分形/複平面迭代/構造

如何構建具有所需屬性的地圖？

形狀

朱利亞集
臨界軌道
落在排斥迴圈上的外部射線（螺旋線）

形狀型別和動力學之間的關係

n-臂螺旋：吸引或排斥 n-週期軌道（迴圈）
封閉曲線：西格爾圓盤（旋轉）
n-臂星形 = 週期 n 拋物線根

建模或造型

通常應該控制 2 個引數

不動點
週期 p 軌道

朱利亞集由臨界軌道建模
朱利亞集由臨界軌道建模
西格爾圓盤臨界軌道由週期 7 排斥軌道建模
西格爾圓盤臨界軌道由週期 2 排斥軌道建模
朱利亞集由週期 6 排斥軌道及其外部射線建模

參見

塑造螺旋線

轉換和演變

從西格爾到拋物線星形
超級吸引透過吸引螺旋到拋物線星形（沿著內部射線）

示例

I. Scherbak 的具有指定臨界點的有理函式
構建具有所需型別的法圖分量的對映^[1]
構建朱利亞集類似於所需形狀的多項式^[2]
構建具有吸引迴圈的多項式，該吸引迴圈訪問平面上預定義的點^[3]
構建包含任何兩個二次朱利亞集副本的四次朱利亞集
構建具有指定組合的臨界有限實多項式對映^[4]

根

代數基本定理指出，每個非零、單變數、n 次復係數多項式都具有（按重數計）正好 n 個復根^[5]^[6]

因式定理^[7]指出，一個多項式 $f(z)$ 有一個因子 $(z-r)^{m}$ 當且僅當 $f(r)=0$ （即 $r$ 是重數為^[8] m 的根）

示例

多項式^[9]

  $f(z)=(z-0)^{2}(z-1)^{2}(z+1)=z^{5}-z^{4}-z^{3}+z^{2}$

多項式

  $f(x)=(x+4)(x-1)^{2}=x^{3}+2x^{2}-7x+4$

有根

1 的重數為 2
-4 的重數為 1

rboyce1000

p(z) = z^4 + O(z^2)，其中 p(z) 的四個根為

一個固定在原點，
其餘三個形成以原點為中心並旋轉的等邊三角形的頂點。

建立具有所需屬性的多項式

- f(z) = z*g(z) 在原點處有根
- g(z) 是 3 次單位根 = $x^{2}+x+1$

f(z) = z(z^2+z+1)

可以使用 Maxima CAS 進行檢查

(%i1) solve([z*(z^2+z+1)=0],[z]);
                      sqrt(3) %i + 1      sqrt(3) %i - 1
(%o1)          [z = - --------------, z = --------------, z = 0]
                            2                   2
(%i2)

為了圍繞原點旋轉，將 1 更改為： $\quad e^{2\pi it}$ 其中 t 是以圈數表示的真分數

f_{t}(z)=z(z^{2}+z+e^{2\pi it})

參見：boyce1000 的向量場短影片

一個引數

二元方程組

${\begin{cases}f_{c}^{p}(z_{p})=z\\{\frac {d}{dz}}f_{c}^{p}(z_{p})=r*e^{2\pi \theta i}\end{cases}}$

其中

$f_{c}$ 是一個帶有一個引數 c 的有理函式
$\ f_{c}^{(p)}(z)$ 是 $p$ 次迭代的 $f_{c}\,$
$z_{p}$ 是一個迴圈點（極限環點）
p 是迴圈的週期
$\lambda =r*e^{2\pi \theta i}$ 是一個乘子^[10]（複數）
$r=|\lambda |$ 是迴圈的穩定性（實數）

輸入

函式 $f$
p（整數）
r（實數）
$\theta$ （實數或有理數）

未知數（解或輸出）

引數 c（複數）
週期點 $z_{p}$ （複數）

Maxima CAS 程式

/*

batch("m.mac");

*/

display2d:false$
kill(all)$
ratprint:false$

/* complex quadratic polynomial */
f(z,c):= z*z+c $

/* iterated function */
F(z, c, n) :=
       if n=1 then f(z,c)
        else f(F(z, c, n-1),c)$
        
        
        
/*  multiplier = first deric=vative */        
m(z,c,p):= diff(F(z,c,p),z,1)$

l(r,t) := float(rectform(r*exp(2*%pi*t*%i)))$

/* input */

p:5$
r:1.0$
t:0$

/* system of equations */
e1: F(z,c,p)=z;
e2: m(z,c,p)=l(r,t);

