Bifurcation fractals discovery

[kram1032]

Oh, well, the whole point of using a color pallette is to get rid of grey-scale colors. I'm not entirely sure if what you propose is exactly like I imagined it but if it is, then all you'd need to do is to use a color pallette, generated by converting wavelength responses to RGB values, or, as a first approximation, to use a rainbow color pallette...

[bkercso]

I see (I think). So I would have to use e.g. HSV colorspace, Hue depends on other physical parameter (time, trajectory etc.) and Value depends on visit events of the pixel?
I wrote a HSVtoRGB converter into my rendering program before, maybe I develop my fractal generator program for store 2 numbers/pixel (for Hue and Value) and try this type of coloring...

[kram1032]

something like that. Note how in the image I posted before (which is just reused from this forum) the Buddhabrot looks like made from a rainbow. I think it's colored by essentially linearly mapping iteration depth (e.g. time) to wavelength. I'd essentially like to see that but applied to your bifurcation map.

[bkercso]

I see no difference between using color palette or grayscale palette. Moreover, at monochrome palette you know the sequence of colors: darker means higher. The novelty would be if I'd color pixels according to 2 parameters not only 1: darker means higher depth and colors means something special.
Buddhabrot shows the same patterns also in grayscale (?):
http://www.fractalfreak.com/BuddhaCompilation/BuddhaGallery.html#row6

[bkercso]

I found a site with a lot of physical simulation ideas (for creating bifurcation maps and other fractals ):
http://www.myphysicslab.com/

[kram1032]

Ok, I'll post the image again:



Note how the blue regions go out really far, the green regions are much more contained but the branches are sort of wide (both cases of low iteration depths), while for yellow and finally red values, the whole structure becomes subtler and subtler.
So I guess, based on those observations, this is done by inversely mapping wavelength to iteration-depth, or stated differently, it maps the corresponding colors' frequencies to iteration depth.

Since you only go through the visible spectrum once, it is similarly readible to a simple grey gradient but it targets a wider section of the human visual system, so the data is more distinquishable to us.
(cbuchner1, if you read this, can you confirm wether that's how you did it?)

[bkercso]

kram1032: I red everything I can found about generating and coloring buddhabrots started from your link. I had no new idea to color my fractals... Bifurcation maps already colored as buddhabrots.
I try to color according to 'actual value - last value' of theta2, found a little bit different pattern but not significant:

Img #19 original coloring. dHavg=1E-8, average 3 points/pixel


Img #20 special coloring detailed above. There are some difference in the middle columns of pixels.


I rendered a 0.001 deg narrow band of chaotic region of Img #6. I think the reason is why we cannot see contoured patterns inside chaotic regions is they are too dense.

Img #21
m1=2, m2=1, L1=L2=1, theta1_initial=0, dHavg=1E-13
x-axis: theta2_initial=179.2 .. 179.201 deg
y-axis: theta2 when omega1=0

[kram1032]

yeah, that sure does look noisy as heck.
Ok, so no new insights here, I guess. Oh well...
As said, though, I wasn't able to find the original thread that lead to the first instance of a wavelength-based Buddhabrot...

[bkercso]

This is a 1E-5 narrow band of chaotic region:

Img #22 Same as Img #21, but theta2_initial=179.200425 .. 179.200435

[kram1032]

nice. Now, patterns slowly become apparent.
It has some pretty arcs in it.

[bkercso]

Yeah. The arcs are trange. I don't know yet what are these. I left the first 10% of points (and there are 3600 points/column), so is not from first iterations.
This is an approximation with dHavg=1E-13. I don't know how much this is similar with "reality", so what would be if I use higher precision.
I zoomed in this region...:

[bkercso]

A 2.7E-9 deg wide section of upper chaotic region:
theta2_initial=179.200425383° ..
                   179.2004253857°

Img #23 2.9 Mpixels. I think my 20 digits- and dHavg=1E-15 precision is not enough for this.

[bkercso]

Img #24 Cowebs of Img #10.
theta2_initial=130.3° .. 133.3°, dHavg=1E-12, 20 points/pixel, 2.5 Mpixels, 2x oversampled
Simulated physical oscillation time of double pendulum: 1 year


A zoom without contrast optimization

[bkercso]

Img #25 A zoom of Img #11.
theta2_initial=77.2° .. 79.8°, Const=0.5, 4 points/pixel, 3.2 Mpixels, dHavg=1E-13 (assume dHavg=1E-11 would be enough)

[bkercso]

Small displacement section

Img #26
theta2_initial=0..20°, 5 pojnts/pixel, 1.1 Mpixels, dHavg=1E-13, others as at Img #6 (but I undrawed fractal to fill image)



Img #27 zoom of Img #26
theta2_initial=0..4°, 10 points/pixel, 2.1 Mpixels