【作者】 张旭；

【导师】 王兴元；

【作者基本信息】 大连理工大学 ， 计算机应用技术， 2006， 硕士

【摘要】 非线性理论是描述复杂系统结构形态的一门新兴边缘科学。它包含了分形、混沌和孤子这三个非常重要的概念。本文侧重研究了分形学中具有重要意义的广义Malldelbrot集（简称广义M集）的分形结构，并取得了一些研究成果。 阐述了一些基本的分形算法和牛顿法。这些分形算法是构造广义M集的算法基础，而牛顿法则是对广义M集中特殊点进行数学分析与求值的重要手段。借助这些强有力的工具，能够更好地对广义M集的分形结构进行研究。 本文首先利用M集和广义M集的分形结构来对一维多次映射进行分析，主要是对一维多次映射的混沌区域的周期倍分岔现象进行了研究，给出了用图形的方法确定一维多次映射混沌区域内的miget或双曲线组分周期的方法，并对一维多次映射的周期倍分岔现象出现的规律性进行了阐述。 对双参数复映射z←（z^a+h₁（c））^β+h₂（c）的参数空间进行了分析，通过台劳公式给出了确定双参数复映射参数空间中类广义M集的位置、尺寸和方向。并与单参数复映射的参数空间进行了比较。 最后，本文给出了一种基于逃逸时间算法的广义M集的渲染方法。该方法根据距离的远近和逃逸时间的不同进行颜色的选取，这种方法能够有效地进行广义M集的内外结构的渲染，从而使广义M集具有三维的立体效果。通过对该方法所绘制的广义M集的局部图形进行放大，会发现一些类似于山体等高线的同心圆环，这些同心圆环的中心点具有着一定的周期性，并且只有那些周期是逃逸时间N的约数的中心点的周围才能够呈现出类似于山体等高线的同心圆环的形状，这些中心点被称为逃逸时间N的约数周期点。更多还原

【Abstract】 The nonlinear theory is a new developing frontier science which describes the complex systematic structure shape. It contains three important concepts: Fractal, Chaos and Soliton. Here we lay a particular emphasis on the studying of the fractal structure of the generalized Mandelbrot set (simplified by the generalized M set). Also we give out some important research results.Some basic fractal algorithms and Newton’s method are introduced here. These fractal algorithms are used to construct the generalized M set, and Newton’s method is a strong tool to analyze and compute the important points in the generalized M set. With these algorithms and methods, the structure of the generalized M set can be understood well.Firstly one-dimensional multiple mappings are studied by the structure of the M set and the generalized M set. The period-doubling bifurcations in the chaotic region are mainly analysed and a set of new rules to graphically determine the period of a midget or a hyperbolic component are provided.The regularity of period-doubling bifurcation is analysed.Secondly the parameter space of two-parameter complex mappings is analyzed. By Taylor expansion a method to calculate the positions, the sizes and the orientations of those small copies of generalized M-like set in the parameter space of two-parameter complex mappings and the comparison to the one-parameter complex mappings are shown.Finally a rendering method based on the escape-time method to draw the generalized M set in different colors is offered. The rendering method chooses different colors according to the distance and the escape-time N, so it can effectively render the generalized M set, which can show us the 3-D view of the generalized M set. According to amplify the part of the generalized M set drawn by the rendering method, some cirques with one heart like the equipotential lines can be found. The center point of the cirques with one heart whose color will be rendered by black has a period, and only those center points, whose periods are the divisors of the escape-time N, will show us the cirques with one heart like the equipotential lines around them. Those center points are named the divisor periodic point of the escape-time N.更多还原