/* 
output = solutions = 2 complex number: c, z 
*/

s:solve([e1,e2])$
s:map('float,s)$
s:map('rectform,s);

示例輸出

對於

p = 3
r=1.0
t=0.0

[
[z = 0.5,c = 0.25],
[z = (-0.4330127018922193*%i)-0.25,c = (-0.6495190528383289*%i)-0.125],
[z = 0.4330127018922193*%i-0.25,c = 0.6495190528383289*%i-0.125],
[z = -0.05495813133539004,c = -1.75],
[z = 1.301937809824245,c = -1.75],
[z = -1.746979634104245,c = -1.75]
]

對於

p = 5
r=1.0
t=0.0

[
[z = 0.5,c = 0.25],
[z = 0.4755282581475767*%i+0.1545084971874737,c = 0.3285819450744551*%i+0.3567627457812099],
[z = 0.1545084971874737-0.4755282581475767*%i, c = 0.3567627457812106-0.3285819450744586*%i],
[z = 0.2938926261462365*%i-0.4045084971874737, c = 0.5316567552200239*%i-0.4817627457812153],
[z = (-0.2938926261462365*%i)-0.4045084971874737, c = (-0.5316567552199369*%i)-0.481762745781224],
[z = -0.003102011282477321,c = -1.985409652076318],
[z = 0.0109289978340113,c = -1.860587002096436],
[z = 8.008393221517376E-4*%i-0.01213161194929343, c = 1.100298437397382*%i-0.1978729466687337],
[z = (-8.008393221517376E-4*%i)-0.01213161194929343, c = (-1.100298437397305*%i)-0.1978729466687667],
        [z = 0.02151217276434695*%i-0.005267866463337371, c = 0.3797412022535638*%i-1.256801993945385],
        [z = (-0.02151217276434695*%i)-0.005267866463337371, c = (-0.3797412022517599*%i)-1.256801993944077],
        [z = 0.02591758988716001*%i+0.0096648625988135, c = 0.9868115621249533*%i-0.04506136597934137],
        [z = 0.0096648625988135-0.02591758988716001*%i, c = (-0.9868115621250132*%i)-0.04506136597930513],
        [z = -0.02506558296814108,c = -1.624396967608546],
        [z = 0.02532354987824971*%i-0.0286751769590709, c = 0.6415066667139064*%i+0.3599331333357185],
        [z = (-0.02532354987824971*%i)-0.0286751769590709,
         c = 0.3599331333357186-0.6415066667139071*%i], [z = 0.7018214526647177,c = -1.860587002096436],
        [z = 0.5745382937725365*%i+0.1798116252110209, c = (-0.379741202251533*%i)-1.25680199394442],
        [z = 0.1798116252110209-0.5745382937725365*%i, c = 0.3797412022514344*%i-1.256801993944486],
        [z = -0.5997918293000261,c = -1.624396967608546],
        [z = 0.6400543521659254*%i+0.3601141169309163, c = 0.6415066667138928*%i+0.3599331333356947],
        [z = 0.3601141169309163-0.6400543521659254*%i, c = 0.3599331333356951-0.6415066667138929*%i],
        [z = 0.747361547631752*%i+0.4122389750905872, c = 0.3599331333377524-0.6415066667118048*%i],
        [z = 0.4122389750905872-0.747361547631752*%i,c = 0.6415066667118131*%i+0.3599331333377574],
        [z = -1.264646754738656,c = -1.624396967608546],
        [z = 0.838427461519175*%i+0.1867295812979602,c = (-0.9868115621248*%i)-0.04506136597962632],
        [z = 0.1867295812979602-0.838427461519175*%i, c = 0.9868115621248269*%i-0.04506136597961512],
        [z = 1.012227741688957,c = -1.624396967608546],
        [z = 0.6736931444481549*%i-0.7131540376767388,  c = 0.9868115621009495*%i-0.04506136566593825],
        [z = (-0.6736931444481549*%i)-0.7131540376767388, c = (-0.9868115621015654*%i)-0.04506136566602404],
        [z = 0.6816651712455555*%i+0.8064792250322852,  c = (-1.100298438532418*%i)-0.1978729463920518],
        [z = 0.8064792250322852-0.6816651712455555*%i,c = 1.100298438531886*%i-0.197872946387467],
        [z = 0.9873125420152975*%i-0.04563967787575593, c = 0.9868115621249436*%i-0.04506136597927069],
        [z = (-0.9873125420152975*%i)-0.04563967787575593, c = (-0.9868115621249249*%i)-0.04506136597929692],
        [z = -1.368033648790746,c = -1.860587002096436],
        [z = -1.623768668573244,c = -1.624396967608546],
        [z = 1.600752508361204,c = -1.860587002096436],
        [z = 0.8177857184842046*%i-0.8491638964763748,  c = 0.6415066726649287*%i+0.3599331357137042],
        [z = (-0.8177857184842046*%i)-0.8491638964763748, c = 0.3599331357115682-0.6415066726792946*%i],
        [z = -1.860467532467532,c = -1.860586580956207], 
        [z = 0.1585230889211015*%i+1.129895436404861,  c = (-0.3797412017812437*%i)-1.256801993890818],
        [z = 1.129895436404861-0.1585230889211015*%i,  c = 0.3797412020742688*%i-1.256801993924219],
        [z = 1.102491882350288*%i+0.07994573682221373, c = 0.641506666713125*%i+0.3599331333375105],
        [z = 0.07994573682221373-1.102491882350288*%i, c = 0.3599331333375118-0.641506666713142*%i],
        [z = 1.10027900645412*%i-0.1977264120044163,c = 1.100298437399976*%i-0.1978729466589521],
        [z = (-1.10027900645412*%i)-0.1977264120044163, c = (-1.100298437392994*%i)-0.1978729466579122],
        [z = 0.3795145554958574*%i-1.257237017109811,  c = 0.3797412012322979*%i-1.256801993538778],
        [z = (-0.3795145554958574*%i)-1.257237017109811, c = (-0.3797412011893692*%i)-1.256801993401957],
        [z = 0.8966903093631682*%i-1.01776444141452, c = 0.986811439368143*%i-0.04506141337632084],
        [z = (-0.8966903093631682*%i)-1.01776444141452, c = (-0.9868114393633113*%i)-0.04506141338736716],
        [z = 1.407944514501891,c = -1.985409652076318],
        [z = 0.7215120925377011*%i+1.234881318742427,  c = (-1.100298500720014*%i)-0.1978727350763138],
        [z = 1.234881318742427-0.7215120925377011*%i,c = 1.100298500782114*%i-0.1978727352231734],
        [z = 0.6651899971189704*%i-1.369391104706556, c = 1.100298438532065*%i-0.1978727774731155],
        [z = (-0.6651899971189704*%i)-1.369391104706556, c = (-1.100298478086625*%i)-0.1978727911942495],
        [z = 0.1731238730127708*%i-1.554564024233688,  c = 0.3797412149717089*%i-1.256801976456581],
        [z = (-0.1731238730127708*%i)-1.554564024233688, c = (-0.3797411926534995*%i)-1.256801968631482],
        [z = 1.842105908761944,c = -1.985410334346504],
        [z = 1.956403762662807,c = -1.985409652076318]
        ]

