### Detachedly computed Lyapunov-based images

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#### marcm200 #### Detachedly computed Lyapunov-based images

« on: November 29, 2018, 07:02:09 PM »
Mario Markus' Lyapunov algorithm uses xn+1=f(xn) for the iteration and the derivative of f for the computation of the resulting Lyapunov exponent value.

Here I introduced a second function g for the computation of the resulting value that will be used to determine the color of the pixel. g does not need to be related to f, at least not in a pure derivative manner.

Currently this is in an experimental stage and I have yet to explore the possibilities in depth. Maybe it will turn out to be a dead end with regard to finding new structures.

I checked with different iteration depths to see whether the resulting image remains largely unchanged, except for the reduction in grainliness.

"CPU"
seen through the heat sink.
f(x)=4.8*sin(x+r)*sin(x-r), AAAAABBBBABB. Center at a=3.00 b=1.43 size=0.63
g(x)=4.8*cos(x+r)*cos(x-r)
Color [-10.00..-6.00..-4.00..-2.00..0.00..0.50..0.80]

"Waver"
And this looks like a waver in different stages of the production process.
f(x)=2.3*sin(x+r)*sin(x-r), AAAAABBBBABB. Center at a=2.22 b=2.22 size=3.19
g(x)=2.3*cos(x+r)*cos(x-r)
Color [-1.55..-0.80..0.00..0.50..0.80]

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #1 on: November 30, 2018, 09:23:47 AM »
"Alien sports arena"

f(x)=2.6*sin(x+r)*sin(x-r), AAAAABBBBABB. Center at a=3.14 b=1.57 size=0.81
g(x)=2.6*cos(x+r)*cos(x-r)
Color [-1.55..-0.80..0.00..0.50..0.80]

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #2 on: December 03, 2018, 09:55:53 AM »
"France"
several flags of France in this picture.

f(x)=5*sin²(x+r)
g(x)=r-2rx
sequence AAAAAB. Center at a=-0.13 b=2.13 size=1.20
Color [-1.55..-0.80..0.00..0.50..0.80]

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #3 on: December 04, 2018, 02:45:17 PM »
Almost symmetrical in the vertical axis.

f(x)=2*sin(x+r)+2*sin²(2*x+r)
g(x)=sin²(x+2*r)-r*x
seq. AABB. Center at a=4.27 b=4.27 size=5.00
Color [-1.55..-0.80..0.00..0.50..0.80]

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #4 on: December 06, 2018, 09:55:11 AM »
"Directions"
The (could be more pointy) arrow shows the way to the (could be more Mandelbrot-like) structure.

f(x)=3*sin(x+r)*sin(x-3*r)
g(x)=3*sin(x+r)
sequence ABAABAAABAAAABAAAAABAAAAAABBBBBBBBBBBBBBBB. Center at a=0.43 b=0.49 size=1.67
Color [-1.55..-0.80..0.00..0.50..0.80]

"The drop of pure chaos"
In a chaotic world there are only small bits of order (purple) - and those might be in danger by a big drop of pure chaos (white).

(same function and coloring)
Center at a=2.56 b=2.59 size=0.49

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #5 on: December 08, 2018, 12:32:12 PM »
"The volcano of chaos"
A volcano (made out of order) erupting with chaotic magma.

f(x)=2.2*sin(x+r)+2.2*sin²(2.2*x+r)
g(x)=sin²(x+2.2*r)-r*x
seq. AAAAAABBBBBB. Center at a=-3.32 b=-3.31 size=0.54
Color [-1.55..-0.80..0.00..0.50..0.80]

"Watch"
with some unknown symbols.

f(x)=3.5*sin(x+r)*sin(x-r)
g(x)=3.5*cos(x+r)*cos(x-r), AAAAABBBBABB. Center at a=1.53 b=3.16 size=0.81
Color [-1.55..-0.80..0.00..0.50..0.80..1.50..2.50]

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #6 on: December 08, 2018, 12:44:46 PM »
"Fractal border"
The border of the turquois object (to the brownish red areas) seems to be of fractal nature. And it indeed has a box dimension of about 1.47. However it does not separate chaos from order but lyapunov order values of less than -1.55 from those above. Chaos is colored in pure black.

