Control schemes for synchronizing two subnetworks with weak couplings

In this paper, we propose a well-designed network model with a parameter and study full and partial synchronization of the network model based on the stability analysis. The network model is composed of a star-coupled subnetwork and a globally coupled subnetwork. By analyzing the special coupling configuration, three control schemes are obtained for synchronizing the network model. Further analysis indicates that even if the inner couplings in each subnetwork are very weak, two of the control schemes are still valid. In particular, if the outer coupling weight parameter θ is larger than (n2 2n)/4, or the subnetwork size n is larger than θ2, the two subnetworks with weak inner couplings can achieve synchronization. In addition, the synchronizability is independent of the network size in case of 0 < θ < n/(n + 1). Finally, we carry out some numerical simulations to confirm the validity of the obtained control schemes. It is worth noting that the main idea of this paper also applies to any network consisting of a dense subnetwork and a sparse network.     

多延时互耦合半导体激光器的实时混沌同步特性

针对双延时和三延时互耦合半导体激光器系统,研究了互耦合延时和互耦合强度对实时混沌同步质量的影响,提出了双延时互耦合系统中可将其中一个互耦合延时看作反馈延时的思想,揭示了多延时互耦合半导体激光器系统实时混沌同步条件和规律.研究结果表明,多延时互耦合系统中,某两条双向链路的互耦合延时比值为2,是实现高品质实时混沌同步的基本条件;增大互耦合强度,可以改善实时混沌同步品质,且在较低的等效耦合强度条件下,双延时互耦合系统较三延时互耦合系统更易于实现良好的实时混沌同步.     

多延时互耦合半导体激光器的实时混沌同步特性

针对双延时和三延时互耦合半导体激光器系统,研究了互耦合延时和互耦合强度对实时混沌同步质量的影响,提出了双延时互耦合系统中可将其中一个互耦合延时看作反馈延时的思想,揭示了多延时互耦合半导体激光器系统实时混沌同步条件和规律.研究结果表明,多延时互耦合系统中,某两条双向链路的互耦合延时比值为2,是实现高品质实时混沌同步的基本条件;增大互耦合强度,可以改善实时混沌同步品质,且在较低的等效耦合强度条件下,双延时互耦合系统较三延时互耦合系统更易于实现良好的实时混沌同步.     

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.     

激光局域网络的混沌控制及并行队列同步

本文提出并研究了"单队列-双参数"光电延时反馈控制条件下的激光局域网络的混沌控制及串联的动力学行为的并行队列"交叉驱动-反馈"网络同步实现,建立了该光学局域网络的数学物理控制模型.通过含时延超越方程理论的分析,预言了该光学局域网络是可以实现混沌控制的,且网络两路结点队列是可以实现实时引导控制到多个类周期状态上的,并通过并行队列同步方程理论证明并行串联队列同步是可以获得的.结果发现在可控的激光局域网络两个并行串联队列光路上,分别实现了网络队列结点的混沌控制并能够实现多个类周期的网络结点的并行串联队列同步,实现了络网结点激光器的2周期、3周期、4周期等状态的并行队列同步,以及其他多个类周期的队列并行同步和动态同步.还发现了两个类周期并行队列网络同步控制区域.本文还给出了激光局域网络"并行多点混沌载波同步发射及其在光学超宽带通信中应用"的一个案例并成功实现.这是一种新型的激光混沌局域网络控制系统,具有光局域网络光传送与光联接核心控制技术要素,具有复杂动力学系统与网络的多变量、多空间维度及并行两路不同队列混沌控制技术特点,还具有光网络超宽带通信功能等.其研究结果对局域网络、光网络的控制与同步、激光技术以及混沌的研究具有重要的参考价值.     

Synchronization by dynamical relaying in electronic circuit arrays

We experimentally study the synchronization of two chaotic electronic circuits whose dynamics is relayed by a third parameter-matched circuit, to which they are coupled bidirectionally in a linear chain configuration. In a wide range of operating parameters, this setup leads to synchronization between the outer circuits, while the relaying element remains unsynchronized. The specifics of the synchronization differ with the coupling level: for low couplings a state of intermittent synchronization between the outer circuits coexists with one of antiphase synchronization. Synchronization becomes in phase for moderate couplings, and for strong coupling identical synchronization is observed between the outer elements, which are themselves synchronized in a generalized way with the relaying element. In the latter situation, the middle element displays a triple-scroll attractor that is not possible to obtain when the chaotic oscillator is isolated.     

Synchronization of asynchronous probabilistic boolean network

This paper researches the synchronization of asynchronous probabilistic Boolean network and synchronization theorems are provided. Inner synchronization and outer synchronization are considered, respectively. It should be pointed that synchronization theorems and corollaries with probability one and probabilities unequal to one are both taken into consideration. What’s more, the asynchronous multivalued Boolean network is also studied as an extension. Finally, some examples for asynchronous probabilistic Boolean network synchronizations are provided in the numerical simulation.     

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.     

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.     

Synchronization in a multiplex network of gene oscillators

Study of the synchronization in the network of gene oscillator network has vital importance in understanding of rhythmicity of molecular and cellular activities. In this paper, we analyze a network of linearly coupled genetic oscillators in a multiplex structure. The coupling strength values are changed and the coupling range is considered to be fixed, but different in the two layers. The analyses are done in two cases of periodic and chaotic oscillations. By computing the statistical measures, the interlayer and intralayer synchronization states are studied. The results show that the layer with higher coupling range has more enhanced synchrony and is less affected by the turbulent behavior of the other layer. On the other hand, the layer with lower coupling range approaches synchronization by strengthening the interlayer and intralayer couplings. The interlayer synchronization is also achieved in high coupling strength values.     

