### Three dimensional Lyapunov space

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#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #15 on: February 15, 2019, 02:18:13 PM »
@sabine62: Lyapunov's equations use a normal function with an argument x and iterate them. So far the same as for Mandelbrot or Julia sets, except, I use real numbers, not complex ones. Inserted into the equation like f(x)=sin(x) there is a disturbance parameter r, so the equation becomes f(x)=sin(x+r). And the value of r depends on the sequence you use, e.g. AB. Usually the A and B values are forming a point in the XY-plane. And the color of that point is calculated by computing the Lyapunov equation for a specific amount of iterations (and calculate the average log of the derivation). To go one dimension higher, I just used a thrid value C that now corresponds to the Z-axis.

As for the voxel-like shape of the points: On one side,I used a very small initial cube (256x256x256 pixels, see below, different coloring) to speed up the computation (took 3 hours anyways). And that pictures shows singleton pixels as well. Additionally so far I haven't encountered any voxel-like parts in other pictures. On the other side, I had to modify the "recomputing larger pictures" program. Just interpolating the necessary coordinates did not incorporate all really necessary points - the images looked somehow "wrong". So I now calculated part of a cube around that pixels of the smaller image, so basically a voxel (in the picture above it was 16x16x16 pixels. However, I still believe that the screen-moving process to interesect with the cube should work correctly, so I tend to answer your question with: The image showed that behavior.

@claude: You used complex numbers for the Lyapunov images. How did you visualize the 4D structures? Did you keep one of ABCD constant and change the others? Did you come across some general findings about the influence as to how the pictures turn out (sth. like: the larger the norm, the more sparse the image (like in the 2D version)) ?

• 3f
• Posts: 1376

#### Re: Three dimensional Lyapunov space

« Reply #16 on: February 15, 2019, 03:54:53 PM »
@claude: You used complex numbers for the Lyapunov images. How did you visualize the 4D structures? Did you keep one of ABCD constant and change the others? Did you come across some general findings about the influence as to how the pictures turn out (sth. like: the larger the norm, the more sparse the image (like in the 2D version)) ?

I visualized it by raytracing (raymarching, sphere marching) with a distance estimate based on the regular Mandelbrot set exterior distance estimate - I don't know if this is 100% correct but it gave interesting images.  I initialized the coordinates by a matrix transformation from (x,y,z,time).  I didn't investigate it all that much as I didn't find the images so very interesting (maybe I should have tried with 3x real coefficients, instead of 2x complex).

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #17 on: February 17, 2019, 10:24:35 AM »
I'm currently collecting shapes of large overview (with respect to the parameter width), so two new ones:

The last full cube looked like being built from sugar pieces. This one here looks like the still unfished concrete skeleton of a building on a construction site: empty apartments - and someone has forgotten to install the window frames. But it shows nicely how to look inside, although there is nothing to be seen there.

The second one is a sphere carved off of a cube. And it demonstrates a weird shape.with missing parts. So One might be able to set the sphere as transparent and only look at the now missing parts. However, I had no luck so far finding the right color scheme for that.

#### Sabine62

• Fractal Freak
• Posts: 667

#### Re: Three dimensional Lyapunov space

« Reply #18 on: February 17, 2019, 11:12:05 AM »
@marcm200 Took a while (my brain is really slow on the math-uptake  ), but it finally all makes sense Thank you!
And thank you also, @claude and @ThunderboltPagoda, for sharing your knowledge!

To thine own self be true

#### ThunderboltPagoda

• Fractal Furball
• Posts: 233

#### Re: Three dimensional Lyapunov space

« Reply #19 on: February 17, 2019, 05:37:16 PM »
The second one is a sphere carved off of a cube. And it demonstrates a weird shape.with missing parts. So One might be able to set the sphere as transparent and only look at the now missing parts. However, I had no luck so far finding the right color scheme for that.

Are your images originally 24 or 32 bit? I guess you should use an alpha channel, which requires 32 bit, for that purpose. JPG doesn't support transparency information, so you might switch to PNG or TIFF. Or you do voxel processing on 32 bit BMP and convert it to 24 bit JPG at the very end. That's just an idea - I have no experience with that voxel stuff.

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #20 on: February 20, 2019, 11:45:06 AM »
@ThunderboltPagoda: I use 24 bit - and when I change the transparency color I just let my screen float through the Lyapunov cube again to create the final image. Takes some time, but was the easiest implementation - and for the 400x400x400 cube I currently use for exploring it is feasible. However, using a somewhat 3D image and then changing the transparency sounds intriguing. But then I have to store the whole cube in the PNG, right? It's about 128 MB for one cube - would that work?

Today I encountered a phenomenon I haven't seen before. The full cube shows three things: some walls, a foggy sphere and some nothingness around. When I carved off some of the x-axis, I could take a look inside the sphere - and that revealed a cube-like object - floating in free air.

But what's most interesting is the connection between the 2D images and the final 3D cube. Usually you get - by varying the C parameter - structures that appear, vansish with different C values, reappear in different shapes and so on. Here however the 2D image changes in one simple manner: The fixed C image shows also three things: Two walls (left and bottom, a circle and a rectangle within. And by varying the C-value the circle gets smaller, the rectangle does so too until it disappears.

And what's even more striking is the resemblance between the C-fixed slice and the non-C containing sequence. Usually there is no direct similarity.

If I now were able to poke a hole into the circle, then I would have a window for the 3D version - so one could look inside without carving off.

That's the first example with which one could (in hindsight) say, it was possible to somehow design the outcome (However, I found the parameters purely by chance).

