Fractal made using strange attractor equations

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Offline Killy

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« on: February 15, 2019, 04:33:02 AM »
A while ago I found a web page which showed a number of different strange attractors created using equations like this:
\[ \begin{equation}
x_{n+1} = C_0x_n + C_1y_n + C_2z_n + C_3x_ny_n + C_4y_nz_n + C_5x_nz_n + C_6x^2 + C_7y^2 + C_8z^2 + C_9\\
y_{n+1} = C_{10}x_n + C_{11}y_n + C_{12}z_n + C_{13}x_ny_n + C_{14}y_nz_n + C_{15}x_nz_n + C_{16}x^2 + C_{17}y^2 + C_{18}z^2 + C_{19}\\
z_{n+1} = C_{20}x_n + C_{21}y_n + C_{22}z_n + C_{23}x_ny_n + C_{24}y_nz_n + C_{25}x_nz_n + C_{26}x^2 + C_{27}y^2 + C_{28}z^2 + C_{29}
\end{equation} \]
I can't seem to find the web page anymore.

These equations produced a wide range of different attractors, here are some examples:






I read somewhere that if you randomly generated sets of C values, that only 1% of these sets would produce chaotic results.  To get these attractors I generated random values, and calculated the Lyapunov exponent in order to quickly find chaotic sets of C.

Animating between the sets produced behaviour reminiscent of bifurcation in the logistic map (or bifuraction diagram).  I even saw periodic windows past chaos.  The bifurcating behaviours were quite diverse.  For example I would see a point split into two points, then split into 4 points, then the 4 points would become 4 loops, then get all twisted until it became an attractor.  Or I would see one point turn into a loop, then the loop would split into two, three or four loops and then become twisted into an attractor.  Any permutation of loops and bifurcations could be seen.

I thought that maybe the sets of values for C0 to C29 that don't diverge, produces a fractal like the Mandelbrot set so I tested it out in some shaders on shadertoy:
https://www.shadertoy.com/view/tsjGWy
https://www.shadertoy.com/view/WsS3Rd
https://www.shadertoy.com/view/Wdj3Rc
https://www.shadertoy.com/view/wsBGzy

Since I can't plot 30 varying values simultaneously I pick two C values at any given time to plot at random.  For the animation, I interpolate between the C values.

So far it seems to produce some interesting results.  I have no idea what kind of fractal this is, nor have I found any information about this online.  Has anyone else explored this before?  Also I am trying to work on a distance estimation to the boundary of the set.   Finding the distance estimation will improve the rendering in 2d but also will allow me to do a ray-marching 3d version with shading.  Feel free to use this thread to have a general discussion about possible fractals made using attractor equations.

Cheers,
Killy

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/fractal-made-using-strange-attractor-equations/2623/
« Last Edit: February 15, 2019, 04:55:59 AM by Killy »

Offline mclarekin

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« Reply #1 on: February 15, 2019, 09:51:35 AM »
HI Killy,

Images remind me of an old program (chaoscope?)

I can figure out most of what you are doing, it is part of the infinite possibilities. In my non-maths way of thinking, I see there are 30 parameters that control quadratic deformations, opposed to a vec3 of 3 parameters performing linear deformation.  Often you will find that a few of the constants produce the main  deformation, with the rest just further tweaking the general shape ( so mathematically correct is 30 parameters, but for making pretty fractal pictures maybe  only 21 parameters or less.)
With 3D rendering, this sort of thing use to be avoided due to large rendering time, but is now more feasible using GPU. You could try with Fragmentarium or MandelbulberV2.


Offline BrutPitt

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« Reply #2 on: February 16, 2019, 08:42:56 AM »
Hi.
This is a strange attractor formula descripted by Julien C.Sprott (Polynomial of degree N)
http://sprott.physics.wisc.edu/sa.htm


If it could be useful, I have developed a function to calculate dynamically all coefficients of any power of N, It is described here:
https://www.michelemorrone.eu/glchaosp/PolynomialN.html
It is written in C/C++, but easily portable in other languages.

Anyhow there is also the program glChAoS.P: it explores polynomial strange attractors until degree N = 20 (until 5313 coefficients) :fp: ... and other 60 "strange" objects, approximately
It Is open source - multiplatform - 3D real time strange attractors scout... and hypercomplex fractals (via stochastic IIM)


There is also a live/online WebGL version, aviable to this page:
https://www.michelemorrone.eu/glchaosp/webGL.html

The full source code is on my github page:
https://github.com/BrutPitt/glChAoS.P

I enclose a "couple" of images of Polynomial attractors of degree N, with N = 7, 11, 11 ... respectively.
(change the rendering type: blended particles, image 1 - solid lighted particles, others)

And other two of a modified bicomplex Julia formula

Greetings
« Last Edit: February 16, 2019, 09:20:48 AM by BrutPitt »
glChAoS.P / wglChAoS.P --> openGL CHaotic Attractor Of Slight (dot) Particles
https://github.com/BrutPitt/glChAoS.P

Offline Killy

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« Reply #3 on: February 16, 2019, 09:55:21 AM »
Any info on the escape time images made using the same equations?  I've never seen the patterns in these in other fractals.  BTW the shadertoy links are NOT for attractors, but the escape time images made using the same equations.  These images map out which sets of coefficients diverge or converge.





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