New eigenvalue based approach to synchronization in asymmetrically coupled networks

Locally and globally exponential stability of synchronization in asymmetrically nonlinear coupled networks and linear coupled networks are investigated in this paper, respectively. Some new synchronization stability criteria based on eigenvalues are derived. In these criteria, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of the sum of the column of the asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability results is that they can be analytically applied to the asymmetrically coupled networks and can overcome the complexity of calculating eigenvalues of the coupling asymmetric matrix. Therefore, these conditions are very convenient to use. Moreover, a necessary condition of globally exponential synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of the coupling matrix.     

Symmetric patterns in linear arrays of coupled cells

In this note we show how to find patterned solutions in linear arrays of coupled cells. The solutions are found by embedding the system in a circular array with twice the number of cells. The individual cells have a unique steady state, so that the patterned solutions represent a discrete analog of Turing structures in continuous media. We then use the symmetry of the circular array (and bifurcation from an invariant equilibrium) to identify symmetric solutions of the circular array that restrict to solutions of the original linear array. We apply these abstract results to a system of coupled Brusselators to prove that patterned solutions exist. In addition, we show, in certain instances, that these patterned solutions can be found by numerical integration and hence are presumably asymptotically stable.     

二维激子极化激元凝聚中涡旋叠加态稳态及动力学特性研究

利用分步Crank-Nicolson方案的虚时和实时有限差分方法求解耗散系统的Gross-Pitaevskii(GP)方程,研究了二维激子极化激元凝聚(exciton-polariton condensates)体系中正反涡旋叠加态的稳态结构并直观地验证这种稳态结构在半导体微腔旋转下的稳定性和动力学特性.通过虚时和实时演化相结合的方法求解出几种角动量情况下所对应的稳定涡旋叠加态.然后利用实时演化方法研究在半导体微腔旋转的情况下,正反涡旋叠加态的稳定性及其旋转角速率和半导体微腔旋转角速率之间的定量关系.最后研究了单涡旋态在半导体微腔旋转时形成涡旋阵列的动力学过程,并给出了泵浦光宽度和增益项对涡旋阵列结构的影响.研究表明,系统的泵浦,损耗和增益对稳定性和动力学特性有重要影响.     

光纤激光器阵列相干合成中的位相探测与校正方法研究

介绍一高速的位相检测与控制方法，可以对高功率光纤激光器阵列中每一路信号光波的位相进行实时探测与控制，从而使整个系统达到锁相输出的目的.以一路信号光为例，给出该方法的实验研究结果，从中可以看出，闭环控制的结果，可以将该信号光的位相稳定在一设定的静态工作点上. 关键词： 光纤激光器阵列 锁相 外差探测     

Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling

A two-dimensional nonlinear Schrödinger lattice with nonlinear coupling, modelling a square array of weakly coupled linear optical waveguides embedded in a nonlinear Kerr material, is studied. We find that despite a vanishing energy difference (Peierls-Nabarro barrier) of fundamental stationary modes the mobility of localized excitations is very poor. This is attributed to a large separation in parameter space of the bifurcation points of the involved stationary modes. At these points the stability of the fundamental modes is changed and an asymmetric intermediate solution appears that connects the points. The control of the power flow across the array when excited with plane waves is also addressed and shown to exhibit great flexibility that may lead to applications for power-coupling devices. In certain parameter regimes, the direction of a stable propagating plane-wave current is shown to be continuously tunable by amplitude variation (with fixed phase gradient). More exotic effects of the nonlinear coupling terms like compact discrete breathers and vortices, and stationary complex modes with nontrivial phase relations are also briefly discussed. Regimes of dynamical linear stability are found for all these types of solutions.     

Dynamics of an array formed by three neutrally buoyant cylindrical cantilevers subjected to tensile follower forces

The flexural-rotational coupled motion of three identical flexible cylindrical cantilevers joined symmetrically to a central head is investigated. Effects of the tensile follower forces and inertia parameters on the natural frequencies of the system are studied. The analysis suggests two types of inplane motion: one corresponding to the oscillation of the cantilevers without any rotation of the central body, while the other involves coupled motion of the array. The former corresponds to the repeated eigenvalues which are identical to those of a single cantilever having the same axial tension parameter, P. Three sets of eigenvalues govern the out-of-plane motion: (a) the central head remaining stationary with no rolling motion of the array; (b) vertical motion of the central body without any rolling motion of the array; and (c) rigid body rolling motion without any vertical motion of the central head. There is a possibility of dynamic instability for small inertia parameters and large axial tension.     

