【作者】 王婷婷；

【导师】 王兴元；

【作者基本信息】 大连理工大学 ， 计算机软件与理论， 2006， 硕士

【摘要】 非线性理论是描述具有无规结构的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牛顿(Newton)迭代M-J集的相关理论和方法,并取得了一些重要的研究成果。 牛顿迭代是求解非线性方程或方程组的一种重要的方法,它将解方程f(x)=0的问题转化为一个动力学过程,求解过程与初值选择有关。牛顿方法并不局限于解复平面上的问题,用复数来讨论问题的好处是,已有十分成熟的数学学科来指导我们的研究,即Julia和Fatou的迭代理论,而关于实函数问题还没有相应的理论。本文中,我们首先将牛顿迭代法应用于非线性方程组,构造并研究了实指数幂多元牛顿变换的Julia集的相关理论。研究中我们发现,随参数实指数幂β值增大,多元牛顿变换的Julia集有一个突变,表现为吸引域的个数加1,且其Julia集的结构依赖于相角主值范围的选取。 接下来我们对Calson,Pickover的轨道陷阱技术进行了研究,提出了使用IFS迭代系统所绘制的自然图形作为轨道陷阱以及双陷阱技术,构造了复多项式的伪3D牛顿变换的M-J集,在得到了艺术的分形图形的同时,我们还发现,广义M-J集中存在具有伪3D效果且与对应陷阱形状相近的大小不同的彩色元素,且广义M集中始终都存在着由坏点组成的经典M集。这一部分的研究工作已被《中国图象图形学报》录用。 最后,我们采用渐变的颜色方案研究了在多种牛顿变换下,根的吸引域的大小及分布与各个根之间的相互距离以及每个根的重数的对应关系。研究发现,对于大多数迭代方法,在根的重数都相同的情况下,根之间的相互距离成为决定根的吸引域的分布的唯一要素;而当三个根之间的距离都相等时,根的重数越大,相应的吸引域也就越大。此外,渐变的颜色方案让我们方便的观察到吸引域内部的结构以及收敛速度的快慢。更多还原

【Abstract】 The nonlinear theory is a new developing frontier science which describes the complex systematic structure shape which has a random structure. The nonlinear theory contains three important concepts: Fractal, Chaos and Soliton. Here we lay a particular emphasis on the studying of the theory and methods of the M-J sets of Newton’s method of fractals. Also we will give out some important research results.Newton’s method is an important method which is used to find the solutions of nonlinear equation or equations. It transforms the solution of the equation f(x) = 0 into a dynamical process and the solution process is related to the initial value. The use of the Newton’s method is not limited in the complex plane. The advantage of the using of the complex value is that there are already well-developed mathematical theories i.e. the iteration theory of Julia and Fatou which will give us a well guide in our study. In this article, we firstly apply the Newton’s method to the nonlinear equations to construct and study the Julia sets of multivariable Newton’s method for real exponent power. And we discover that as β increases, there is a sudden change in the Julia sets which expresses by the addition of the number of the attraction areas, and the structure of the Julia sets depends on the main range of the phase angle.Then we give a research into the orbit trap technique in Newton’s method. And the images generated by IFS system are used as the orbit trap in our method with which we construct the pseud 3D M-J sets of the Newton’s method of the complex polynomial. It always is found that there is standard Mandelbrot structure in the 3D M-set, which is formed by "bad" points. In generalized M-J set, there are various 3D self-similarity color cells that correspond with the shape of trap unit. And the work of this part has been accepted by the journal.Finally, a gradual changed color plan is adopted in generalized Newton’ method to show the inner structure of the attraction area. And we study on how the distances between the roots and the order of the roots affect the distribution of the attraction area. By our study, we discover that when the orders of the roots are the same, the distances between the roots is the only factor determining the attraction area; when the distances are the same, the size of the attraction depends only on the order of the roots.更多还原