耦合含时滞的相互依存网络的局部自适应异质同步

针对由两个子网络构成的耦合含时滞的相互依存网络,研究其局部自适应异质同步问题.时滞同时存在于两个子网络的内部耦合项和子网络间的一对一相互依赖耦合项中,且网络的耦合关系满足非线性特性和光滑性.基于李雅普诺夫稳定性理论、线性矩阵不等式方法和自适应控制技术,通过对子网络设置合适的控制器,提出了使得相互依存网络的子网络分别同步到异质孤立系统的充分条件.针对小世界网络和无标度网络构成的相互依存网络进行数值模拟,验证了提出理论的正确性和有效性.     

Outer Synchronization of Complex Networks by Impulse

This paper investigates outer synchronization of complex networks, especially, outer complete synchronization and outer anti-synchronization between the driving network and the response network. Employing the impulsive control method which is uncontinuous, simple, efficient, low-cost and easy to implement in practical applications, we obtain some sufficient conditions of outer complete synchronization and outer anti-synchronization between two complex networks. Numerical simulations demonstrate the effectiveness of the proposed impulsive control scheme.     

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.     

The role of degree-weighted couplings in the synchronous onset of Kuramoto oscillator networks

This paper investigates the role of asymmetrical degree-dependent weighted couplings in synchronization of a network of Kuramoto oscillators, where the conditions of coupling criticality for the onset of phase synchronization in degree-weighted complex networks are arrived at. The numerical simulations visualize that for networks having power-law or exponential degree distributions, asymmetrical degree-weighted couplings (with increasing weighting exponent β) increases the critical coupling to achieve the onset of phase synchronization in the networks.     

Outer synchronization between two nonidentical networks with circumstance noise

This paper regards the outer synchronization between two delay-coupled complex dynamical networks with nonidentical topological structures and a noise perturbation. Considering one network as the drive network and the other one as the response network, the drive-response system achieves synchronous states through a suitably designed adaptive controller. The stochastic LaSalle invariance principle is employed to theoretically prove the almost sure synchronization between two networks. Finally, two numerical examples are examined in order to illustrate the proposed synchronization scheme.     

Electric and magnetic dipole couplings in split ring resonator metamaterials

In this paper, the electric and the magnetic dipole couplings between the outer and the inner rings of a single split ring resonator (SRR) are investigated. We numerically demonstrate that the magnetic resonance frequency can be substantially modified by changing the couplings of the electric and magnetic dipoles, and give a theoretical expression of the magnetic resonance frequency. The results in this work are expected to be conducive to a deeper understanding of the SRR and other similar metamaterials, and provide new guidance for complex metamaterials design with a tailored electromagnetic response.     

非线性函数耦合的Chen吸引子网络的混沌同步

利用非对称非线性函数耦合混沌同步方法,讨论了Chen吸引子的混沌同步问题,数值模拟分析初始值和耦合强度因子的选择对于实现混沌同步的影响. 将非对称非线性函数耦合同步方法进一步推广发展到完全连接网络和由星形子网络构成的复杂大网络混沌同步的研究中. 提供了确定网络中神经元之间混沌同步状态稳定性的误差发展方程,并讨论各个耦合强度因子对网络同步稳定性过程的影响,给出了相应的稳定性范围. 通过数值模拟证明利用非线性函数作为耦合函数,实现完全连接网络、星形子网络构成大网络的混沌同步是有效的. 可以预测在网络的混沌同步 关键词： 非线性耦合函数 Chen吸引子 混沌同步 网络     

Novel criteria for exponential synchronization of inner time-varying complex networks with coupling delay

This paper mainly investigates the exponential synchronization of an inner time-varying complex network with coupling delay. Firstly, the synchronization of complex networks is decoupled into the stability of the corresponding dynamical systems. Based on the Lyapunov function theory, some sufficient conditions to guarantee its stability with any given convergence rate are derived, thus the synchronization of the networks is achieved. Finally, the results are illustrated by a simple time-varying network model with a coupling delay. All involved numerical simulations verify the correctness of the theoretical analysis.     

Analytic treatment of the network synchronization problem with time delays

Motivated by novel results in the theory of network synchronization, we analyze the effects of nonzero time delays in stochastic synchronization problems with linear couplings in an arbitrary network. We determine analytically the fundamental limit of synchronization efficiency in a noisy environment with uniform time delays. We show that the optimal efficiency of the network is achieved for λτ = π(3/2)/(2√π + 4) ≈ 0.738, where λ is the coupling strength (relaxation coefficient) and τ is the characteristic time delay in the communication between pairs of nodes. Our analysis reveals the underlying mechanism responsible for the trade-off phenomena observed in recent numerical simulations of network synchronization problems.     

Robust projective outer synchronization of coupled uncertain fractional-order complex networks

In this work, we propose a novel projective outer synchronization (POS) between unidirectionally coupled uncertain fractional-order complex networks through scalar transmitted signals. Based on the state observer theory, a control law is designed and some criteria are given in terms of linear matrix inequalities which guarantee global robust POS between such networks. Interestingly, in the POS regime, we show that different choices of scaling factor give rise to different outer synchrony, with various special cases including complete outer synchrony, anti-outer synchrony and even a state of amplitude death. Furthermore, it is demonstrated that although stability of POS is irrelevant to the inner-coupling strength, it will affect the convergence speed of POS. In particular, stronger inner synchronization can induce faster POS. The effectiveness of our method is revealed by numerical simulations on fractional-order complex networks with small-world communication topology.