时滞模型与McMullen函数族映射的复动力系统研究

发布时间：2018-01-04 11:24

  本文关键词：时滞模型与McMullen函数族映射的复动力系统研究　出处：《大连理工大学》2015年硕士论文　论文类型：学位论文

  更多相关文章： 分形 M集 Julia集 时滞 McMullen

【摘要】：1975年,美籍数学家Mandelbrot正式提出“分形”的概念,从此,分形成为诸多领域科学家热衷于研究的一门学科。广大科学研究人员运用分形成功的解释生活中和自然界用传统学科无法解释的问题和现象。随着计算机科学的发展,特别是计算机图形学的发展使复杂而又奇妙的分形图形得以重现,如今,计算机图形学结合复动力系统理论已经成为当今科学界研究分形理论的主要方法。本文采用上述方法重点研究了时滞迭代下的复杂动力系统和McMullen有理函数族映射的M-J集的特性。主要内容如下：基于二次多项式映射的时滞复动力系统。研究了时滞迭代条件下的f(z)=z2+c映射,分别对迭代方程的横轴和纵轴迭代方程的坐标进行短暂性时滞和持续性时滞,时滞发生的时间设定为系统的初始状态、不稳定状态和稳定状态,并使用逃逸时间算法构造Julia集。通过对实验结果的研究和对时滞迭代方程的理论分析,分别得出了横、纵坐标轴时滞条件下,复动力系统保持稳定的条件。同时通过对时滞Julia集的研究得出了一些关于无时滞状态下Julia集的特性。McMullen有理函数族映射Mandelbrot集。研究了McMullen有理函数族映射f(z)=zm+c/zd Mandelbrot集的N周期稳定区域问题,得出了N(N1)周期稳定区域数量的和一周期稳定中心点、稳定区域边界的计算方法。同时研究了自由临界点的问题,通过实验验证了当m=d时,自由临界点不影响周期稳定区域分布的结论,且重点分析了当m≠d时,自由临界点对一周期稳定区域分布的影响,并找到其分布规律。McMullen函数族映射Julia集性质分析。重点研究了连通状态下的McMullen映射的填充Julia集。使用不同颜色区分Julia集的不同区域,精细的刻画了Julia集的内部结构,计算出Julia集中共形同胚于f(z)=z2+c映射Julia集的最大稳定区域的几何对称中心点。且证明了其中心点的分布只由m和d决定,不受C值得影响。
[Abstract]:In 1975, American mathematician Mandelbrot formally put forward the concept of "fractal", since then. Fractal has become a subject that scientists in many fields are keen to study. The majority of scientific researchers have successfully used fractal to explain problems and phenomena in life and nature that cannot be explained by traditional disciplines. Development. Especially with the development of computer graphics, the complex and wonderful fractal graphics can be reproduced. Computer graphics combined with complex dynamical system theory has become the main research method of fractal theory in the scientific community. In this paper, the complex dynamical systems with time-delay iteration and McMullen are studied with emphasis on the above methods. The properties of M-J sets of rational function family mappings. The main contents are as follows:. The time-delay complex dynamical system based on quadratic polynomial mapping is studied in this paper. ZZ2 c mapping. The coordinates of the horizontal and longitudinal axis iterative equations are respectively carried out with transient and persistent delays, and the time of delay is set as the initial state, unstable state and stable state of the system. The escape time algorithm is used to construct the Julia set. Based on the experimental results and the theoretical analysis of the iterative equations with time delay, the horizontal and vertical coordinate axes with time delay are obtained respectively. By studying the delay Julia set, we obtain some properties of the Julia set with no delay. McMullen rational function family map Man. Delbrot set. The mapping of McMullen rational function family f (. The problem of N-periodic stable region of zm / zd Mandelbrot set. The method of calculating the number of stable center points and the boundary of stable region is obtained. The problem of free critical point is also studied and verified by experiments. The conclusion that free critical point does not affect the distribution of periodic stable region is concluded, and the influence of free critical point on the distribution of periodic stable region is analyzed emphatically when m 鈮，

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