Synchronization in the network of chaotic microwave oscillators

Time scale synchronization in networks of chaotic microwave oscillators with the different topologies of the links between nodes has been studied. As a node element of the network the one-dimensional distributed model of the low-voltage vircator has been used. To characterize the degree of synchronization in the whole network the synchronization index has been introduced. The transition to the synchronous regime is shown to take place via cluster time scale synchronization. Meanwhile, the spectral structure of the output signals is complicated sufficiently which allows using such devices in a number of practical applications.     

振幅耦合动态网络中相邻结点间的相同步

研究用适当的量化指标来刻画动态网络的相同步,为此定义了新的量化指标:相邻结点的网络平均锁相值和网络平均相频差.动态网络结点选择的是多旋转中心的Lorenz混沌振子,对Lorenz系统进行柱面坐标变换,用振幅耦合方法构造动态网络.分别对星形网络和小世界网络进行了仿真计算,结果表明随着耦合强度的增大,网络中相邻结点的两个系统之间存在相同步现象,而且相同步行为与定义的量化指标之间存在较准确的对应关系.     

Amplitude and phase effects on the synchronization of delay-coupled oscillators

We consider the behavior of Stuart-Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding self-feedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behavior.     

Explosive synchronization: From synthetic to real-world networks

Synchronization is a widespread phenomenon in both synthetic and real-world networks. This collective behavior of simple and complex systems has been attracting much research during the last decades. Two different routes to synchrony are defined in networks; first-order, characterized as explosive, and second-order, characterized as continuous transition. Although pioneer researches explained that the transition type is a generic feature in the networks, recent studies proposed some frameworks in which different phase and even chaotic oscillators exhibit explosive synchronization. The relationship between the structural properties of the network and the dynamical features of the oscillators is mainly proclaimed because some of these frameworks show abrupt transitions. Despite different theoretical analyses about the appearance of the first-order transition, studies are limited to the mean-field theory, which cannot be generalized to all networks. There are different real-world and man-made networks whose properties can be characterized in terms of explosive synchronization, e.g., the transition from unconsciousness to wakefulness in the brain and spontaneous synchronization of power-grid networks. In this review article, explosive synchronization is discussed from two main aspects. First, pioneer articles are categorized from the dynamical-structural framework point of view. Then, articles that considered different oscillators in the explosive synchronization frameworks are studied. In this article, the main focus is on the explosive synchronization in networks with chaotic and neuronal oscillators. Also, efforts have been made to consider the recent articles which proposed new frameworks of explosive synchronization.     

The effect of time-delay on anomalous phase synchronization

Anomalous phase synchronization in nonidentical interacting oscillators is manifest as the increase of frequency disorder prior to synchronization. We show that this effect can be enhanced when a time-delay is included in the coupling. In systems of limit-cycle and chaotic oscillators we find that the regions of phase disorder and phase synchronization can be interwoven in the parameter space such that as a function of coupling or time-delay the system shows transitions from phase ordering to disorder and back.     

Global synchronization in general complex delayed dynamical networks and its applications

The main objective of the present paper is further to investigate global synchronization of a general model of complex delayed dynamical networks. Based on stability theory on delayed dynamical systems, some simple yet less conservative criteria for both delay-independent and delay-dependent global synchronization of the networks are derived analytically. It is shown that under some conditions, if the uncoupled dynamical node is stable itself, then the network can be globally synchronized for any coupling delays as long as the coupling strength is small enough. On the other hand, if each dynamical node of the network is chaotic, then global synchronization of the networks is heavily dependent on the effects of coupling delays in addition to the connection configuration. Furthermore, the results are applied to some typical small-world (SW) and scale-free (SF) complex networks composing of coupled dynamical nodes such as the cellular neural networks (CNNs) and the chaotic FHN neuron oscillators, and numerical simulations are given to verify and also visualize the theoretical results.     

Adaptive synchronization of a class of fractional-order complex-valued chaotic neural network with time-delay

This paper is concerned with the adaptive synchronization of fractional-order complex-valued chaotic neural networks (FOCVCNNs) with time-delay. The chaotic behaviors of a class of fractional-order complex-valued neural network are investigated. Meanwhile, based on the complex-valued inequalities of fractional-order derivatives and the stability theory of fractional-order complex-valued systems, a new adaptive controller and new complex-valued update laws are proposed to construct a synchronization control model for fractional-order complex-valued chaotic neural networks. Finally, the numerical simulation results are presented to illustrate the effectiveness of the developed synchronization scheme.     

Engineering synchronization of chaotic oscillators using controller based coupling design

We propose a general formulation of coupling for engineering synchronization in chaotic oscillators for unidirectional as well as bidirectional mode. In the synchronization regimes, it is possible to amplify or to attenuate a chaotic attractor with respect to other chaotic attractors. Numerical examples are presented for a Lorenz system, Ro?ssler oscillator, and a Sprott system. We physically realized the controller based coupling design in electronic circuits to verify the theory. We extended the theory to a network of coupled oscillators and provided a numerical example with four Sprott oscillators.     

Synchronization in asymmetrically coupled networks with node balance

We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.     

Synchronization of impulsively coupled complex systems with delay

This paper investigates the synchronization of complex systems with delay that are impulsively coupled at discrete instants only. Based on the comparison theorem of impulsive differential system, a distributed impulsive control scheme is proposed to achieve the synchronization for systems with delay. In the control strategy, the influence of all nodes to network synchronization relies on its weight. The proposed control scheme is applied to the chaotic delayed Hopfield neural networks and numerical simulations are presented to demonstrate the effectiveness of the proposed scheme.     

