广义Mandelbrot和Julia组合集

Based on the switched mapping advanced by autbor,the method constructing the general Mandelbort and Julis combination sets was elaborated,and a series of the general Mandelort and Julis combination sets were constructed.The trajectories of a starting point in the complex z-plane under switch mapping were analyzed,the construction characteristics of the general Mandelbort and julia combination sets were described.The algorithm constructing the general Mandelbort and Julia combination sets was researched,the conclusions that the evoutions of the general Mandelbort and Julia combination sets rely on the choice of principal range for the phase angle were given.     

The Computational Complexity of Some Julia Sets

Numerous computer programs have been written to compute sets of points which approximate Julia sets [4]. Usually, no error estimations are added so that it remains unclear, how good such approximations are. Furthermore, high precision pictures are unreliable because of rounding errors, since the realizing computer programs use fixed length floating point numbers. Computable error estimation w.r.t. the Hausdorff metric d_H means that the set is recursive [10]. Many Julia sets J are recursive [11]. Recursive compact subsets of the Euclidean plane have a computable Turing machine time complexity [10]. In this paper we prove that the Julia set of a complex function f(z) = z² + c for c < 1/4 can be computed locally in time O(k²M(k)) (where M(k) is a time bound for multiplication of k-bit integers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide for a single point whether it belongs to a grid K_k (2^−k · )² such that d_H(K_k,J) ≤ = 2^−k.     

构造球面对称充满Julia集

为连续构造球面上具有多吸引周期轨道的充满Julia集图形,提出了一个关于构造任意投影面的半球面图形的算法.首先自动挑选可使动力系统迭代轨道的平均离散速度指标值L＜0的参数;其次,用建立在以球心为原点的坐标系下的迭代映射,通过坐标变换,跟踪与任意坐标系相应的投影面的上半球面上各点的迭代轨道,搜索出半球面上所有互不相同的吸引周期轨道,并建立轨道链表;进一步计算投影面的上半球面各点的迭代轨道到达吸引周期轨道所需要的迭代次数,根据迭代次数为球面上的相应点指定颜色并构造出半球面上的对称充满Julia集图形.研究结果表明,采用该算法可以大量生成对称的球面充满Julia集图形.     

一类复指数映射的复合的广义M-J集

一种常见的构造分形图象的方法是利用复映射的迭代,所使用的一般是复幂映射和复指数映射.Shirriff 在文献[1]中描述了一种利用两个复映射的复合构造分形图象的方法,不过在那里只使用了两个简单的复多项式映射.本文将Shirriff的方法进行了推广,构造了两个复指数映射的复合的广义M集和广义J集,证明了广义M集和广义J集的一些性质.通过大量的计算机实验,总结了广义M集和广义J集的演化规律,并从理论上分析了当两个复映射的参数变化时,广义J集的变化规律.     

Optimal Control and Synchronization of Alternated Julia Sets

In this paper, spatial‐alternated Julia sets are discussed, which are obtained by alternating the spatial‐alternated system, and plane‐alternated Julia sets are discussed, which are obtained by alternating the plane‐alternated system. The control of spatial‐alternated Julia sets and plane‐alternated Julia sets is achieved via the optimal control. Moreover, the synchronization of two different plane‐alternated Julia sets is implemented via the optimal control. The simulations illustrate the effectiveness of the control method.     

基于Newton迭代法的Julia集的生成方法

介绍了Newton迭代法的基本思想及其改进方法，研究了用于复多项式的Cayley问题，并且从理论上讨论了Newton迭代法在Julia集研究中的重要作用，给出了用Newton迭代代法和成Julia集的计算机快速算法。     

Control of Julia Sets in Generalized Alternated System

In this paper, we first introduce the generalized alternated system. The definition of the Julia set in the generalized alternated system is given, which is called a generalized alternated Julia set. Then, we achieve the control of generalized alternated Julia sets by applying the classic control methods, which are gradient control and optimal control. In addition, the synchronization between two different generalized alternated Julia sets is implemented using gradient control and optimal control. The simulations illustrate the effectiveness and correctness of these two control methods, and the results are displayed in 2D computer graphics.     

