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# 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|MATH3062 Basic Concepts and Examples

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## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|Basic Concepts and Examples

a. A threefold cord: fractals, dynamics, and chaos. The word “fractal” is one which has wriggled its way into the popular consciousness over the past few decades, to the point where a Google search for “fractal” yields over 12 million results (at the time of this writing), more than six times as many as a search for the rather more fundamental mathematical notion of “isomorphism”. With a few clicks of a mouse and without any need to enter the jargon-ridden world of academic publications, one may find websites devoted to fractals for kids, a blog featuring the fractal of the day, photo galleries of fractals occurring in nature, online stores selling posters brightly emblazoned with computer-generated images of fractals, … the list goes on.

Faced with this jungle of information, we may rightly ask, echoing Paul Gauguin, “What are fractals? Where do they come from? Where do we go with them?”

The answers to the second and third questions, at least as far as we are concerned, will have to do with the other two strands of the threefold cord holding this book together-namely, dynamical systems and chaos. ${ }^1$ As an initial, naïve formulation, we may say that in most cases where a dynamical system exhibits chaotic behaviour, this behaviour is associated with the presence of a fractal. For our purposes, fractals will come from particular dynamical systems, and will lead us to an understanding of certain aspects of chaos.

But all in good time. We must begin by addressing the first question, “What are fractals?”
b. Fractals: intricate geometry and self-similarity. Consider an oak tree in the dead of winter, ${ }^2$ viewed from a good distance away, as in the first panel of Figure 1.1. Its trunk rises from the ground to the point where it bifurcates into two large boughs; each of these boughs leads away from the centre of the tree and eventually sends off smaller branches of its own. Walking closer to the tree, one sees that these branches in turn send off still smaller branches, which were not visible from further away, and more careful inspection reveals a similar branching structure all the way down to the level of tiny twigs only an inch or two long. The various scales are shown in the second and third panels of Figure 1.1.

The key points to observe are as follows. First, the tree has a complicated and intricate shape, which is not well captured by more familiar geometric objects, such as lines, circles, polygons, and so on. Second, we see the same sort of shape on all scales: whether we view the tree from fifty yards away or from fifty inches, we will see a branching structure in which the largest branch (or trunk) in our field of view splits into smaller branches, which then divide themselves, and so on. This self-similarity is one of the key characteristics of fractals.

## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|Population models and the logistic map

b.1. A rather unrealistic population model. Consider a population of duck-billed platypi (or bacteria, or whatever species you fancy), whose size will be represented by a variable $x$. Given the size of the population at the present time, we want to predict the size of next year’s population (or perhaps the next hour’s, in the case of bacteria). So if there are $x$ platypi this year, there will be $f(x)$ next year, where $f$ is a suitable function which models the change in the platypus population from year to year. Of course, since we cannot have a negative number of platypi, we must restrict $x$ to lie in the interval $[0, \infty)$, which will be the domain of definition for $f$.

What form should $f$ take? As a first (simplistic) approximation, we may suppose that the platypi reproduce at a constant rate, and so if there are $x$ of them this year, there will be $r x$ next year, where $r>1$ is a real number, and $r-1$ represents the proportion of newborns each year.

# 分形几何和混沌系统

## 数学代写|分形几何和混沌系统代考FRACTAL GEOMETRY \& CHAOTIC DYNAMICS代写|BASIC CONCEPTS AND EXAMPLES

atthetimeofthiswriting，是寻找更基本的数学概念“同构”的六倍多。只需点击几下鼠标，无需进入充满行话的学术出版物世界，您就可以找到专门为孩子们设 计的分形网站、一个以当今分形为特色的博宮、自然界中发生的分形照片库、在线商店出售的海报上印有计算机生成的分形图像，….不胜枚举。

## 数学代写|分形几何和混沌系统代考FRACTAL GEOMETRY \&CHAOTIC DYNAMICS代写|POPULATION MODELS AND THE LOGISTIC MAP

b.1。一个相当不切实际的人口模型。考虑一群鸭哺鸭orbacteria, orwhateverspeciesyou fancy，其大小将由一个变量表示 $x$. 㧛于目前的人口规模，我们想预测

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。