本文關(guān)鍵詞：時滯模型與McMullen函數(shù)族映射的復動力系統(tǒng)研究　出處：《大連理工大學》2015年碩士論文　論文類型：學位論文

  更多相關(guān)文章： 分形 M集 Julia集 時滯 McMullen

【摘要】：1975年,美籍數(shù)學家Mandelbrot正式提出“分形”的概念,從此,分形成為諸多領(lǐng)域科學家熱衷于研究的一門學科。廣大科學研究人員運用分形成功的解釋生活中和自然界用傳統(tǒng)學科無法解釋的問題和現(xiàn)象。隨著計算機科學的發(fā)展,特別是計算機圖形學的發(fā)展使復雜而又奇妙的分形圖形得以重現(xiàn),如今,計算機圖形學結(jié)合復動力系統(tǒng)理論已經(jīng)成為當今科學界研究分形理論的主要方法。本文采用上述方法重點研究了時滯迭代下的復雜動力系統(tǒng)和McMullen有理函數(shù)族映射的M-J集的特性。主要內(nèi)容如下：基于二次多項式映射的時滯復動力系統(tǒng)。研究了時滯迭代條件下的f(z)=z2+c映射,分別對迭代方程的橫軸和縱軸迭代方程的坐標進行短暫性時滯和持續(xù)性時滯,時滯發(fā)生的時間設(shè)定為系統(tǒng)的初始狀態(tài)、不穩(wěn)定狀態(tài)和穩(wěn)定狀態(tài),并使用逃逸時間算法構(gòu)造Julia集。通過對實驗結(jié)果的研究和對時滯迭代方程的理論分析,分別得出了橫、縱坐標軸時滯條件下,復動力系統(tǒng)保持穩(wěn)定的條件。同時通過對時滯Julia集的研究得出了一些關(guān)于無時滯狀態(tài)下Julia集的特性。McMullen有理函數(shù)族映射Mandelbrot集。研究了McMullen有理函數(shù)族映射f(z)=zm+c/zd Mandelbrot集的N周期穩(wěn)定區(qū)域問題,得出了N(N1)周期穩(wěn)定區(qū)域數(shù)量的和一周期穩(wěn)定中心點、穩(wěn)定區(qū)域邊界的計算方法。同時研究了自由臨界點的問題,通過實驗驗證了當m=d時,自由臨界點不影響周期穩(wěn)定區(qū)域分布的結(jié)論,且重點分析了當m≠d時,自由臨界點對一周期穩(wěn)定區(qū)域分布的影響,并找到其分布規(guī)律。McMullen函數(shù)族映射Julia集性質(zhì)分析。重點研究了連通狀態(tài)下的McMullen映射的填充Julia集。使用不同顏色區(qū)分Julia集的不同區(qū)域,精細的刻畫了Julia集的內(nèi)部結(jié)構(gòu),計算出Julia集中共形同胚于f(z)=z2+c映射Julia集的最大穩(wěn)定區(qū)域的幾何對稱中心點。且證明了其中心點的分布只由m和d決定,不受C值得影響。
[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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