Mandelbrot 集合 - 海馬谷中 P/Q 分支的收斂進化

Mandelbrot 集合 - 由 izaytsev0 繪製的海馬谷中 P/Q 分支的收斂進化

主心形海馬谷 = 頭部（週期 2 區域）和主體（或肩膀 = 主心形）之間的間隙。特別是上部的一部分
2 個視窗
- 左：來自週期 2 區域的分支
- 右：來自週期 1 區域的分支
在每個視窗中，可以看到從 14/30 開始並隨著 p 增加的 p/q 分支？

與 SeryZone Arts 的“真實密集分形縮放！第二部分” 進行比較

另請參見

參考文獻

[1] ractalforums：julia-sets-true-shape-and-escape-time

[2] ractalforums : constructing-polynomials-whose-julia-set-resemble-a-desired-shape

[3] ractalforums : constructing-polynomials-with-attracting-cycles

[4] W. Thurston 演算法應用於實多項式對映 Araceli Bonifant、J. Milnor、S. Sutherland 發表日期：2020 年 5 月 15 日

[5] 維基百科中的代數基本定理

[6] Ed Pegg Jr “代數基本定理” http://demonstrations.wolfram.com/TheFundamentalTheoremOfAlgebra/ Wolfram 演示專案發表日期：2011 年 11 月 10 日

[7] 維基百科中的因子定理

[8] 維基百科上的多項式根的多重性

[9] ractalforums.org : julia-sets-true-shape-and-escape-time

[10] 維基百科上的週期點（軌道）的穩定性——乘子

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[3]

[4]

[5]

[6]

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[9]

[10]