Filtering the eigenvalues at infinite from the linear stability analysis of incompressible flows

Steady state, two-dimensional flows may become unstable under two and three-dimensional disturbances if the flow parameters exceed some critical values. In many practical situations, determining the parameters at which the flow becomes unstable is essential. Linear hydrodynamic stability of a laminar flow leads to a generalized eigenvalue problem (GEVP) where the eigenvalues correspond to the rate of growth of the disturbances and the eigenfunctions to the amplitude of the perturbation. Solving GEVP’s is challenging, because the incompressibility of the liquid gives rise to singularities leading to non-physical eigenvalues at infinity that require substantial care. The high computational cost of solving the GEVP has probably discouraged the use of linear stability analysis of incompressible flows as a general engineering tool for design and optimization.In this work, we propose a new procedure to eliminate the eigenvalues at infinity from the GEVP associated to the linear stability analysis of incompressible flow. The procedure takes advantage of the structure of the matrices involved and avoids part of the computational effort of the standard mapping techniques used to compute the spectrum of incompressible flows. As an example, the method is applied in the solution of linear stability analysis of plane Couette flow.     

Conditional stability of diatomic molecule driven by a weak laser field

Using a direct perturbation method, we investigate the stability of a diatomic molecule modelled by a weakly laser-driven Morse oscillator. It is shown that stationary state solution of the system is stable in the sense of Lyapunov and the periodical one possesses conditional stability, namely its stability depends on the initial conditions and system parameters. The corresponding sufficient and necessary conditions are established that indicate the stable states associated with some discrete energies. The results reveal how a diatomic molecule can be stabilized or dissociated with a weak laser, and demonstrate that the mathematical conditional stability works in the considered physical system.     

Fiber Optic Coupling of CW Linear Laser Diode Array

Based on a set of microoptics the output radiation from a continuous wave (CW) linear laser diode array is coupled into a multi-mode optical fiber of 400 μm diameter.The CW linear laser diode array is a 1 cm laser diode bar with 19 stripes with 100 μm aperture spaced on 500 μm centers.The coupling system contains packaged laser diode bar,fast axis collimator,slow axis collimation array,beam transformation system and focusing system.The high brightness,high power density and single fiber output of a laser diode bar is achieved.The coupling efficiency is 65% and the power density is up to 1.03×104 W/cm2.     

盘型激光焊金属蒸汽特征色调-色饱和度-亮度分析法研究

对于大功率盘型激光焊,金属焊件表面在激光束辐射下强烈汽化并形成等离子体状的金属蒸汽羽状物. 该金属蒸汽羽状物可逆向激光束传输,对激光有明显的屏蔽作用, 降低激光辐射至焊件的能量密度,影响焊接效率和质量. 因此研究金属蒸汽特征变化规律及其与焊接质量之间的关联 ,可实现由金属蒸汽特征实时监测激光焊接状态. 以10 kW大功率盘型连续激光焊接304不锈钢钢板为试验对象, 应用高速摄像机摄取金属蒸汽动态图像,将其转换至色调-色饱和度-亮度空间, 提取金属蒸汽面积、激光束受影响路径长度等相关特征量, 以焊缝熔宽的变化作为衡量焊接状态稳定性的参数. 通过金属蒸汽特征值的均值统计和方差分析,试验证明根据金属蒸汽面积和激光束受影响路径长度等金属蒸汽特征可有效地反映熔宽质量, 从而对焊接状况做出动态评估.     

Synchronization of Complex Network with Drivingly Coupled Scheme

We present a network model with a new coupled scheme which is the generalization of drive-response systems called a drivingly coupled network. The synchronization of the network is investigated by numerical simulations based on Lorenz systems. By calculating the largest transversal Lyapunov exponents of such network, the stable and unstable regions of synchronous state for eigenvalues in such network can be obtained and many kinds of drivingly coupled arrays based on Lorenz systems such as all-to-all, star-shape, ring-shape and chain-shape networks are considered.     

Fiber Optic Coupling of CW Linear Laser Diode Array

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On instability of excited states of the nonlinear Schrödinger equation

We introduce a new notion of linear stability for standing waves of the nonlinear Schrödinger equation (NLS) which requires not only that the spectrum of the linearization be real, but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive. We prove that excited states of the NLS are not linearly stable in this more restrictive sense. We then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability.     

