Cantor Type Fixed Sets of Iterated Multifunction Systems Corresponding to Self-Similar Networks

We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph models introduced by Barabási, Ravasz and Vicsek ([1] [2]). We generalize the graph models using stars and cliques: both algorithm construct graph sequences such that the next iteration is always based on n replicas of the current iteration, where n is the size of the initial graph structure, being a star or a clique. We analyze these self-similar graph sequences using IMSs in function of the size of the initial star and clique, respectively. Our research uses the Cantor set for the description of the fixed set of these IMSs, which we interpret as the limit object of the analyzed self-similar networks.

Simulation for chaos game representation of genomes by recurrent iterated function systems

Chaos game representation (CGR) of DNA sequences and linked protein sequences from genomes was proposed by Jeffrey (1990) and Yu et al. (2004), respectively. In this paper, we consider the CGR of three kinds of sequences from complete genomes: whole genome DNA sequences, linked coding DNA sequences and linked protein sequences. Some fractal patterns are found in these CGRs. A recurrent iterated function systems (RIFS) model is proposed to simulate the CGRs of these sequences from genomes and their induced measures. Numerical results on 50 genomes show that the RIFS model can simulate very well the CGRs and their induced measures. The parameters estimated in the RIFS model reflect information on species classification.

ERGODIC THEOREM FOR INFINITE ITERATED FUNCTION SYSTEMS

A set of contraction maps of a metric space is called an iterated function systems. Iterated function systems with condensation, can be considered infinite iterated function systems. Infinite iterated function systems on compact metric spaces were studied. Using the properties of Banach limit and uniform contractiveness, it was proved that the random iterating algorithms for infinite iterated function systems on compact metric spaces-satisfy ergodicity. So the random iterating algorithms for iterated function systems with condensation satisfy ergodicity, too.

AFFINE TRANSFORMATION IN RANDOM ITERATED FUNCTION SYSTEMS

Random iterated function systems (IFSs) is discussed, which is one of the methods for fractal drawing. A certain figure can be reconstructed by a random IFS. One approach is presented to determine a new random IFS, that the figure reconstructed by the new random IFS is the image of the origin figure reconstructed by old IFS under a given affine transformation. Two particular examples are used to show this approach.

SOME NEW ITERATED FUNCTION SYSTEMS CONSISTING OF GENERALIZED CONTRACTIVE MAPPINGS

Iterated function systems （IFS） were introduced by Hutchinson in 1981 as a natural generalization of the well-known Banach contraction principle. In 2010, D. R. Sahu and A. Chakraborty introduced K-Iterated Function System using Kannan mapping which would cover a larger range of mappings. In this paper, following Hutchinson, D. R. Sahu and A. Chakraborty, we present some new iterated function systems by using the so-called generalized contractive mappings, which will also cover a large range of mappings. Our purpose is to prove the existence and uniqueness of attractors for such class of iterated function systems by virtue of a Banach-like fixed point theorem concerning generalized contractive mappings.

Fixed Point Theorems of the Iterated Function Systems

In this paper, we present some fixed point theorems of iterated function systems consisting of α-ψ-contractive type mappings in Fractal space constituted by the compact subset of metric space and iterated function systems consisting of Banach contractive mappings in Fractal space constituted by the compact subset of generalized metric space, which is Mso extensively applied in topological dynamic system.

Iterated Function System-Based Crossover Operation for Real-Coded Genetic Algorithm

An iterated function system crossover (IFSX) operation for real-coded genetic algorithms (RCGAs) is presented in this paper. Iterated?function system (IFS) is one type of fractals that maintains a similarity characteristic. By introducing the IFS into the crossover operation, the RCGA performs better searching solution with a faster convergence in a set of benchmark test functions.