Does anyone have experience in tweaking functions/sequences in a way like "I want a rectangle to appear - now I have to add r*x-sin" or something like that ?

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #21 on: February 26, 2019, 10:10:18 AM »
Two more semi-transparent cube shapes I encountered. Grey-scaling from order (-2) to chaos (+2) seems to be good as a standard colouring scheme to get some transparency into the cube. The second one showed a nice double feature: The overall thread-y somewhat chaotic fabric of the cube - and the pink/orange blobs at the corners. But up to now all the cubes looked more or less like those thread-y parts - when a specific colouring was used. That seems to be the general case. It's quite different from the 2D images.

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #22 on: February 27, 2019, 01:44:56 PM »
Some new cube shapes I found: A metal salt grid, an uncomfortable bed and part of a rail.

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #23 on: February 28, 2019, 02:53:00 PM »
I checked whether the x0 parameter can be used as a third axis to generate a different kind of 3D Lyapunov spaces. I started with the easiest parameters I could think of: the logistic equation with sequence ABC in the range 0.01 <= A,B,C <= 3.99 (I usually omit 0 and 4 because the Lyapunov values go to infinity and give a one-colored 2D image that blocks the view into the cube).

At first glance the 3D cube looked like a stacked version of the same 2D image. But on the right hand side there are some black holes, or better, black traingles that suggest an x0-dependency of the outcome value. Looking from the side showed them more clearly.

However I'm not sure what that means. I am using only 200 initial iterations and 400 for the computation for speed reasons. So I checked two x0 values that show a different outcome in higher iteration depths (10000/20000) and found no difference. Clearly, that's no proof, just numerical evidence for a difference. I'm not sure if those x0 dependencies would vanish by vastly increasing the iteration depths (only applicable with arbitrary precision I guess) or whether they really show that the x0 value sometimes is of importance. But since the Lyapunov value is defined as the limes, there should be no such thing as an x0-dependency in the long-run, so I suspect that those points simply converge to a fixed value (or diverge to infinity) way too slowly for me to detect. Or are there even points for which neither occurs?

Any thoughts on that matter?

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #24 on: March 04, 2019, 02:09:28 PM »
Experimenting with a third axis, I tried something different here. I wanted to see how "connected" the order parts of a 2D image were, so I checked for every pixel (in the plane) how long is the horizontal (in pixel coordinates) streak of consecutive order points and how long is the vertical one. Then I virtually put arcs with the respective diameter over the streak and took the lower of the two heights for the given pixel. Drawing  a line with the color of the 2D pixel then finished the image. I expected some sort of round objects surfaces extending more or less "into the air". Well, some images looked like that, some not. Overall it was a nice experiment but the images do not look very promising from an artistic point of view. But maybe the next idea will be better (speed of convergence).

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #25 on: March 08, 2019, 07:10:19 AM »
"Bob, the builder"

To build a house one needs a metal scaffold (dont't know if this is accurate in real life, but in Lyapunov space...), and then you can build concrete walls with weirdly shaped windows to get some appartments. I tried a lot of different color schemes to get inhabitants appearing, but all I could get was fog.

And I did a first overview picture of some cubic shapes I encountered (I have seen this idea lately in the forum, but can't remember where, so credit to whom it belongs!).

#### Sabine62

• Fractal Freak
• Posts: 667

#### Re: Three dimensional Lyapunov space

« Reply #26 on: March 08, 2019, 01:14:07 PM »
Quote
Bob, the Builder

Great results, again!

#### gerson

• Fractal Furball
• Posts: 293

#### Re: Three dimensional Lyapunov space

« Reply #27 on: March 08, 2019, 05:59:22 PM »
I think chaospro works similar to ldn88_streaks to generate pseudo 3d images, see:
www.deviantart.com/gs13/art/gs-deviant07-98939198

#### marcm200

• Fractal Frankfurter
• Posts: 631

#### Re: Three dimensional Lyapunov space

« Reply #28 on: March 10, 2019, 10:50:30 AM »
@sabine: Thanks. It seems that recently I find a lot of Lyapunov images that somehow lead to kid's games.

@gerson: How is the height of the purple object calculated? Since I enjoy the programming part of generating fractal images much (and was never good in reading software documentation), I think that in the future it might happen again that I find something that is new to me but "old news" to others (reinventing the wheel, you know).

Here are some new cube shapes. Especially the first one is very intersting to me. It is the full cube (in a specific colouring), but one can only see one 2D slice, meaning that in other C-values, the Lyapunov exponents are way different than in that one particular plane. I've not seen this in the 2D images (hence there should be a colouring to produce an image with only one horizontal or vertical streak). I'm currently checking this in a more in-depth and automated manner to see if I missed something.

Does anyone have encountered a similar phenomenon in their type of fractals? Are their, e.g. in the Mandelbort set, specific escape times that only occur in a particular spot - and nowhere else?

• 3f
• Posts: 1376

#### Re: Three dimensional Lyapunov space

« Reply #29 on: March 10, 2019, 11:18:19 AM »
Are their, e.g. in the Mandelbort set, specific escape times that only occur in a particular spot - and nowhere else?
No, the (renormalized) escape times form a smooth continuum which loops around the whole Mandelbrot set: from any escape time value at a point there are two directions you can go that keep the escape time constant, clockwise or anticlockwise around the Mandelbrot set.  These curves of constant escape time are called equipotentials.  For connected Julia sets the same property holds, for disconnected Julia sets the curves eventually split up into multiple parts.

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