and control of chaos are also well investigated using Field Programmable Gate Array (FPGA), circuit realizations and chemical reactions.     

Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks

We study projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, where projective-anticipating and projective-lag synchronization can be achieved only on two coupled chaotic systems. In this paper, we realize projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In this paper, the dynamics of the nodes of the complex networks are time-delayed chaotic systems without the limitation of the partial linearity. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its existence and sufficient stability conditions. The validity of the proposed method is demonstrated and verified by examining specific examples using Ikeda and Mackey-Glass systems on Erdos-Renyi networks.     

Phase synchronization of bursting neural networks with electrical and delayed dynamic chemical couplings

Diffusive electrical connections in neuronal networks are instantaneous, while excitatoryor inhibitory couplings through chemical synapses contain a transmission time-delay.Moreover, chemical synapses are nonlinear dynamical systems whose behavior can bedescribed by nonlinear differential equations. In this work, neuronal networks withdiffusive electrical couplings and time-delayed dynamic chemical couplings are considered.We investigate the effects of distributed time delays on phase synchronization of burstingneurons. We observe that in both excitatory and Inhibitory chemical connections, the phasesynchronization might be enhanced when time-delay is taken into account. This distributedtime delay can induce a variety of phase-coherent dynamical behaviors. We also study thecollective dynamics of network of bursting neurons. The network model presents theso-called Small-World property, encompassing neurons whose dynamics have two time scales(fast and slow time scales). The neuron parameters in such Small-World network, aresupposed to be slightly different such that, there may be synchronization of the bursting(slow) activity if the coupling strengths are large enough. Bounds for the criticalcoupling strengths to obtain burst synchronization in terms of the network structure aregiven. Our studies show that the network synchronizability is improved, as itsheterogeneity is reduced. The roles of synaptic parameters, more precisely those of thecoupling strengths and the network size are also investigated.     

Understanding the enhanced synchronization of delay-coupled networks with fluctuating topology

We study the dynamics of networks with coupling delay, from which the connectivity changes over time. The synchronization properties are shown to depend on the interplay of three time scales: the internal time scale of the dynamics, the coupling delay along the network links and time scale at which the topology changes. Concentrating on a linearized model, we develop an analytical theory for the stability of a synchronized solution. In two limit cases, the system can be reduced to an “effective” topology: in the fast switching approximation, when the network fluctuations are much faster than the internal time scale and the coupling delay, the effective network topology is the arithmetic mean over the different topologies. In the slow network limit, when the network fluctuation time scale is equal to the coupling delay, the effective adjacency matrix is the geometric mean over the adjacency matrices of the different topologies. In the intermediate regime, the system shows a sensitive dependence on the ratio of time scales, and on the specific topologies, reproduced as well by numerical simulations. Our results are shown to describe the synchronization properties of fluctuating networks of delay-coupled chaotic maps.     

Synchronization in complex networks by time-varying couplings

Synchronization in networks of complex topologies using couplings of time-varying strength is numerically investigated. The time-dependencies of coupling strengths are coupled to the dynamics of the nodes in a way to enhance synchronization. By time-varying couplings, oscillators are found to take quite a short time to reach synchronization state when the couplings are relatively strong. Even when a nearly regular networks of large-size with few shortcuts is difficult to be synchronized by fixed couplings, the time-varying couplings can easily enhance the emergence of synchronization.     

Finite-time complex projective synchronization of fractional-order complex-valued uncertain multi-link network and its image encryption application

Multi-link networks are universal in the real world such as relationship networks, transportation networks, and communication networks. It is significant to investigate the synchronization of the network with multi-link. In this paper, considering the complex network with uncertain parameters, new adaptive controller and update laws are proposed to ensure that complex-valued multilink network realizes finite-time complex projective synchronization (FTCPS). In addition, based on fractional-order Lyapunov functional method and finite-time stability theory, the criteria of FTCPS are derived and synchronization time is given which is associated with fractional order and control parameters. Meanwhile, numerical example is given to verify the validity of proposed finite-time complex projection strategy and analyze the relationship between synchronization time and fractional order and control parameters. Finally, the network is applied to image encryption, and the security analysis is carried out to verify the correctness of this method.     

Synchronization criterion for Lur'e type complex dynamical networks with time-varying delay

In this Letter, the synchronization problem for a class of complex dynamical networks in which every identical node is a Lur'e system with time-varying delay is considered. A delay-dependent synchronization criterion is derived for the synchronization of complex dynamical network that represented by Lur'e system with sector restricted nonlinearities. The derived criterion is a sufficient condition for absolute stability of error dynamics between the each nodes and the isolated node. Using a convex representation of the nonlinearity for error dynamics, the stability condition based on the discretized Lyapunov-Krasovskii functional is obtained via LMI formulation. The proposed delay-dependent synchronization criterion is less conservative than the existing ones. The effectiveness of our work is verified through numerical examples.     