Observer-based Adaptive Iterative Learning Control for Nonlinear Systems with Time-varying Delays

An observer-based adaptive iterative learning control （AILC） scheme is developed for a class of nonlinear systems with unknown time-varying parameters and unknown time-varying delays. The linear matrix inequality （LMI） method is employed to design the nonlinear observer. The designed controller contains a proportional-integral-derivative （PID） feedback term in time domain. The learning law of unknown constant parameter is differential-difference-type, and the learning law of unknown time-varying parameter is difference-type. It is assumed that the unknown delay-dependent uncertainty is nonlinearly parameterized. By constructing a Lyapunov-Krasovskii-like composite energy function （CEF）, we prove the boundedness of all closed-loop signals and the convergence of tracking error. A simulation example is provided to illustrate the effectiveness of the control algorithm proposed in this paper.

An Exploration on Adaptive Iterative Learning Control for a Class of Commensurate High-order Uncertain Nonlinear Fractional Order Systems

This paper explores the adaptive iterative learning control method in the control of fractional order systems for the first time. An adaptive iterative learning control(AILC) scheme is presented for a class of commensurate high-order uncertain nonlinear fractional order systems in the presence of disturbance.To facilitate the controller design, a sliding mode surface of tracking errors is designed by using sufficient conditions of linear fractional order systems. To relax the assumption of the identical initial condition in iterative learning control(ILC), a new boundary layer function is proposed by employing MittagLeffler function. The uncertainty in the system is compensated for by utilizing radial basis function neural network. Fractional order differential type updating laws and difference type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. The hyperbolic tangent function and a convergent series sequence are used to design robust control term for neural network approximation error and bounded disturbance, simultaneously guaranteeing the learning convergence along iteration. The system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapnov-like composite energy function(CEF)containing new integral type Lyapunov function, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

Generalized Product of Iteration Function System

The definition of generalized product of fractal is first put forward for the research on the relations between original fractal and its product of fractal when the transformations of iteration function system （IFS） are incomplete. Then the representations of generalized product of IFS are discussed based on the theory of the product of fractal. Furthermore, the dimensional relations between the product of fractal and its semi-product are obtained. The dimensional relations of self-similar set are discussed. Finally, the examples for rendering fractal graphs are given. These results posses potentials in image compression and pattern recognition.

迭代函数系IFS吸引子的参数控制与树木的模拟

文章讨论了迭代函数系ＩＦＳ吸引子的参数控制，讨论了ＩＦＳ参数控制实现模拟树木的方法。

基于迭代函数系统IFS的动态树木模拟

植物作为自然景物中最常见的现象之一,模拟的方法是应用数学和图形学领域的一个重要课题。迭代函数系统IFS是分形理论的重要分支,由于植物结构的自相似性,利用IFS(Iterated Function System)可以逼真地模拟各植物形态,简述几种模拟植物的方法,主要研究迭代函数系统IFS模型,并在VC++6.0环境下基于IFS模型构造出静态蕨叶和树木,详细讨论利用带参量的IFS随机系统实现动画的过程,并利用双缓冲技术,形象逼真地模拟随风摇摆的蕨叶和生长树木的动画效果。实验结果表明,带参数的IFS可使图像发生预期的变化,如果让参数在适当的范围保持连续变化,则动画效果良好。

基于IFS块的快速图象编码算法

文中首先叙述了目前分形块编码研究领域的大致情况，然后，提出了一种新的快速编码算法．用Ｌ１距离替换了以往算法所用的Ｌ２距离，把匹配过程转换成类似于矢量量化的搜索过程，因而采纳了许多矢量量化的加速算法．还讨论了对平坦区的处理并提出了新的剖分方法，取得了较好的效果．同其它的分形块编码方法相比，该算法大大缩短了编码时间，改善了压缩图象的质量，特别是它能较大程度地消除块效应，压缩比也有了进一步的提高．

迭代函数系统IFS吸引子图像局部控制方法的研究

针对目前的 IFS吸引子随机迭代算法在 IFS吸引子图像控制中存在的不足 ,本文给出了两种 IFS吸引子随机迭代算法的改进方法 ,其方法可以对 IFS吸引子图像实现其局部细节与色彩的控制 。

