### Three dimensional Lyapunov space

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#### marcm200

• Fractal Frankfurter
• Posts: 632

#### Three dimensional Lyapunov space

« on: February 09, 2019, 01:31:53 PM »
I recently started experimenting with three-dimensional Lyapunov spaces. In addition to A and B in the sequence, I introduced a new parameter C, the z coordinate. Here are the first examples of the logistic equation and the sequence AAAAAACCCCCCBBBBBB.

Technically I am still computing 2D-images in the AB-plane with a fixed C value, varying it slightly from picture to picture. I stack these files into a cube and visualize it using a self-coded rudimentary program that still lacks perspective information, so the cubes look "too wide" in the far. Technically I am moving a screen bitmap through space and when it intersects with the cube it takes on the color value - if that color had not been defined as transparent to be able to somehow look into the cube. I included the coloring scheme into the image (however, not all intervals contain Lyapunov exponents, I have still to work on a better visualization).

The first cube shows just the total cube (A,B,C from 0.1 to 3.99) with full coloring.

With the second cube it becomes more interesting. Just coloring part of chaos (and letting the rest be transparent) lets one look inside the object, but shows no additional colored structures.

The third cube is the most interesting one. Coloring part of the order realm reveals an interesting inner net like structure. In the upper right quadrant, it looks like a snowflake.

Has anyone else done 3D-work on (stacks of) Lyapunov images?

The fourth picture ("cross") is the start of a new experiment. Instead of looking inside the cube, I wanted to insert other (smaller) 3D objects, carve off of the cube what lies outside that smaller volume and look on the remaining parts. Here I inserted a 3D-cross into the center of the cube.

#### ThunderboltPagoda

• Fractal Furball
• Posts: 233

#### Re: Three dimensional Lyapunov space

« Reply #1 on: February 09, 2019, 02:50:41 PM »
Has anyone else done 3D-work on (stacks of) Lyapunov images?

Take a look at Tom Gidden's work. Article:

https://medium.com/@gid/exploring-lyapunov-space-b810a8bed153

Very short video:

#### marcm200

• Fractal Frankfurter
• Posts: 632

#### Re: Three dimensional Lyapunov space

« Reply #2 on: February 09, 2019, 04:24:16 PM »
Thanks!

So I am pretty much in his volumetric stage - but it's good to know that there lies a solution for biggen pictures (mine are currently only about 400 pixels, because otherwise it'd take me a day or so to calculate one cube).

#### Sabine62

• Fractal Freak
• Posts: 667

#### Re: Three dimensional Lyapunov space

« Reply #3 on: February 09, 2019, 09:08:05 PM »
Claude has coded a 3D lyapunov DE for Fragmentarium, so maybe this is a helpful link for you: https://fractalforums.org/fragmentarium/17/lyapunov-example/1989/msg9938#msg9938

I'm very much looking forward to more experiments, your examples up here look really cool!
To thine own self be true

#### marcm200

• Fractal Frankfurter
• Posts: 632

#### Re: Three dimensional Lyapunov space

« Reply #4 on: February 10, 2019, 10:23:32 AM »
@sabine62: Thanks! Glad you liked it. It's always amazing what nowadays can be done with the computing power.

@ThunderboltPagoda: Your article was quite helpful. Using two ideas there (storing intermediate data and the raytracing princile "compute only what you can see") I was able to devise an algorithm to be able to compute much larger final pictures.

I use the Lyapunov coordinates of the pixels in the final (small) screen and double the resulution, filling in the voids with interpolated coordinates. Then I only recalculate those Lyapunov values with high iteration depth. I'm pretty sure it would not be exactly the same as doubling the initial 3D space because I do not consider screen intersection angle when interpolating the new coordinates, but I guess the gain in speed and availability outweighs that by far. So thanks again!

#### Sabine62

• Fractal Freak
• Posts: 667

#### Re: Three dimensional Lyapunov space

« Reply #5 on: February 10, 2019, 11:15:58 AM »
Now, of course, you must oblige us with some examples, Marc!

#### marcm200

• Fractal Frankfurter
• Posts: 632

#### Re: Three dimensional Lyapunov space

« Reply #6 on: February 11, 2019, 04:11:10 PM »
"Let's play Tetris"

The initial cube had 400x400x400 resolution, therein I carved out some Tetris pieces. The final picture was computed in 4 times the resoltion. The left tetris piece shows a zig-zag chainsaw-like edge on the rght upper hand. I wonder whether this feature actually is real or an artefact of the coordinate interpolation I perform.

