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The cubic polynomial has many faults;
amoung them are:
1) size ( cubic Julias are usuaklly bigger in size)
2) placement of bilateral bulbs in the y direction
3) lack of the antenna structure associated with the Mandelbrot set.
The general polynomial simulation model goes like this :
Limit[x^2+Sum[a(i)*x^i,{i,0,Infinity}]-> f(x)=x^2+c
The theory goes that the polynomial:
p(x)=Sum[a(i)*x^i,{i,0,Infinity}]
is a reconstruction of the attractor roots such that the Julia iteration
mimics the Mandelbrot set exactly.
Two polynomial Julias that demonstrate this approach are:
cubic: ( based loosely on the Boris Solomyak function: * On the
`Mandelbrot set' for pairs of linear maps: asymptotic self-similarity,
<http://www.math.washington.edu/%7Esolomyak/PREPRINTS/asymp2.ps>
*/
Nonlinearity /* 18 * (2005), 1927--1943.
http://www.math.washington.edu/~solomyak/PREPRINTS/fractal.html)
f(r)=(r^2 + (1 - r - r^2 + r^3)/3)
pentic:
f(r)=(r^2 + (1 - r - r^2 + r^3 - r^4 + r^5)/5)
Another less successful polynomial I found is:
g[x_] = ExpandAll[(x + .6557)*(x - 0.2616574221163778` -
0.8516078436906926*I*(x - 0.2616574221163778` + 0.8516078436906926`*I)]
An animation of a Bezier between a cubic and a pentic:
a0 = 1 + 5/9;
sd = Sqrt[7];
f0[r_] := p*(r^3 + (a0)*r^2 - (r/(
a0) - 1))/sd + (1 - p)*(r^2 + (1 - r - r^2 + r^3 - r^4 + r^5)/5)
3D Julia of Mandelbrot like cubic to pentic Bezier
(*Julia with SQRT(x^2 + y^2) limited measure*)
(*by R. L. BAGULA 21 July 2007 © *)
numberOfz2ToEscape[z_] := Block[
{escapeCount, nz = N[z], nzold = 0},
For[
escapeCount = 0,
(Sqrt[Re[nz]^2 + Im[nz]^2] < 16) && (
escapeCount < 255) && (Abs[nz - nzold] > 10^(-3)),
nzold = nz;
nz = f0[nz];
++escapeCount
];
escapeCount
]
FractalPureM[{{ReMin_, ReMax_, ReSteps_},
{ImMin_, ImMax_, ImSteps_}}] :=
Table[
numberOfz2ToEscape[x + y I],
{y, ImMin, ImMax, (ImMax - ImMin)/ImSteps},
{x, ReMin, ReMax, (ReMax - ReMin)/ReSteps}
]
p = n/26
Table[ListDensityPlot[FractalPureM[{{-3.5, 1.5, 100}, {-2.5, 2.5, 100}}],
Mesh -> False,
AspectRatio -> Automatic,
ColorFunction -> (Hue[2#] &)];, {n, 1, 25}]
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The cubic polynomial has many faults;<br>
amoung them are:<br>
1) size ( cubic Julias are usuaklly bigger in size)<br>
2) placement of bilateral bulbs in the y direction<br>
3) lack of the antenna structure associated with the Mandelbrot set.<br>
<br>
The general polynomial simulation model goes like this :<br>
<br>
Limit[x^2+Sum[a(i)*x^i,{i,0,Infinity}]-> f(x)=x^2+c<br>
<br>
The theory goes that the polynomial:<br>
<br>
p(x)=Sum[a(i)*x^i,{i,0,Infinity}]<br>
<br>
is a reconstruction of the attractor roots such that the Julia
iteration <br>
mimics the Mandelbrot set exactly.<br>
<br>
Two polynomial Julias that demonstrate this approach are:<br>
cubic: ( based loosely on the Boris Solomyak function: <b> <a
href="http://www.math.washington.edu/%7Esolomyak/PREPRINTS/asymp2.ps">
On the `Mandelbrot set' for
pairs of linear
maps: asymptotic self-similarity,
</a> </b><i> Nonlinearity </i><b> 18 </b> (2005), 1927--1943.<br>
<a class="moz-txt-link-freetext"
href="http://www.math.washington.edu/~solomyak/PREPRINTS/fractal.html">http://www.math.washington.edu/~solomyak/PREPRINTS/fractal.html</a>)<br>
f(r)=(r^2 + (1 - r - r^2 + r^3)/3)<br>
pentic:<br>
f(r)=(r^2 + (1 - r - r^2 + r^3 - r^4 + r^5)/5)<br>
<br>
Another less successful polynomial I found is:<br>
g[x_] = ExpandAll[(x + .6557)*(x - 0.2616574221163778` -
0.8516078436906926*I*(x - 0.2616574221163778` +
0.8516078436906926`*I)]<br>
<br>
An animation of a Bezier between a cubic and a pentic:<br>
a0 = 1 + 5/9;<br>
sd = Sqrt[7];<br>
<br>
f0[r_] := p*(r^3 + (a0)*r^2 - (r/(<br>
a0) - 1))/sd + (1 - p)*(r^2 + (1 - r - r^2 + r^3 - r^4 +
r^5)/5)<br>
3D Julia of Mandelbrot like cubic to pentic Bezier<br>
(*Julia with SQRT(x^2 + y^2) limited measure*)<br>
(*by R. L. BAGULA 21 July 2007 © *)<br>
numberOfz2ToEscape[z_] := Block[<br>
{escapeCount, nz = N[z], nzold = 0},<br>
For[<br>
escapeCount = 0,<br>
(Sqrt[Re[nz]^2 + Im[nz]^2] < 16) && (<br>
escapeCount < 255) && (Abs[nz - nzold] >
10^(-3)),<br>
nzold = nz;<br>
nz = f0[nz];<br>
++escapeCount<br>
];<br>
escapeCount<br>
]<br>
FractalPureM[{{ReMin_, ReMax_, ReSteps_},<br>
{ImMin_, ImMax_, ImSteps_}}] :=<br>
Table[<br>
numberOfz2ToEscape[x + y I],<br>
{y, ImMin, ImMax, (ImMax - ImMin)/ImSteps},<br>
{x, ReMin, ReMax, (ReMax - ReMin)/ReSteps}<br>
<br>
]<br>
<br>
p = n/26<br>
Table[ListDensityPlot[FractalPureM[{{-3.5, 1.5, 100}, {-2.5, 2.5,
100}}],<br>
Mesh -> False,<br>
AspectRatio ->
Automatic,<br>
ColorFunction ->
(Hue[2#] &)];, {n, 1,
25}]<br>
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