用迭代法绘制高阶M—J分形混沌图

介绍用迭代法绘制高阶M-J分形混沌图的原理，方法及分形图的特征，并用计算机模拟。     

Newton分形的原理及Matlab实现

详细推导了复平面上Newton迭代法的原理和计算公式,用MATLAB编制程序实现了Newton迭代算法,得到了一些奇异、绚丽的分形图形。对《数学实验》课程有一定的参考价值。     

具有平面晶体对称性混沌吸引子的分形极限图

提出用混沌函数构造具有平面晶体群对称性混沌吸引子的上半平面、方极限和圆极限分形图的简便方法.通过确定单个混沌吸引子所在的基本区域,利用同胚仿射变换,把基本区域中的混沌吸引子自相似地映射到上半平面,进而生成方极限、圆极限分形图.这些图案具有很好的艺术渲染力.该方法使单个混沌吸引子保持几何结构不变,但图形在整体上却具有很好分形自相似的性质.     

混沌分形艺术与产品设计实践

运用数学和计算机等现代科学技术手段生成的混沌分形图除了给人带来视觉上的美感享受外,也使人们产生了在产品设计中应用这些图案的愿望。文章在构造具有鲜明艺术特征的混沌吸引子、分形图和双曲极限圆图案基础上,将混沌分形图案作为装饰图案,结合案例探索了在产品设计中的应用实践。     

Newton迭代及其生成的3D分形图象

探讨Newton求根迭代及其生成的3D分形图象。采用轨迹井的跟踪和调色板技术来生成具有3D效果的分形图象,这类分形有很漂亮的显示,并且生成算法简单,快速。     

Sierpinski三角形分形图及其推广

论文通过对Sierpinski三角形的随机陛迭代算法进行分析，进而推广到多边形分形图，并给出了用VB语言实现的通用程序。     

分形自相似性实现技术的研究

分形理论是数学与计算机相结合的产物,本文主要研究分形图形自相似性的实现方法与技术,通过采用面向对象的方法,对分形中常见的算法实现的图形进行放大,实现了包括Mandelbrot集、Julie集、以及Newton Raphson迭代的放大技术,并根据放大情况自动调节迭代次数,对分形自相似性的研究提供实现方法,并在VC环境中...     

基于随机聚类的结肠镜图像分割技术的研究

针对彩色结肠镜图像，提出一种融合颜色、亮度、空间距离和纹理信息，采用随机聚类的彩色图像分割新算法。该算法采用分形维作为图像纹理特征的度量，用一种基于元素间的相似性的随机聚类方法对特征空间进行聚类。该聚类算法是基于对图中的切割进行采样的新的图论算法，可以自动获得最佳的分类数目。     

基于分形几何的信息隐藏技术

本算法借鉴了信息隐藏的原理,以分形几何作为数学工具,利用Mandelbrot集迭代的混沌特性,通过将文字信息映射至原始图像中并改变图像的微小细节实现信息隐藏及加密过程。本算法将图像作为隐藏信息的载体,并结合了分形几何的特性,是一种全新的加密思路,具有较高的安全性,隐蔽性及实用性。     

基于分形特征和阈值分析的图像边缘检测

针对传统的边缘检测算法对噪声敏感且伪边缘较多,提出一种基于分形特征和阈值分析的图像边缘检测方法。该方法利用改进的毯覆盖算法计算出图像的分形特征,将图像的灰度分布映射到分形维数空间上,再根据提取的分形特征图进行阈值分析,获得高低两个阈值,将像素值分为非边缘、弱边缘和强边缘3类,再对弱边缘像素进一步加以判断。实验结果与其他算法相比较表明,该算法检测出来的图像边缘伪边缘和噪声最少。     

Chaotic interactions of self-replicating RNA

A general system of high-order differential equations describing complex dynamics of replicating biomolecules is given. Symmetry relations and coordinate transformations of general replication systems leading to topologically equivalent systems are derived. Three chaotic attractors observed in Lotka-Volterra equations of dimension n = 3 are shown to represent three cross-sections of one and the same chaotic regime. Also a fractal torus in a generalized three-dimensional Lotka-Volterra Model has been linked to one of the chaotic attractors. The strange attractors are studied in the equivalent four-dimensional catalytic replicator network. The fractal torus has been examined in adapted Lotka-Volterra equations. Analytic expressions are derived for the Lyapunov exponents of the flow in the replicator system. Lyapunov spectra for different pathways into chaos has been calculated. In the generalized Lotka-Volterra system a second inner rest point--coexisting with (quasi)-periodic orbits--can be observed; with an abundance of different bifurcations. Pathways from chaotic tori, via quasi-periodic tori, via limit cycles, via multi-periodic orbits--emerging out of periodic doubling bifurcations--to "simple" chaotic attractors can be found.     

