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Author Topic: Bifurcation fractals discovery  (Read 13620 times)
Description: Hi res. images and equations of Logistic map etc.
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fractower
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« Reply #15 on: October 24, 2012, 02:03:51 AM »

Quote
This gives a 4D fractal: C(re), C(im), Z(re), Z(im). Interesting...

I tried taking some projections of the complex logistic map and it looked pretty boring. I should learn to test first post second.
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bkercso
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« Reply #16 on: October 24, 2012, 02:31:01 AM »

I tried taking some projections of the complex logistic map and it looked pretty boring. I should learn to test first post second.
Have you images to share? Maybe boring area did you see, or in poor quality...
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kram1032
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« Reply #17 on: October 24, 2012, 09:05:35 AM »

If I recall correctly, I saw some images of the complex logistic map and it essentially showed Antibuddhabrot-like extensions. z-axis was mapped in the same way as the typical logistic map, which caused those extensions to be multiplied along several heights. It looked fairly pretty.
Also, if you look at the right planes of the Buddhabrot set, a logistic map is clearly visible.
Like this:
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bkercso
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« Reply #18 on: October 24, 2012, 12:10:50 PM »

Wow, it's wonderful!

OFF:
And what about delayed Mandelbrot? It would be:
Z(1)=C
Z(2)=Z(1)^2+C
Z(i+2)=Z(i)^2+Z(i-1), or Z(i+2)=Z(i-1)^2+Z(i)...
Anybody? smiley
« Last Edit: October 24, 2012, 01:57:14 PM by bkercso » Logged
fractower
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« Reply #19 on: October 24, 2012, 08:14:59 PM »

Quote
Quote
I tried taking some projections of the complex logistic map and it looked pretty boring. I should learn to test first post second.
Have you images to share? Maybe boring area did you see, or in poor quality...


The primary limitation appears to be for most phases of C (the complex value formally known as r) the iterations become unbounded. The first picture is the real map between 0 to 4 and is provided for reference. The second shows the same range for a phase shift of C of 0.52.


* p0.jpg (7.56 KB, 300x300 - viewed 903 times.)

* p0.52.jpg (4.43 KB, 300x300 - viewed 893 times.)
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bkercso
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« Reply #20 on: October 25, 2012, 12:34:01 AM »

Ok, but you see a 4D fractal on a 2D plane. Did you choose the right section?
And I suggest to use more iterations, log coloring, oversampling and higher resolution for higher quality images.
« Last Edit: October 25, 2012, 12:40:24 AM by bkercso » Logged
hobold
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« Reply #21 on: October 25, 2012, 02:36:44 AM »

OFF:
And what about delayed Mandelbrot? It would be:
Z(1)=C
Z(2)=Z(1)^2+C
Z(i+2)=Z(i)^2+Z(i-1), or Z(i+2)=Z(i-1)^2+Z(i)...
Anybody? smiley
Offtopic answer: didn't work too well, in terms of what the resulting image looks like when displayed as an escape time fractal. A bit mysterious, though, because the result has a triple symmetry, despite the numbers being squared.

I changed the definition slightly to
Z(0) := 0,
Z(1) := C,
z(n) := Z(n-1)^2 + Z(n-2),
which is equivalent, except that it starts at zero. This would have opened up the possibility of rendering Mandelbrot (starting value 0, C pixel dependent) and Julia (starting value pixel dependent, C invariant) variations.
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bkercso
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« Reply #22 on: October 25, 2012, 03:20:39 AM »

Great!
Have you pics for post? smiley I'm curious...
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bkercso
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« Reply #23 on: October 25, 2012, 03:31:07 AM »

I found a Henon bifurcation map: http://commons.wikimedia.org/wiki/File:Henon_bifurcation_map_b%3D0.3.png
This is high resolution: 6,5 Mpix
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hobold
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« Reply #24 on: October 25, 2012, 09:52:23 PM »

Have you pics for post? smiley I'm curious...
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bkercso
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« Reply #25 on: October 26, 2012, 03:16:50 AM »

Interesting section.
Quote
A bit mysterious, though, because the result has a triple symmetry, despite the numbers being squared.
Cube also has tripple symmetry...
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bkercso
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« Reply #26 on: October 28, 2012, 08:39:21 PM »

I continue the simulation of double pendulum and create bifurcation map. Changed from Euler method to 4th order Runge-Kutta method, which taken approx. 50 times acceleration (!). (The method described with source code in this article: https://freddie.witherden.org/tools/doublependulum/report.pdf, and on wikipedia: http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods).

Setup was as at the first image, except the x-axis is linear.
Calculation time was the same (17 hours @ 2.5GHz), but the quality is much better.
Image is 1200x1800 pix, 4800 points/column, dt=1.3E-4.
I took some measurements and I found that with RK4 method dt is proportional with maxiter^(-0.4) at constant image quality (and with maxiter^(-1) with Euler method). Conclusion: higher resolution image not requires as more calculation time as with Euler method.
I measured the relative change of Hamiltonian (=sum of mechanical energy of the pendulum) after maxiter, because I experienced image quality depends on this parameter. This relates the precision of the simulation: theoretically the Hamiltonian are const. in time.
The relative change of energy after 4800 oscillations (not iterations!) of upper pendulum was: abs(dH/H0)=1E-17 at left side (small displacements) and 6E-14 at right side (large displacements).

I think the vertical widenings are from the first few iterations. I left the first 1% of points, but I took 20 times less iterations as for the other bifurcation fractals I posted. This one still requires a lot of processor time: one oscillation (means one point in the fractal) requires hundreds or thousands iterations. This image are made from 8.64 million oscillations.

Img #3 with log coloring


with linear coloring with saturation
« Last Edit: May 27, 2015, 03:45:06 PM by bkercso » Logged
kram1032
Fractal Senior
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Posts: 1863


« Reply #27 on: October 29, 2012, 09:23:00 AM »

that delayed Mset might be a distant relative of certain Julia-sets, Cantor-dust and/or the plot of p-adic numbers:
http://en.wikipedia.org/wiki/P-adic_number - image on the right.
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Ryan D
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« Reply #28 on: October 29, 2012, 01:09:11 PM »

Quote
that delayed Mset might be a distant relative of certain Julia-sets
It's a very close relative to the Manowar fractal in Fractint.

manowar
c = z1(0) = z(0) = pixel;  z(n+1) = z(n)^2 + z1(n) + c;  z1(n+1) = z(n);
Parameters: real & imaginary perturbations of z(0)

manowarj
z1(0) = z(0) = pixel;  z(n+1) = z(n)^2 + z1(n) + c;  z1(n+1) = z(n);
Parameters: real & imaginary parts of c

Here's a test animation I did a while back of the Julia set, tracing a spiral path for the parameters.

<a href="http://vimeo.com/moogaloop.swf?clip_id=49998325&amp;amp;server=vimeo.com&amp;amp;fullscreen=1" target="_blank">http://vimeo.com/moogaloop.swf?clip_id=49998325&amp;amp;server=vimeo.com&amp;amp;fullscreen=1</a>

Ryan
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bkercso
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« Reply #29 on: October 30, 2012, 12:37:43 PM »

New bifurcation map of double pendulum:
m1=10, m2=1, L1=L2=1, theta1_initial=0, theta2_initial=0..179.9 deg
Values: theta2 when m1 stops.
Calculation time: 18 hours @2.5GHz

Img #4 log coloring with optimized contrast
« Last Edit: May 27, 2015, 03:58:00 PM by bkercso » Logged
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