Circles and Periodic Points in Quaternionic Julia Sets

In this paper, the quadratic family fQ(X) X2 + Q is considered, where Q is a quaternionic parameter. Iteration of this quadratic function gives rise to a discrete dynamical system, its Julia set being the boundary of the basin of attraction of infinity. Three interesting types of circles appear in this Julia set.     

The Effect of System Stochastic Noise on SINS Initial Alignment

捷联惯导系统初始对准系统状态方程所描述的状态统计信息,受系统随机噪声驱动,系统随机噪声通过噪声驱动阵作用于状态。常用系统状态方程的系统噪声驱动阵为单位阵。经过分析发现系统噪声驱动阵为单位阵是有条件的;通过推导给出了系统状态方程可以简化此种形式的条件;指出了在水平陀螺、加速度计随机噪声水平不同时,若使用简化系统模型会引起状态估计误差;并通过单轴旋转多位置对准仿真实验进行了验证。仿真结果表明：若水平陀螺随机噪声水平不同,采用简化模型进行初始对准会引起状态估计误差。     

基于三维多项式映射的广义Julia集表示与绘制

研究了基于三维多项式映射的三维广义Julia集表示方法.从理论上分析并证明了三维多项式映射满足等变的条件,精确地给出了关于正四面体群和正八面体群具有旋转不变对称性的两类三维等变映射的具体公式,在此基础上讨论并证明了三维多项式映射的广义Julia集所具有的性质.提出了基于逃选距离色彩调配的光线跟踪体绘制算法,对给定三维空间中属于Julia集的离散点根据其逃逸距离赋予颜色和不透明度,并采用光线跟踪法进行体绘制.实验结果表明,利用三维多项式映射来构造三维Julia集,不仅可以根据映射的性质预知Julia集的总体结构特征,并且能够通过调控映射的参数来获得多种具有不同旋转对称结构的Julia集,因而有效地克服了现有三维分形集生成方法所构造的分形集包含信息量少、形状结构单一和分形形状无法预测等缺陷.进一步地,三维多项式映射可以应用于其他三维分形的构造,从而为三维分形的生成提供一个新的有效途径.     

负实数阶广义J集的演化

在阐述由复映射z←zα+c(α<0)所构造的广义Julia集(简称广义J集)定义的基础上,通过改变参数α,作出了一系列负实数阶的广义J分形图,当α为负整数时,广义J分形图呈现为一个由|α|个卫星群环绕中央行星的星群结构;而当α为负小数时,广义J分形图中则出现尺寸与α的小数部分成比例的部分卫星群.同时利用复变函数理论和计算机制图相结合的实验数学方法,研究了广义J集的分形结构特征及其演化过程,进而发现相角θ范围的不同选取导致了广义J集的不同演化,并首次给出了广义J集的4种演化过程.     

Stochastic Processes for Bounded Noise

The scalar differential inclusionx f(x) + g(x) u, u [-1,1], x(0) = x ₀ (0.1) is considered as a model of the dynamical system x = f(x) perturbed by the bounded noise g(x)u, u [-1,1], and the problem of constructing a nontrivial probability measure on the set {\cal S} of solutions to (0.1) is studied. In particular, it is shown that: (i) every Markov process whose probability measure is supported on {\cal S} is degenerate, in a sense to be specified (see Theorem 3.1); (ii) given a flow of probability measures _t on the reachable sets R _t of (0.1), satisfying a certain compatibility condition, a Markov process X _t is constructed such that its marginals are exactly _t and (0.1) is satisfied from one side (see Theorem 4.1); its finite-dimensional distributions are computed and the regularity of its sample paths is investigated (see Section 5.2); (iii) given a process of a type previously considered, another process Y _t is constructed through its finite-dimensional distributions, and its distribution is shown to be supported exactly on {\cal S}. Finally, a model example is considered (see Section 7). Date received: July 18, 2000. Date revised: July 18, 2002. The work of G.C. was supported by EC Grant ERB-CIPA-CT-93-1554; that of V.K. by EC Grant ERB-CIPA-CT-92-0370 and GA R 201/98/0227. The stay of V.K. and I.V. at the Faculty of Biological Sciences was supported by MMT Grant No. VS96086. I.V. was partly supported by GA R Grant No. 201/95/0629.     