Synchronization and modularity in complex networks

We investigate the connection between the dynamics of synchronization and the modularity on complex networks. Simulating the Kuramoto's model in complex networks we determine patterns of meta-stability and calculate the modularity of the partition these patterns provide. The results indicate that the more stable the patterns are, the larger tends to be the modularity of the partition defined by them. This correlation works pretty well in homogeneous networks (all nodes have similar connectivity) but fails when networks contain hubs, mainly because the modularity is never improved where isolated nodes appear, whereas in the synchronization process the characteristic of hubs is to have a large stability when forming its own community.     

Synchronization in multilayer networks through different coupling mechanisms

In recent years, most studies of complex networks have focused on a single network and ignored the interaction of multiple networks, much less the coupling mechanisms between multiplex networks. In this paper we investigate synchronization phenomena in multilayer networks with nonidentical topological structures based on three specific coupling mechanisms:assortative, disassortative, and anti-assortative couplings. We find rich and complex synchronous dynamic phenomena in coupled networks. We also study the behavior of effective frequencies for layers I and II to understand the underlying microscopic dynamics occurring under the three different coupling mechanisms. In particular, the coupling mechanisms proposed here have strong robustness and effectiveness and can produce abundant synchronization phenomena in coupled networks.     

Synchronization processes in complex networks

During the last decades the emergence of collective dynamics in large networks of coupled units has been investigated in fields such as optics, chemistry, biology and ecology. Recently, complex networks have provided a challenging framework for the study of synchronization of dynamical units, based on the interplay between complexity in the overall topology and local dynamical properties of the coupled units. In this work, we review the constructive role played by such complex wirings for the synchronization of networks of coupled dynamical systems. We review the main techniques that have been proposed for assessing the propensity for synchronization (synchronizability) of a given networked system. We will also describe the main applications, especially in the view of selecting the optimal topology in the coupling configuration that provides enhancement of the synchronization features.     

Synchronization in complex delayed dynamical networks with impulsive effects

The present paper is mainly concerned with the issues of synchronization dynamics of complex delayed dynamical networks with impulsive effects. A general model of complex delayed dynamical networks with impulsive effects is formulated, which can well describe practical architectures of more realistic complex networks related to impulsive effects. Based on impulsive stability theory on delayed dynamical systems, some simple but less conservative criterion are derived for global synchronization of such dynamical network. It is shown that synchronization of the networks is heavily dependent on impulsive effects of connecting configuration in the networks. Furthermore, the theoretical results are applied to a typical SF network composing of impulsive coupled chaotic delayed Hopfield neural network nodes, and are also illustrated by numerical simulations.     

Evolution of microscopic and mesoscopic synchronized patterns in complex networks

Previous studies about synchronization of Kuramoto oscillators in complex networks have shown how local patterns of synchronization emerge differently in homogeneous and heterogeneous topologies. The main difference between the paths to synchronization in both topologies is rooted in the growth of the largest connected component of synchronized nodes when increasing the coupling between the oscillators. Nevertheless, a recent study focusing on this same phenomenon has claimed the contrary, stating that the statistical distribution of synchronized clusters for both types of networks is similar. Here we provide extensive numerical evidences that confirm the original claims, namely, that the microscopic and mesoscopic dynamics of the synchronized patterns indeed follow different routes.     

Synchronization Transition of Time Delayed Complex Dynamical Networks with Discontinuous Coupling

This paper investigates the synchronization of time delayed complex dynamical networks with periodical on-off coupling. Both the theoretical and numerical results show that, in spite of time delays and on-off coupling, two networks may synchronize if the coupling strength and the on-off rate are large enough. It is shown that, for undirected and strongly connected networks, the upper bound of time delays for synchronization is a decreasing function of the absolute value of the minimum eigenvalue of the adjacency matrix. The theoretical analysis confirms the numerical results and provides a better understanding of the influence of time delays and on-off coupling on the synchronization transition. The influence of random delays on the synchronization is also discussed.     

Synchronization in uncertain complex networks

We consider the problem of synchronization in uncertain generic complex networks. For generic complex networks with unknown dynamics of nodes and unknown coupling functions including uniform and nonuniform inner couplings, some simple linear feedback controllers with updated strengths are designed using the well-known LaSalle invariance principle. The state of an uncertain generic complex network can synchronize an arbitrary assigned state of an isolated node of the network. The famous Lorenz system is stimulated as the nodes of the complex networks with different topologies. We found that the star coupled and scale-free networks with nonuniform inner couplings can be in the state of synchronization if only a fraction of nodes are controlled.     

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.     

Multi-time-scale synchronization and information processing in bursting neuron networks

We analyze the effect of synchronization in networks of chemically coupled multi-time-scale (spiking-bursting) neurons on the process of information transmission within the network. Although, synchronization occurs first in the slow time-scale (burst) and then in the fast time-scale (spike), we show that information can be transmitted with low probability of errors in both time scales when the bursts become synchronized. Furthermore, we show that for networks of non-identical multi-time-scales neurons, complete synchronization is no longer possible, but instead burst phase synchronization. Our analysis shows that clusters of burst phase synchronized neurons are very likely to appear in a network for parameters far smaller than the ones for which the onset of burst phase synchronization in the whole network takes place.