Random delays and the synchronization of chaotic maps

We investigate the dynamics of an array of chaotic logistic maps coupled with random delay times. We report that for adequate coupling strength the array is able to synchronize, in spite of the random delays. Specifically, we find that the synchronized state is a homogeneous steady state, where the chaotic dynamics of the individual maps is suppressed. This synchronization behavior is largely independent of the connection topology and depends mainly on the average number of links per node. We carry out a statistical linear stability analysis that confirms the numerical results and provides a better understanding of the nontrivial roles of random delayed interactions.     

Gross-Pitaevskii方程的数值解及其动力学研究

采用辛-打靶法求解了简谐势阱中碱金属玻色-爱因斯坦凝聚原子在T=0时的基态波函数,检验了波函数的稳定性,并研究了当简谐势阱突然改变时波函数随时间演化的动力学.     

Eigenstructure of the 1-particle density operator corresponding to the first excited state of two coupled oscillators

The eigenstructure of the 1-particle density operator ? corresponding to the first excited state of two coupled oscillators is investigated. It is shown that ? is strictly positive and thus may be considered as the canonical density matrix of some effective Hamiltonian system. The form of the eigenfunctions and their dependence on the eigenvalues is found. A number of properties of the spectrum are proved and a method allowing to find a finite number of exact terms of power expansions of the eigenvalues in a parameter λ∈(0, 1) is given.     

Photovoltaic spatial soliton pairs in two-photon photorefractive materials

We report the existence of incoherently coupled bright-bright steady state photovoltaic soliton pairs in two-photon photorefractive material under open circuit conditions. Based on WKBJ method and paraxial ray approximation, we have obtained coupled equations describing dynamical evolution of spatial soliton pairs. In the steady state regime, the present analysis leads to the identification of existence equation of bright-bright solitons, which captures a plethora of soliton pairs. We have undertaken linear stability analysis which shows that these solitons are stable.     

Statistically stable ultrasonic imaging in random media

Analysis of array data from acoustic scattering in a random medium with a small number of isolated targets is performed in order to image and, thereby, localize the spatial position of each target. Because the host medium has random fluctuations in wave speed, the background medium is itself a source of scattered energy. It is assumed, however, that the targets are sufficiently larger and/or more reflective than the background fluctuations so that a clear distinction can be made between targets and background scatterers. In numerical simulations nonreflective boundary conditions are used so as to isolate the effects of the host randomness from those of the spatial boundaries, which can then be treated in a separate analysis. It is shown that the key to successful imaging is finding statistically stable functionals of the data whose extreme values provide estimates of scatterer locations. The best ones are related to the eigenfunctions and eigenvalues of the array response matrix, just as one might expect from prior work on array data processing in complex scattering media having homogeneous backgrounds. The specific imaging functionals studied include matched-field processing and linear subspace methods, such as MUSIC (MUtiple SIgnal Classification). But statistical stability is not characteristic of the frequency domain, which is often the province of these methods. By transforming back into the time domain after first diagonalizing the array data in the frequency domain, one can take advantage of both the time-domain stability and the frequency-domain orthogonality of the relevant eigenfunctions.     

New approach to synchronization in asymmetrically coupled networks

Global exponentially synchronization in asymmetrically coupled networks is investigated in this Letter. We extend eigenvalue based method to synchronization in symmetrically coupled network to synchronization in asymmetrically coupled network. A new stability criterion of eigenvalue based is derived. In this criterion, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of sum of column of asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability result is that it can analytical be applied to the asymmetrically coupled networks and overcome the complexity on calculating eigenvalues of coupling asymmetric matrix. Therefore, this condition is very convenient to use. Moreover, a necessary condition of this synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of coupling matrix.     

频率漂移对耦合激光器阵列锁相状态的影响

耦合激光器阵列可以通过激光器之间的倏逝波耦合实现相位锁定,是有望实现光束相干合成的方案之一。为了探究该阵列锁相状态的稳定性,分析了频率漂移对两路激光器阵列相位锁定状态的影响。通过对耦合激光器阵列动力学方程的数值研究发现,频率漂移将破坏激光器阵列的锁相状态,并引入相位噪声,导致输出远场光强分布不稳定; 研究还发现：倏逝波耦合能够对频率漂移的影响起到一定的抑制作用,但这种抑制只在频率漂移较弱时才有效,因此,如果将该阵列用于光束合成,加入相应的抗干扰措施还是很必要的。数值分析的结果还表明：频率漂移的低频成分对于阵列相位锁定状态的影响更为剧烈,因此,抗干扰措施应主要针对低频干扰来进行。