正实数阶广义J集内部结构的探讨

1 引言复映射f:z←z~α+c(α=2)对不同的c值(c∈C),经过迭代能生成各种形状奇特的分形,这些集合被称为Julia集。而如果根据不同的c值对应的Julia集的连通性对参数c进行分类,还可在参数空间做出称为Mandelbort集的c的点集。目前人们对Julia集已进行了深入研究,发现其中深藏着规律性的结构,从而大     

D4对称平面排列映射广义充满Julia集

为了更加直观、有效地在参数空间挑选参数构造出具有D4对称特性的平面排列映射的混沌吸收子和广义充满Julia集，在参数空间任选两个实参数构造参数断面，构造其上的广义M集。在这种广义M集的周期区域中挑选参数，可以由计算机生成大量新颖的广义充满Julia集。为了揭示出这种广义充满Julia集内部的复杂结构，给出了两种构造方法。为具有平面对称特性的动力系统的计算机图形化研究工作增添了新形式的艺术图像。     

Fast Generation of Approximate Blue Noise Point Sets

Poisson‐disk sampling is a popular sampling method because of its blue noise power spectrum, but generation of these samples is computationally very expensive. In this paper, we propose an efficient method for fast generation of a large number of blue noise samples using a small initial patch of Poisson‐disk samples that can be generated with any existing approach. Our main idea is to convolve this set of samples with another to generate our final set of samples. We use the convolution theorem from signal processing to show that the spectrum of the resulting sample set preserves the blue noise properties. Since our method is approximate, we have error with respect to the true Poisson‐disk samples, but we show both mathematically and practically that this error is only a function of the number of samples in the small initial patch and is therefore bounded. Our method is parallelizable and we demonstrate an implementation of it on a GPU, running more than 10 times faster than any previous method and generating more than 49 million 2D samples per second. We can also use the proposed approach to generate multidimensional blue noise samples.     

有色噪声扰动下的随机控制问题研究

本文研究了有色噪声扰动下的线性二次随机控制问题,探讨了有色噪声与白色噪声随机控制问题的等效化方法,利用准最优性能指标综合了准最优输出反馈控制,最后,利用原始系统的系数矩阵导出了带有普遍意义的准最优控制算法框图.     

双稳随机共振系统信号调制噪声效应用于弱信号检测

通过对双稳系统随机共振模型的数值分析，得出在双稳系统输出信号中，有一个正弦信号成分和一个表现为维纳过程的噪声成分分别与输入的正弦信号和白噪声相对应。通过选择合适的系统参数，可以减小系统输出中信号和噪声之间的耦合效应。该系统可以大大抑制噪声，并在双稳系统中产生信号调制噪声效应。然后对双稳系统的输出信号作功率谱分析。不但可以辨识出淹没在白噪声中的微弱正弦信号的频率，还可以较精确地估算出微弱正弦信号的幅值。数值仿真表明，双稳系统的信号调制噪声效应可用于多个微弱正弦信号的检测。     

基于模糊集的数字图像脉冲噪声滤除

模糊集理论是基于人的认知特性的,以模糊集理论为基础的图像处理算法具有较强的智能性和鲁棒性.本文使用隶属度对像素的污染程度进行刻画,从而得到一个基于模糊集的数字图像脉冲噪声滤波算法.与现有的一些经典算法相比,本文算法的滤波效果更好.最后通过实验仿真表明了该算法的可行性和有效性.     

非周期随机共振抑噪应用研究

研究了噪声和随机二进制信号同时激励双稳系统时的输出响应，并测量了输入输出互相关系数。结果表明，随机共振并不限于周期信号，对于非周期信号也广泛存在。利用双稳系统中的非周期随机共振效应，可以减小随机信号传输中的噪声水平，改善输出信号质量，这在数字通信领域具有十分重要的意义。     

Fractal Dimension and Synchronization of the Controlled Julia Sets of a Reaction–Diffusion System

This paper is concerned with the fractal dynamics of a reaction–diffusion system, – the forced Brusselator model. The Julia set of the discrete version of the model is established. Then, the control of the Julia set is realized by combining the parameter perturbation control method and feedback control method. The box‐counting dimensions of the Julia sets of the controlled system for different control parameters are computed, which is used to describe the complexity and irregularity of the Julia sets. Finally, nonlinear coupling items are designed to make one Julia change to be another. The simulations illustrate the efficacy of these methods.