Noisy IFS fractals: new patterns

[bkercso]

I open a new topic for this, but I think this theme is related to coupled systems (http://www.fractalforums.com/mandelbrot-and-julia-set/coupled-mandelbrot-sets-%28cms%29/), because of the similarity of patterns.

The idea is: add random noise to every variables after each iteration. You have to use the same series of random numbers for each pixel to get a coherent image.

Code:

{randomize;}   {NOT USED, because of repeatebility!}
for i:=0 to 10000000 do 
for j:=1 to 4 do 
rnd[i,j]:=Amplitude*(random-0.5)*2;   {random is in 0..1}


Iterations:
... {the iterated functions}
x:=x+rnd[iterationNumber,1];
y:=y+rnd[iterationNumber,2];
x2:=x2+rnd[iterationNumber,3];   { in the case of coupled systems }
y2:=y2+rnd[iterationNumber,4];   { in the case of coupled systems }

Noisy Mandelbrot (not coupled, just one single set):

Img #1: the neck (top) of period-3 bulb: random wormholes (Ampl=1E-3)


Img #2: a zoom of Img #1: empty patterns. Only the size of patterns is fractal-like, but the figures are filled with a single color, not with patterns (as we saw at coupled systems) (Ampl=1E-3)

[bkercso]

Zoom series #1: (Ampl=1E-3)


secoundary pattern lane, and non-smooth surface:







Zoom series #2: (Ampl=1E-3)




wormholes


empty patterns





Zoom series #3: (Ampl=1E-3)


disappeared black area












There are no nondivergent areas here

[bkercso]

Zoom series #4: (Ampl=1E-3)
Burning ship, inner coloring (http://www.fractalforums.com/new-theories-and-research/a-new-look-into-escapetime-fractals-using-abs-and-inner-coloring/msg53013/#msg53013)






You have to increase the iteration number to get a more detailed image


Empty areas between the pillars (normally not common inside the miniships)


Overlapping periodic point and fractal pillars


ImageX


The fractal patterns continue


Get noisy: overlapping random patterns and periodic points


There are still fractal patterns; zoom: 3E10 times


But if you zoom into a different area of imageX...


...empty patterns found (really empty: no patterns even at very large iteration numbers)

[Chillheimer]

this is sooo cool!
I love it how the mandelbrot set becomes even more alive and "more realistic" through this!
these are awesome:

[bkercso]

cheers!
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Here is an article from the scientific side of the world (requires access to ScienceDirect):
Research on characteristics of noise-perturbed M–J sets based on equipotential point algorithm, 2016:
http://www.sciencedirect.com/science/article/pii/S1007570415002221

I think fig. 3.a)-c) shows the evolution of the empty patterns: as the added noise increases, the fractal becames disconnected and a part of it slowly escapes.

[bkercso]

Some free articles:

Role of Superior Iterates in Optimizing the Dynamic Noise, 2013
http://research.ijcaonline.org/volume64/number20/pxc3885618.pdf

The Generalized Julia Set Perturbed by Composing Additive and Multiplicative Noises, 2009
http://www.emis.de/journals/HOA/DDNS/Volume2009/781976.pdf

[cKleinhuis]

the question is:

CAN YOU ANIMATE THIS ?

[bkercso]

Sure. I plan to make animation about increse the noise amplitude, but before that I play a little bit more with it.

[Max Sinister]

Mandelbrot with tentacles, so to speak... I don't know, I prefer the original.

Although some of the pictures in Zoom series #4 are quite aesthetic too.

[bkercso]

An animation is made of Img #1, with increasing amplitude of noise from 1E-4 to 5E-2 (on log. allocation).
Video #1:
<a href="http://www.youtube.com/v/F7Wm7IFYQGY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/F7Wm7IFYQGY&rel=1&fs=1&hd=1</a>

[quaz0r]

that video is what happens when you take a hit of dmt and then look at the standard mandelbrot 

[bkercso]

 
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Img #3: a Julia set of Mandelbrot + noise system