连续时间系统二维不稳定流形的异构算法

非线性系统的二维流形通常具有复杂几何结构和丰富动力学信息,因此在流形计算与可视化时存在大量的不可避免的数值计算.因此,如何高效地完成这些计算就成了关键问题.鉴于当今计算机的异构发展趋势(包含多核CPU和通用GPU),本文在兼顾精度和通用性的基础上,提出了适用于新一代计算平台的快速流形计算方法.本算法将计算任务分为轨道延伸和三角形生成两部分,前者运算量大而单一适合GPU完成,后者运算量小而复杂适合CPU执行.通过对Lorenz系统原点稳定流形的计算,表明本算法能充分发挥异构平台的综合性能,可大幅度提高计算速 关键词： 不稳定流形 流形计算 异构计算 Lorenz系统     

广义Birkhoff系统动力学的一类逆问题

针对广义Birkhoff系统动力学,提出广义Birkhoff系统动力学的一类逆问题,研究由已知积分流形来建立广义Birkhoff方程. 这类逆问题的解通常不是唯一的,需给出必要的补充要求. 最后举例说明结果的应用. 关键词： 广义Birkhoff系统 动力学逆问题 积分流形     

广义混沌同步中的多稳定同步流形

从构造驱动系统和响应系统之间的函数关系出发,一般性地研究了广义混沌同步中同步流形的多值性问题,并对同步流形的稳定性进行了分析,提出了多稳定同步流形存在的条件.利用该稳定性条件对两个实例进行了分析,一个为Genesio-Rssler耦合系统,另一个为具有二次、三次非线性的耦合Duffing系统,结果表明前者仅有唯一的稳定同步流形,而后者随着耦合强度的增加,从具有两个稳定的同步流形变为只存在一个稳定同步流形. 关键词： 混沌同步 多稳定性 同步流形     

不确定非线性系统混沌反控制

基于滑模变结构控制,研究了不确定非线性系统模型到达混沌反控制,保证经有限时间,使受控不确定非线性系统达到与混沌参考系统数学模型完全一致,并就到达的快速性,鲁棒性进行了讨论,最后以Lorenz混沌系统为混沌参考模型的情况进行了仿真,仿真结果表明所给方法是有效的. 关键词： 混沌系统 Lorenz系统 变结构控制 滑动流形     

基于Foliation条件的离散动力系统二维流形计算

研究离散动力系统双曲不动点的二维流形计算,利用不变流形轨道上Jacobian矩阵能够传递导数这一特殊性质,提出一种新的一维流形计算方法,通过预测-校正两个步骤迅速确定流形上新网格点,避免重复计算,并简化精度控制条件.在此基础上,将基于流形面Foliation条件进行推广,推广后的Foliation条件能够控制二维流形上的一维子流形的增长速度,从而实现二维流形在各个方向上的均匀增长.此外,算法可以同时用于二维稳定和不稳定流形的计算.以超混沌三维Hénon映射和具有蝶形吸引子的Lorenz系统为例验证了算法的有效性.     

基于自适应因子轨道延拓法的不变流形计算

提出自适应因子和轨道延拓相结合的二维流形计算方法,利用以平衡点为中心的椭圆对局域流形的近似,通过轨道的等距延拓和椭圆初始点的自适应调节,在精度要求下自适应的添加轨道,完成二维双曲不变流形的计算.此方法比"轨道弧长法"精度高,包含更多细节信息;同时要比"盒子细分法"更能反映流形的延拓趋势.     

基于Stiefel流形的粒子滤波器研究

为了解决粒子滤波的粒子退化和粒子多样性丧失问题,提出了一种基于Stiefel流形的粒子滤波算法.该算法将系统模型置于Stiefel流形上,用朗之万分布描述过程转移概率分布,用矩阵正态分布表示似然函数分布,在流形分布上进行粒子采样.在计算加权粒子的均值时,将流形嵌入到欧氏空间中,先计算欧氏空间中的粒子均值,再将计算结果投影到嵌套流形上,这就排除了噪声统计特性对粒子权重方差的影响,得到了一种受系统状态模型限制较少的重要性概率密度函数通用选择方案.仿真时选取单变量非静态增长模型,仿真结果验证了该算法的实时性、鲁棒性,滤波精度和滤波效率均比无味粒子滤波算法更好.     

