### Lyapunov diagrams

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#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Lyapunov diagrams

« on: November 09, 2018, 06:55:56 PM »
I recently found my old Lyapunov diagrams and got interested again in computing new ones.

Here the first:

Title: Saber fight
Parameters: f(x)=2.84*sin²(x+r), sequence AAAAAABBBBBB, zoom in the upper right length 4-quadrant.
Exponents between -0,25 and 0.2 are colored using 4 adjacent intervals with linear RGB interpolation.

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #1 on: November 10, 2018, 11:14:59 AM »
Title: Waves in space time

To me this looks like two non-interfering (gravitational) waves changing the fabric of space time.
Parameters: f(x)=2.84*sin²(x+r), sequence AAAAAAAAAABBBBBBBBBB. Zoom in the section around a=2.2 and b=5.9.
Exponents between -1.92 and 0 are colored with 2 adjacent intervals using linear RGB interpolation.

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #2 on: November 10, 2018, 02:31:45 PM »
Title: Scarabaeus
(while its brain is working in blue/white)

Parameters: f(x)=2.461538*sin²(x+r), sequence AAAAAABBBBBB. Around center point a=0.9, b=0.9
Exponents between -1.55 and 0.8, 4 intervals, linear color interpolation.

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #3 on: November 11, 2018, 09:33:07 AM »
Title: Fish
Parameters: f(x)=2.84*sin²(x+r), sequence AAAAAABBBBBB.
Exponents between -0.3 and 0.2, 4 color intervals

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #4 on: November 11, 2018, 03:29:33 PM »
Title: Alien ringfight
Two alien creatures sitting in opposite corners of a boxing ring, ready to fight.

Parameters: f(x)=2.923077*sin²(x+r), sequence AAAAAABBBBBB. Zoom in around a=0.85 ad b=0.85.
Exponents between -1.55 and 0.8, 4 color intervals

#### ThunderboltPagoda

• Fractal Furball
• Posts: 233

#### Re: Lyapunov diagrams

« Reply #5 on: November 12, 2018, 02:12:45 AM »
Nice to see more Lyapunov stuff here, from you and vasyan. Very beautiful images! Inspired by them, I added the b*sin²(r+x) formula to my own program and posted the first results.

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #6 on: November 12, 2018, 09:24:02 AM »
Thanks!

A zoom of the intersection of the two waves in one of the previous images, "Waves in space time".

Title: Borg cube scaffold
That's how I would imagine a Borg starship construction site in Star Trek - dark and green.

Parameter: f(x)=2.84*sin²(x+r), sequence AAAAAAAAAABBBBBBBBBB.
Exponents between -1.92 and 0 are colored with linear RGB interpolation.

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #7 on: November 12, 2018, 04:40:45 PM »
Title: Entry to hell
A flying creature from hell guarding the entrance

Parameters: f(x)=3.692308*sin²(x+r), sequence AAAAAABBBBBB. Zoom in around a=1.7 and b=2.9.
Exponents between -1.55 and 0.8, 4 intervals, linear RGB interpolation again

#### marcm200

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• Posts: 618

#### Re: Lyapunov diagrams

« Reply #8 on: November 13, 2018, 09:40:23 AM »
Title: Fingerprint
Looks like a thumbs' fingerprint on a sheet of paper

f(x)=2.461538*sin²(x+r), sequence AAAAAABBBBBB. Center a=1.9 / b=3.1
Exponents between -1.92 and 0, 2 intervals in black to white. Non-negative values are uniformly white.

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #9 on: November 13, 2018, 02:52:01 PM »
Title: Ghost gown / Handle
The same image is painted by two different color schemes.
The reddish picture looks to me like a ghost gown, where the ghost has left and the gown remained in shape. The greener image resembles a handle to climb up a slippery wall.

Parameters: f(x)=2*sin²(x+r), sequence AAAAAABBBBBB. Center around a=0.7 and b=0.8
Exponents between -2.55 and 0.8 (reddish), 0.3 and 0.8 (greenish), 4 intervals.

With today's speedy computers it's much more fun exploring the sequence and parameters in detail. 20 years ago it would've taken days to compute an image with decent resolution and quality.

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #10 on: November 14, 2018, 11:43:13 AM »
Title: Pure order
Quite astonishing, the first two are exactly the same picture, just with different coloring intervals, the 3rd is a zoom.

The first one looks like a tile in a kitchen. It is common to see regular patterns like these when using small (i.e. smaller than around 1.5) values for the factor b.
The second coloring looks like subatomic particles interacting while circling around a nucleus.
And by zooming in and changing the colors of that subatomic region, one gets something like a brain MRI as seen in medical drama TV shows.

Parameters: f(x)=1.05*sin²(x+r), sequence AABB. Center point at a=0.00 and b=0.00
1st image: Exponents' color intervals [-1.55..-0.80..0.00..0.50..0.80]
2nd: [-0.25..-0.20..-0.10..0.00..0.10..0.20]
3rd: [-0.40..-0.30..-0.20..-0.10..0.00]

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #11 on: November 14, 2018, 04:10:55 PM »
Parameters: f(x)=2.35*sin²(x+r), sequence AAABBB. Center point at a=0.96 and b=2.05
Color [-1.55..-0.80..0.00..0.50..0.80]

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #12 on: November 15, 2018, 09:09:01 AM »
"Hexagon"
A quadratic magnifying glass lies in the upper left corner that exhibits some optical errors to have the hexagon border appear to be broken.

f(x)=2*sin²(x+r), sequence AAAAAABBBBBB. Zoom in around a=0.52 and b=2.17
Exponents between -0.3 and 0.2, again 4 intervals using linear RGB interpolation.

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #13 on: November 15, 2018, 03:49:56 PM »
Title: Magnetic spot
Looks like the surface of the sun with swirling hot gas and a dark spot

f(x)=2*sin²(x+r), sequence AAAAAABBBBBB. Zoom in around a=0.75 and b=1.65.
Exponents between -0.5 and 0.7, 4 intervals

#### marcm200

• Fractal Frankfurter
• Posts: 618

#### Re: Lyapunov diagrams

« Reply #14 on: November 16, 2018, 07:45:56 AM »
"Karyotyping"
This is a collage out of 4 images showing two X and two Y chromosomes that are paired with each other. The slight shape difference was deliberately computed using small differences in the factor (5.8 or 5.828).

f(x)=5.828 / 5.8 * sin²(x+r), sequence AABABB. Center points at a=1.32 b=0.73 / a=0.39 b=0.49
Color [-1.55..-0.80..0.00..0.50..0.80]

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