Trajectory function f(x)=2.435*sin²(x+r)
sequence ABCAABBCCAAABBBCCCAAAABBBBCCCCAAAAABBBBBCCCCCAAAAAABBBBBBCCCCCCAAAAAAABBBBBBBCCCCCCC
0 <= A, B <= 4
0.1 <= C <= 3.99

#### marcm200

• Fractal Frankfurter
• Posts: 632

#### Re: Three dimensional Lyapunov space

« Reply #7 on: February 12, 2019, 10:29:10 AM »
The two following pictures are what I had initially in mind when trying to go into three dimensions with the Lyapunov system - somehow being able to see inside. I wondered what lies between the walls here. Looking from the bottom up revealed a rather feather-y light substance floating between.

It is based on a series of detachedly computed 2D images (the France picture I posted a couple of month ago in the other thread).

Trajectory function f(x)=5*sin²(x+r)
Computing function g(x)=r-2rx

Sequence AAAAABCC
-0.73 <= A <= 0.47
1.33 <= B <= 2.73
0.1 <= C <= 3.99

#### Sabine62

• Fractal Freak
• Posts: 667

#### Re: Three dimensional Lyapunov space

« Reply #8 on: February 12, 2019, 11:18:03 AM »
Those are great!

#### marcm200

• Fractal Frankfurter
• Posts: 632

#### Re: Three dimensional Lyapunov space

« Reply #9 on: February 14, 2019, 05:06:47 PM »
Thanks! It's not as easy getting a look inside as I hoped/imagined. Maybe it happens more regularly when I switch from those large overview pictures with wide parameter ranges to more condensed areas. At least the 2D Lyapunov images get more and more sparse there, which might help looking "inside".

The next one is quite interesting, because it hides something inside, at least so it seems. Eight pieces of sugar stacked free-floating in a cube. If a 3D cross is put inside and everything outside of it carved off of the cube, one gets the "Life on Phobetor" image in the gallery.

#### Sabine62

• Fractal Freak
• Posts: 667

#### Re: Three dimensional Lyapunov space

« Reply #10 on: February 14, 2019, 07:55:00 PM »
Both very nice images!

The 'inside' of Life on Phobetor is very "voxelesque", is that by design or a choice to limit rendertimes or are the emerging forms really cube-shaped?

And before I forget: thank you for your Lyapunov-explanantion. I never understood that it is much more a method than a formula...
One more question: what do the "AAABBBCCCAACC"-sequences relate to? claude uses ABCD to help make up the last part of a 4D vector. It does not look to me that you are doing anything like that?
I should say that I do not know your software, so have no idea how things are implemented. Also, sorry if these are very dumb questions, I have no background in maths to fall back on and school was a long time ago.. ;}

• 3f
• Posts: 1378

#### Re: Three dimensional Lyapunov space

« Reply #11 on: February 14, 2019, 10:29:01 PM »
claude uses ABCD to help make up the last part of a 4D vector

if I remember right it was (A+Bi, C+Di) so there are still 2 values to choose from, but they are complex, so the overall parameter space is 4D

#### Sabine62

• Fractal Freak
• Posts: 667

#### Re: Three dimensional Lyapunov space

« Reply #12 on: February 14, 2019, 11:13:06 PM »
@claude I misread the line of code, forgot to really count parenthesis (really embarassing ), you multiplied the vec4 by a matrix mat4(A,B,C,D)

Code: [Select]
float DE(vec3 p){ vec4 q = vec4(p, mix(mix(-2.0, 2.0, 110.0 / 600.0), mix(-2.0, 2.0, 490.0 / 600.0), time / 600.0)) * mat4(A, B, C, D);

#### ThunderboltPagoda

• Fractal Furball
• Posts: 233

#### Re: Three dimensional Lyapunov space

« Reply #13 on: February 14, 2019, 11:41:22 PM »
I never understood that it is much more a method than a formula... One more question: what do the "AAABBBCCCAACC"-sequences relate to?

Sabine, to get the idea behind Lyapunov fractals, you can read this article by their inventor. Alas, some parts of the original article are missing here.

#### Sabine62

• Fractal Freak
• Posts: 667

#### Re: Three dimensional Lyapunov space

« Reply #14 on: February 14, 2019, 11:57:28 PM »
@ThunderboltPagoda, thank you very much for that link!
I will read and re-read and let it ferment a bit ...

Bierhefe...

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