广义Birkhoff自治系统平衡状态的流形稳定性

研究广义Birkhoff自治系统平衡状态流形稳定性.建立广义Birkhoff自治系统的受绕运动方程和平衡方程.由Liapunov稳定性理论给出广义Birkhoff自治系统的平衡状态流形稳定性的有关判据.举例说明结果的应用.     

基于线性嵌入和张量流形的高光谱特征提取

为了使降维结果更好地体现高光谱数据的空间结构信息,并进一步提高分类精度,提出了一种基于线性嵌入和张量流形的高光谱特征提取算法。不同于其他流形结构的表达方法,所提算法采用协同表示理论求解全局线性嵌入的权重矩阵,更有利于保持高维数据的全局信息,提高了流形结构表达的准确性。同时,建立了基于多特征描述的张量流形降维框架,得到的显式映射具有较强的可靠性和全局适应性。实验结果表明:与主成分分析、局部线性嵌入、拉普拉斯特征映射和线性保留投影等算法相比,所提算法表现出了更优越的分类性能。     

重离子碰撞中的热化、膨胀和径向流

在BUU方程框架下讨论中能重离子碰撞过程中的热华、膨胀和径向流.低能时,由四极矩和密度分布的计算知道系统的整体平衡可以达到.对于能量较高时理论计算表明,系统的整体平衡被破坏,但局域平衡可以达到.同时计算还指出重离子碰撞过程 中存在着径向膨胀.在Skyrme有效相互作用下,讨论了状态方程对径向流的影响,证明径向流对状态方程中有效质量较灵敏,对压缩系数不灵敏.     

Slow integral manifolds for Lagrangian fluid dynamics in unsteady geophysical flows

The authors consider Lagrangian motion of fluid particles in unsteady gravity currents in geophysical flows. The vertical motion of fluid particles, especially the induced vertical mixing in these currents, is partially responsible for the ocean thermohaline circulation, and thus plays a role in the global climate dynamics.First, a reduced dynamic system for slow variables is derived for a nonautonomous multiscale system. The reduced system, still nonautonomous, is the original system restricted to a centre-like nonautonomous invariant manifold (so-called slow manifold) which holds slow motions of the system. An algorithm is also presented to obtain an approximation of the nonautonomous slow manifold. A novelty here is that the reduction principle applies to nonautonomous multiscale systems which satisfy conditions that are true only locally in space (as in many physical cases). This makes the reduction principle applicable to real physical systems.Then, this invariant manifold reduction principle is applied to an approximate conceptual Lagrangian model of gravity currents and a reduced nonautonomous system for slow vertical motion is obtained. This reduced system may be useful as a conceptual tractable tool for understanding some features of vertical mixing in unsteady gravity currents.     

二维不稳定流形的计算

提出了动力系统中稳定流形和不稳定流形的一种实用的快速算法,可以求得稳定流形和不稳定流形的直观图像,从而从几何角度研究动力系统的动态行为和稳定性区域的边界特征.算法由两步构成:①在不稳定流形上求得一些分布均匀的点,以精确反映流形的每个细节;②借助三角形剖分或二维单纯形剖分利用①的算法将这些点画出直观流形图像.     

The dynamics of the k–ϵ mix model toward its self-similar Rayleigh–Taylor solution

We propose an analysis of the non-linear system of partial differential equations for the k–? model expressing the evolution of a turbulent mixing zone induced by the Rayleigh–Taylor instability. The method developed in this work is based on dynamical system theory. Our objective is to prove the global stability of the self-similar solution and at the same time to investigate the dynamics of transient phases. In fact, it is possible to show the existence of a central manifold allowing to reduce the dimension of the problem to a set of two ordinary differential equations.We establish that this simplified non-linear system globally converges toward a fixed point representing the self-similar solution by application of the Poincaré–Bendixson theorem. In addition, we shed light on the existence of a second fixed point which influences the trajectories in the phase space and leads to a non-physical enhanced growth rate in some cases explicitly detailed.     

