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.     

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.     

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.     

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.     

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     

基于二阶离散变分方法的非线性映射参数估计

提出一种估计非线性映射未知参数的二阶离散变分方法.首先针对非线性离散混沌系统, 利用变分方法导出了伴随方程和目标泛函梯度, 以此为基础利用二阶离散变分方法给出了二阶伴随方程和精确计算Hessian矩阵-向量乘积的显式表达式; 其次设计了估计非线性映射未知参数的新算法, 并以此对Hyperhenón映射和二维抛物映射中的未知参数进行了精确的估计. 数值仿真结果表明了该方法的有效性和优点. 关键词： 非线性映射 参数估计 二阶离散变分方法 伴随方程     

Duffing单边碰撞系统的混沌鞍合并激变

以典型的Duffing单边碰撞系统为研究对象,对系统中的混沌鞍进行了细致的分析.研究表明,系统的混沌鞍同样存在合并激变,合并激变是由连接两个混沌鞍的周期鞍的稳定流形与不稳定流形相切所诱发,相切使得边界上的混沌鞍与内部的混沌鞍发生碰撞而突然合并为一个较大的边界混沌鞍.混沌鞍的合并激变行为最终会诱导混沌吸引子的合并激变发生. 关键词： Duffing碰撞系统 混沌鞍 周期鞍 稳定与不稳定流形     

混沌吸引子的对称破缺激变

许多非线性动力系统都有某种对称性,在不同情形下可有不同的表现形式,但始终保持其对称的特点.不同对称形式间的转变导致对称破缺分岔或激变.关于非线性动力系统中相空间运动轨道的对称破缺分岔,已有大量研究工作,但绝大多数是指周期或拟周期相轨的对称破缺,偶尔提到对称系统中的混沌相轨也存在“对偶性”.最近,在简谐外激Duffing系统周期轨道对称破缺引发鞍-结分岔的研究中,得到了分岔后由Poincaré映射点间断流构成的图像,其中包括两个稳定周期结点、一个周期鞍点,及其稳定流形与不稳定流形,均较规则.本工作研究了正弦 关键词： 对称破缺 混沌 激变 分形吸引域     

截断误差导致的非双曲不动点邻域拓扑变异

提供一个关于截断误差使简单系统复杂化的直接实验证据,以此证明存在混沌抗退化机理.分别构造了一个一维圆弧迭代系统和一个一维抛物线迭代系统,两者均有一个非双曲不动点,其迭代序列被证明是简单极限序列,数字计算实验显示这两个迭代系统都存在可以越过不动点的序列.采用计算"元胞"分析方法清晰地展示了截断误差导致非双曲不动点邻域拓扑变异:形成第I类阵发混沌通道,或产生纹波分岔.     

非线性系统混沌运动的神经网络控制

设计前馈反传神经网络控制非线性系统混沌运动的新方法.根据扰动参数模型输入输出数据,按照非线性学习算法训练网络产生系统稳定所需的小扰动控制信号,去镇定混沌运动,使嵌入在混沌吸引子中的不稳定周期轨道回到稳定不动点上.Hnon映射数值仿真结果表明,这种方法控制非线性混沌系统响应速度快、控制精度高 关键词： 混沌控制 神经网络 吸引子 非线性     

Heteroclinic bifurcations and chaotic transport in the two-harmonic standard map

We study a two-parameter family of standard maps: the so-called two-harmonic family. In particular, we study the areas of lobes formed by the stable and unstable manifolds. Variational methods are used to find heteroclinic orbits and their action. A specific pair of heteroclinic orbits is used to define a difference in action function and to study bifurcations in the stable and unstable manifolds. Using this idea, two phenomena are studied: the change of orientation of lobes and tangential intersections of stable and unstable manifolds.     

Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps

We study families of volume preserving diffeomorphisms in R(3) that have a pair of hyperbolic fixed points with intersecting codimension one stable and unstable manifolds. Our goal is to elucidate the topology of the intersections and how it changes with the parameters of the system. We show that the "primary intersection" of the stable and unstable manifolds is generically a neat submanifold of a "fundamental domain." We compute the intersections perturbatively using a codimension one Melnikov function. Numerical experiments show various bifurcations in the homotopy class of the primary intersections. (c) 2000 American Institute of Physics.     