非解析复映射构造IFS的参数研究

为了采用非解析复映射构造分形或奇怪吸引子,研究了复映射f(z)=e^(iπ/2)z^n+c的广义M集的1周期参数对构造非线性IFS的影响.在该复映射的M集1周期区域随机选取参数;根据M集的对称性,用与所选参数在M集中对称位置的参数构成迭代函数系;在动力平面上构造出迭代函数系中的所有迭代函数的充满Julia集以及它们的公共吸引域;将随机选出参数所构造出的迭代函数的吸引不动点作为初始迭代点,通过在迭代函数系中连续随机选取一个迭代函数,跟踪这个吸引不动点在动力平面上的公共吸引域内的迭代轨道.通过实验,找到了可以生成分形的非线性IFS的参数选取方法.结果表明:当n取不同值时,非解析复映射族f(z)=e^(iπ/2)z^n+c的广义M集的1周期参数可以用于构造非线性IFS,这种IFS可以大量生成分形山以及具有Z_(n+1)和D_(n+1)对称特性的新分形.

IFS分形吸引子的连续变形技术

针对迭代函数系统(IFS)分形吸引子变形时出现失真、断裂、特征退化等问题,分析现有变形技术的缺陷.在分形、数学等理论基础上,提出基于IFS的交互式分形造型生成算法,即点变换算法,以及点变换之间的相互转化(升级)技术,包括两点变换、三点变换(其二维情况为仿射变换)和四点变换(其二维情况为双线性变换).该算法解决了吸引子变形时的连通性问题,并在吸引子连续变化中采用对点交互式插值的方法获得高效快速的实时动画效果.

基于分形IFS的运动目标RCS序列分解

针对运动目标 Ｒ Ｃ Ｓ 变化规律，提出将目标 Ｒ Ｃ Ｓ 变化序列分解为确定变化序列（ Ｃｅｒｔａｉｎ Ｖａｒｉａｔｉｏｎ Ｓｅｑｕｅｎｃｅ， Ｃ Ｖ Ｓ） 和随机变化序列（ Ｒａｎｄｏｍ Ｖａｒｉａｔｉｏｎ Ｓｅｑｕｅｎｃｅ ， Ｒ Ｖ Ｓ） 的设想．其中 Ｃ Ｖ Ｓ 敏感于姿态，在姿态域表现为丰富的细节特征，而 Ｒ Ｖ Ｓ则表现为近似正态分布的非平稳随机过程．本文用实验数据证明这两种变化序列均具有分形的特征．由以上的分析结果，利用分形迭代函数系统将这两种变化序列进行有效分解，其分解效果明显优于传统的滤波方法．

IFS系统在植物形态模拟中的应用

IFS系统是模拟植物形态的一门重要技术。首先分析迭代函数系统中IFS码产生原理,阐述IFS如何模拟植物形态,然后依据拼贴定理,研究IFS码的变换特点,并利用IFS码的连续性探讨IFS码变换的规律及表现效果。另外通过增加IFS码,对分形图进行变换,逼真模拟森林、草地等自然景观。

三维分形树木IFS生成算法

给出了生成单轴分枝和合轴分枝两类树木的递归算法,经过计算机实验得出了单轴分枝和合轴分枝两类树木的IFS码,并进行了正确性证明和真实感加工.

改进的广义置信度自适应IFS图象压缩编码算法

通过对分块 IFS图象压缩编码过程中匹配误差的分析 ,提出了广义置信度的概念 ,并据此提出了对输入图象进行四叉树分块的自适应匹配门限 (AT)算法 .根据排列块的相对复杂程度 ,修正了自适应匹配门限的公式 ,提出了改进的自适应门限 (RAT) IFS图象压缩编码算法 .在对输入图象进行四叉树分块编码过程中 ,该方法可以根据当前排列块的统计特征确定匹配门限 ,从而使分块编码过程自动地适应输入图象 .实验结果表明 ,这种新的编码方法可以自适应地对输入图象进行编码 ,且压缩比较高 。