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 C

_{0} to C

_{29} 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/tsjGWyhttps://www.shadertoy.com/view/WsS3Rdhttps://www.shadertoy.com/view/Wdj3Rchttps://www.shadertoy.com/view/wsBGzySince 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

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