f(x)=r*sin²(x-r)+3*sin3(x+2r)
g(x)=r*x-3*sin4(rx-3)
seq. ABAABAAABAAAABAAAAABAAAAAABBBBBBBBBBBBBBBB. Center at a=0.00 b=0.00 size=6.40
Color [-2.55..-1.55..-0.80..0.00]

and a second one.
Function (value 2.8 instead of 3) and center/size the same.
seq. AAAAABBABBBB. Color [-1.92..-1.00..0.00]

#### 3DickUlus

• • 3f
•      • Posts: 1647 #### Re: Detachedly computed Lyapunov-based images

« Reply #7 on: December 08, 2018, 01:03:48 PM »
Some interesting work here Fragmentarium is not a toy, it is a very versatile tool that can be used to make toys #### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #8 on: December 18, 2018, 09:44:16 AM »
The image here shows quite a nice feature: The same structure (the puzzle piece like object int the 4 corners of the 3x3 grid) shows up but has different color features in the same picture. Normally, reappearing structures are identical.

Here I used a function which is periodic and only r-dependent in the argument (thanks to ThunderboltPagoda and pauldelbrot for pointing that out) which fills the image with a repetitive occurance of that puzzle piece-like object. But to calculate the resulting coloring value, I used the logistic function (non-periodic and directly r-dependent), hence the value changes in accordance with the position in the AB-plane.

f(x)=4.8*sin(x+r)*cos(x+r) ; g(x)=r*x*(1-x) ; seq. AAAAAABBBBBB. Center (-1.76/1.76) size=4.49.

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #9 on: December 26, 2018, 03:03:58 PM »
Very interesting pattern here, looks like a set of different watch bracelets. The image has some kind of qualitative self-similarity, not in the strict mathematical sense however. Narrow coloring intervals used - the more negative the fewer values fall within. Values above zero are colored in white.

Trajectory function f(x)=r*sin²(x-r)-2*sin3(x+2*r)
Computing function g(x)=-2*sin(r*x)
seq. AAAAAABBBBBB. Center (1.67/2.94) size=0.04

0 values at infinity, all values between -1.54562 and 0.5155554, larger than zero: 42.26%

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #10 on: January 17, 2019, 10:31:10 AM »
Another quite weird shape here. Looks like multiple audio signals being read into the sound chip (corresponds a bit to the earlier image "Sound" I posted in the original Lyapunov thread that used a completely different function).

Trajectory function f(x)=1*sin(2x+3r)
Computing function g(x)=r*x*(1-x)+1*sin²(x+r)
sequence ABAABAAABAAAABAAAAABAAAAAABBBBBBBBBBBBBBBB.
Center (-0.47/3.32) size=0.20

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #11 on: January 26, 2019, 11:51:56 AM »
"Detachedly" seems to be a synonym for weird pictures. Here I encountered one (first time with an exp-function) that the eyes have difficulties to focus onto. It looks somewhat "box-y", like someone has spread out a number of tiles and smeared paint over the edges.

Trajectory function f(x)=-8*sin(x+r)*cos(x+r)
Computing function g(x)=4*exp(-x+r)*(sin(x+r)-cos(x+r))
Sequence AAAAAABBBBBB. Center (3.18/-5.30) size=9.0
Coloring with large (size=1 or 2) intervals spanning Lyapunov's values from -10 to 6.

#### marcm200 #### Re: Detachedly computed Lyapunov-based images

« Reply #12 on: March 11, 2019, 12:54:22 PM »
In my 3D Lyapunov thread I wondered about Lyapunov values that only occured in confined regions of the whole image. I checked my 2D images via a script and surprisingly (because I haven't noticed that before) I could actually find 3 images that showed such a phenomenon. All three contain a horizontal or vertical line (red-colored in the attached image) with Lyapunov values that do not occur anywhere else in the image. I haven't noticed that before because the values lie way outside the region I normally set the coloruing limits to. This might be a method to construct a new type of "heat"-image: how far from the average Lyapunov value (of a row, column or the entire picture) is the current one in location a,b.

And, as a nice side note, in the first image "A crack in spacetime" (detachedly computed, entered previously into the image of the month contest) it illustrates the name: The central crack contains a vertical line of red Lyapunov values that are so strange to the rest of the image, that they might actually come from another universe.

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