Dynamic transitions through scattors in dissipative systems

Scattering of particle-like patterns in dissipative systems is studied, especially we focus on the issue how the input-output relation is controlled at a head-on collision where traveling pulses or spots interact strongly. It remains an open problem due to the large deformation of patterns at a colliding point. We found that a special type of unstable steady or time-periodic solutions called scattors and their stable and unstable manifolds direct the traffic flow of orbits. Such scattors are in general highly unstable even in the one-dimensional case which causes a variety of input-output relations through the scattering process. We illustrate the ubiquity of scattors by using the complex Ginzburg-Landau equation, the Gray-Scott model, and a three-component reaction diffusion model arising in gas-discharge phenomena.     

Invariant curves,attractors, and phase diagram of a piecewise linear map with chaos

We introduce equations describing the invariant curves associated with periodic points in a wide class of two-dimensional invertible maps, which in the special case of the mapT(x, z)=(1?a¦x¦+bz,x) can be solved by analytical methods. In the dissipative case several branches of the separatrices of the fixed points, as well as, of the period-2 and -4 points, are constructed. The regions of the parameter space where a given type of strange attractor exists are located. We point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period-2 points and the stable manifold of the fixed point may occur separately, and the latter leads already to the appearance of a two-piece strange attractor. This phenomenon may happen at weak dissipation in other maps, too. In the conservative caseb=1 separatrices and certain invariant tori are calculated.     

Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Henon map as an example

The subject of this paper is the construction of the exponential asymptotic expansions of the unstable and stable manifolds of the area-preserving Henon map. The approach that is taken enables one to capture the exponentially small effects that result from what is known as the Stokes phenomenon in the analytic theory of equations with irregular singular points. The exponential asymptotic expansions were then used to obtain explicit functional approximations for the stable and unstable manifolds. These approximations are compared with numerical simulations and the agreement is excellent. Several of the main results of the paper have been previously announced in A. Tovbis, M. Tsuchiya, and C. Jaffe ["Chaos-integrability transition in nonlinear dynamical systems: exponential asymptotic approach," Differential Equations and Applications to Biology and to Industry, edited by M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme (World Scientific, Singapore, 1996), pp. 495-507, and A. Tovbis, M. Tsuchiya, and C. Jaffe, "Exponential asymptotic expansions and approximations of the unstable and stable manifolds of the Henon map," preprint, 1994]. (c) 1998 American Institute of Physics.     

New type of heteroclinic tangency in two-dimensional maps

A new mechanism of heteroclinic tangency is investigated by using two-dimensional maps. First, it is numerically shown that the unstable manifold from a hyperbolic fixed point accumulates to the stable manifold of a nearby period-2 hyperbolic point in a piecewise linear map and that the unstable manifold from a hyperbolic fixed point accumulates to the accumulation of the stable manifold of a nearby period-2 hyperbolic point in a cubic map. Second, a theorem on the impossibility of heteroclinic tangency (in the usual sense) is given for a particular type of map. The notions ofdirect andasymptotic heteroclinic tangencies are introduced and heteroclinic tangency is classified into four types.     

On a simple recursive control algorithm automated and applied to an electrochemical experiment

We review a simple recursive proportional feedback (RPF) control strategy for stabilizing unstable periodic orbits found in chaotic attractors. The method is generally applicable to high-dimensional systems and stabilizes periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The goal of the control scheme is the fixed point itself rather than a stable manifold and the controlled system reaches the fixed point in d+1 steps, where d is the dimension of the state space of the Poincare map. We provide a geometrical interpretation of the control method based on an extended phase space. Controllability conditions or special symmetries that limit the possibility of using a single control parameter to control multiply unstable periodic orbits are discussed. An automated adaptive learning algorithm is described for the application of the control method to an experimental system with no previous knowledge about its dynamics. The automated control system is used to stabilize a period-one orbit in an experimental system involving electrodissolution of copper. (c) 1997 American Institute of Physics.     

Dynamical ordering of symmetric non-Birkhoff periodic points in reversible monotone twist mappings

We study the coexistence of symmetric non-Birkhoff periodic orbits of C(1) reversible monotone twist mappings on the cylinder. We prove the equivalence of the existence of non-Birkhoff periodic orbits and that of transverse homoclinic intersections of stable and unstable manifolds of the fixed point. We derive the positional relation of symmetric Birkhoff and non-Birkhoff periodic orbits and obtain the dynamical ordering of symmetric non-Birkhoff periodic orbits. An extension of the Sharkovskii ordering to two-dimensional mappings has been carried out. In the proof of various properties of the mappings, reversibility plays an essential role. (c) 2002 American Institute of Physics.     

Intersection properties of invariant manifolds in certain twist maps

We consider the spaceN ofC ² twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem). In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.