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     

Synchronization of impulsively coupled complex networks

We investigate the synchronization of complex networks,which are impulsively coupled only at discrete instants.Based on the comparison theory of impulsive differential systems,a distributed impulsive control scheme is proposed for complex dynamical networks to achieve synchronization.The proposed scheme not only takes into account the influence of all nodes to network synchronization,which depends on the weight of each node in the network,but also provides us with a flexible method to select the synchronized state of the network.In addition,it is unnecessary for the impulsive coupling matrix to be symmetrical.Finally,the proposed control scheme is applied to a chaotic Lorenz network and Chua’s circuit network.Numerical simulations are used to illustrate the validity of this control scheme.     

Adaptive Synchronization between Two Different Complex Networks with Time-Varying Delay Coupling

A new general network model for two complex networks with time-varying delay coupling is presented. Then we investigate its synchronization phenomena. The two complex networks of the model differ in dynamic nodes, the number of nodes and the coupling connections. By using adaptive controllers, a synchronization criterion is derived. Numerical examples are given to demonstrate the effectiveness of the obtained synchronization criterion. This study may widen the application range of synchronization, such as in chaotic secure communication.     

Synchronization of spatiotemporal chaos in a class of complex dynamical networks

This paper studies the synchronization of complex dynamical networks constructed by spatiotemporal chaotic systems with unknown parameters. The state variables in the systems with uncertain parameters are used to construct the parameter recognizers, and the unknown parameters are identified. Uncertain spatiotemporal chaotic systems are taken as the nodes of complex dynamical networks, connection among the nodes of all the spatiotemporal chaotic systems is of nonlinear coupling. The structure of the coupling functions between the connected nodes and the control gain are obtained based on Lyapunov stability theory. It is seen that stable chaos synchronization exists in the whole network when the control gain is in a certain range. The Gray--Scott models which have spatiotemporal chaotic behaviour are taken as examples for simulation and the results show that the method is very effective.     

Zero-lag long-range synchronization via dynamical relaying

We show that isochronous synchronization between two delay-coupled oscillators can be achieved by relaying the dynamics via a third mediating element, which surprisingly lags behind the synchronized outer elements. The zero-lag synchronization thus obtained is robust over a considerable parameter range. We substantiate our claims with experimental and numerical evidence of such synchronization solutions in a chain of three coupled semiconductor lasers with long interelement coupling delays. The generality of the mechanism is validated in a neuronal model with the same coupling architecture. Thus, our results show that zero-lag synchronized chaotic dynamical states can occur over long distances through relaying, without restriction by the amount of delay.     

Rich-club network topology to minimize synchronization cost due to phase difference among frequency-synchronized oscillators

As exemplified by power grids and large-scale brain networks, some functions of networks consisting of phase oscillators rely on not only frequency synchronization, but also phase synchronization among the oscillators. Nevertheless, even after the oscillators reach frequency-synchronized status, the phase synchronization is not always accomplished because the phase difference among the oscillators is often trapped at non-zero constant values. Such phase difference potentially results in inefficient transfer of power or information among the oscillators, and avoids proper and efficient functioning of the networks. In the present study, we newly define synchronization cost by using the phase difference among the frequency-synchronized oscillators, and investigate the optimal network structure with the minimum synchronization cost through rewiring-based optimization. By using the Kuramoto model, we demonstrate that the cost is minimized in a network with a rich-club topology, which comprises the densely-connected center nodes and low-degree peripheral nodes connecting with the center module. We also show that the network topology is characterized by its bimodal degree distribution, which is quantified by Wolfson’s polarization index.     

Synchronization of time-delay chaotic systems on small-world networks with delayed coupling

By using the well-known Ikeda model as the node dynamics, this paper studies synchronization of time-delay systems on small-world networks where the connections between units involve time delays. It shows that, in contrast with the undelayed case, networks with delays can actually synchronize more easily. Specifically, for randomly distributed delays, time-delayed mutual coupling suppresses the chaotic behaviour by stabilizing a fixed point that is unstable for the uncoupled dynamical system.     

一类节点结构互异的复杂网络的混沌同步

提出了一种实现节点结构互异的复杂网络的混沌同步方法.以异结构混沌系统作为节点构造复杂网络,基于Lyapunov稳定性定理确定了复杂网络中连接节点的耦合函数的形式.以Rssler系统、Coullet系统以及Lorenz系统作为网络节点构成的复杂网络为例,仿真模拟发现,整个复杂网络存在稳定的混沌同步现象.此方法不但可以实现任意混沌系统作为节点的网络混沌同步,而且网络节点数对整个复杂网络同步的稳定性也无影响,因而,具有一定的普适性. 关键词： 混沌同步 复杂网络 异结构 Lyapunov稳定性定理     

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.     

Two chaos synchronization schemes and public-channel message transmission in a mutually coupled semiconductor lasers system

Chaos synchronization and message transmission of a mutually coupled system consisting of two semiconductor lasers (SLs) and a partially transparent mirror (PTM) in between are investigated theoretically. Analytical results show that two types of chaos synchronization schemes, named as isochronal synchronization (IS) and leader/laggard synchronization (LLS), can be achieved by adjusting the reflectivity and position of PTM. By establishing SIMULINK model, numerical simulations illustrate that as the PTM is positioned at the center of two lasers, IS is available when the reflectivity of PTM is moderate. The LLS is achieved when the reflectivity of PTM equals to 0.5, which means feedback strength equals to coupling strength. Its lag time is just determined by the difference of feedback delay time. The investigations of mutual chaos pass filtering (MCPF) effects and the secure chaotic communication simulations indicate that IS allows real-time bidirectional message transmission on a public-channel, while LLS can achieve higher security chaotic communication by using its lag time as cryptography key. The demonstrated system can be used as a rudiment of array chaos communications system.