基于神经网络误差修正的混沌控制与同步

针对混沌系统非线性强、多变量耦合等特点,提出了一种基于神经网络误差修正的自适应多变量混沌系统的广义预测控制算法,用线性广义预测控制器控制混沌系统,用神经网络对模型预测误差进行修正。算法中辩识过程模型用递推最小二乘法(RLS)、神经网络权值用Davidon最小二乘法(DLS)训练。这种算法对被控混沌系统的先验知识要求较少,无需知道被控系统的精确模型,数值仿真显示可实现混沌系统的宽范围控制与同步。     

基于重根牛顿迭代法和比较算法实现的分形图形的研究

介绍了一种运用重根牛顿迭代法在复数范围内进行反复迭代运算求根,然后根据求根的结果采用比较的方法绘制分形图形的算法,并从数学上验证了算法的几何意义。绘制出的分形图优美而玄妙,几何意义明显。同时该算法可以绘制出高次数根值复杂的分形图形。     

Fourth-order derivative-free methods for solving non-linear equations

Two new one-parameter families of methods for finding simple and real roots of non-linear equations without employing derivatives of any order are developed. Error analysis providing the fourth-order convergence is given. Each member of the families requires three evaluations of function per step, and therefore the method has an efficiency index of 1.587. Numerical examples are presented and the performance of the method presented here is compared with methods available in the literature.     

Global analyses of crisis and stochastic bifurcation in the hardening Helmholtz-Duffing oscillator

Crisis and stochastic bifurcation of the hardening Helmholtz-Duffing oscillator are studied by means of the generalized cell mapping method using digraph.For the system subject to a single deterministic harmonic excitation,our study reveals that a series of crisis phenomena can occur when the system parameter passes through different critical values,including chaotic boundary crisis,regular boundary crisis and interior crisis.A chaotic boundary crisis due to the collision of regular attractor with chaotic s...     

Meta-Patterns and Higher Order Meta-Patterns in Cellular Systems

A new method for function approximation and classification in cellular systems is developed. The solution of a function approximation or a classification problem does not correspond to a connection matrix, but to a pattern in a cellular automaton. The pattern is obtained by a procedure of chaotic growth. It acts as a map between input- and ouput-patterns, and in this sense it is a meta-pattern. It is shown that problems with symmetry are typically solved by fractal patterns with aesthetic attractiveness. Also meta-meta-patterns are introduced, which are patterns that solve sets of problems instead of a single problem. Such patterns specify sets of sets of couples of input- and ouput-patterns. This schema is generalized straightforwardly for higher order meta-patterns.     

环形天线结构的复杂动力学研究

部分改进高维系统的广义Melnikov方法研究含参非线性动力系统的混沌问题,并应用于研究环形天线结构的混沌运动等复杂非线性动力学行为.通过定义恰当的横截面,发展适用于研究五维含参非线性动力系统的Melnikov方法,获得系统发生Smale马蹄意义下混沌运动区域及判定定理.将所得理论结果应用于研究面内激励与横向激励共同作用下环形天线结构的混沌运动,得到系统发生混沌运动的不稳定区域及相应的参数控制条件.探讨阻尼系数、参数激励对系统动力学行为的影响,并利用数值模拟方法给出其相图构型,验证理论结果的正确性.     

Chaos induced by the generalized Euler method

In this paper, bifurcation and the chaotic phenomenon in numerical computation are investigated using the generalized Euler method. In this method, the forward and backward Euler methods and the trapezoidal method can be represented as special cases. It will be made clear that, for some computation step size, the solution of the difference equation does not always approximate to the solution of the original differential equation; period-doubling bifurcations and the chaotic phenomenon occur in the solution of the difference equation. Two examples are shown to illustrate the effect of step size.     

环境干扰下分形生长的定方位控制

针对复杂无序的实际分形生长系统,研究了环境干扰下分形生长的定方位控制模型．不同于以往对分形生长的定方位的研究仅限于对它的实验方面的讨论,本文利用范数理论得到了干扰项为广义函数时控制生长粒子凝聚到不同方位的定量方法．利用所提出的控制方法解决了干扰项为更一般函数形式的定方向生长问题,并以非线性项为二次多项式、正弦函数为例,运用所得的定量关系对该系统不同作用区域的生长粒子进行了控制,仿真结果表明了该控制方法的有效性．