Adaptively locating unknown steady states: Formalism and basin of attraction

The adaptive technique, which includes both dynamical estimators and coupling gains, has been recently verified to be practical for locating the unknown steady states numerically. This Letter, in the light of the center manifold theory for dynamical systems and the matrix spectrum principle, establishes an analytical formalism of this adaptive technique and reveals a connection between this technique and the original adaptive controller which includes only the dynamical estimator. More interestingly, in study of the well-known Lorenz system, the selections of the estimator parameters and initial values are found to be crucial to the successful application of the adaptive technique. Some Milnor-like basins of attraction with fractal structures are found quantitatively. All the results obtained in the Letter can be further extended to more general dynamical systems of higher dimensions.     

一种二维不稳定流形的新算法及其应用

提出了连续时间系统二维(不)稳定流形的一种新数值算法,不但可以快速地求得流形的直观图像,而且能够准确地获取流形上各点的位置、时间、轨道距离等丰富的信息,从而有利于人们从几何上去研究系统的全局行为,如边界特征、演化过程、奇异环等等.本算法首先通过解初值问题求出均匀分布的相邻轨道,然后连接这些轨道既得流形面.Lorenz系统原点的稳定流形的计算表明本算法快速有效.此外,通过试着寻找异宿轨道,还研究了一个三维神经网络中的混沌产生机理.     

基于广义Foliation条件的非线性映射 二维流形计算

主要研究非线性映射函数双曲不动点的二维流形计算问题. 提出了推广的Foliation条件, 以此来衡量二维流形上的一维流形轨道的增长量, 进而控制各子流形的增长速度, 实现二维流形在各个方向上的均匀增长. 此外, 提出了一种一维子流形轨道的递归插入算法, 该算法巧妙地解决了二维流形面上网格点的插入、前像搜索, 以及网格点后续轨道计算问题, 同时插入的轨道不必从初始圆开始计算, 避免了在初始圆附近产生过多的网格点. 以超混沌三维Hénon映射和具有蝶形吸引子的Lorenz系统为例验证了算法的有效性.     

Pitchfork bifurcation and circuit implementation of a novel Chen hyper-chaotic system

In this paper,a novel four dimensional hyper-chaotic system is coined based on the Chen system,which contains two quadratic terms and five system parameters.The proposed system can generate a hyper-chaotic attractor in wide parameters regions.By using the center manifold theorem and the local bifurcation theory,a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point.Numerical analysis demonstrates that the hyper-chaotic system can generate complex dynamical behaviors,e.g.,a direct transition from quasi-periodic behavior to hyper-chaotic behavior.Finally,an electronic circuit is designed to implement the hyper-chaotic system,the experimental results are consist with the numerical simulations,which verifies the existence of the hyper-chaotic attractor.Due to the complex dynamic behaviors,this new hyper-chaotic system is useful in the secure communication.     

Pitchfork bifurcation and circuit implementation of a novel Chen hyper-chaotic system

In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-chaotic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-chaotic system is useful in the secure communication.     

Generalized synchronization of two unidirectionally coupled discrete stochastic dynamical systems

The existence of two kinds of generalized synchronization manifold in two unidirectionally coupled discrete stochastic dynamical systems is studied in this paper. When the drive system is chaotic and the modified response system collapses to an asymptotically stable equilibrium or asymptotically stable periodic orbit, under certain conditions, the existence of the generalized synchronization can be converted to the problem of a Lipschitz contractive fixed point or Schauder fixed point. Moreover, the exponential attractive property of generalized synchronization manifold is strictly proved. In addition, numerical simulations demonstrate the correctness of the present theory. The physical background and meaning of the results obtained in this paper are also discussed.%vspace1mm