Bifurcation fractals discovery

[bkercso]

I recalculated Img #3 with dt=1.5E-5 (10 times smaller), and recolored it. It shows a little bit more of the fractal. But the hard chaotic regions are still blurred. These are the most complex areas of the picture, I think I need more points for this not less dt. On the firs page's fractals I used average 300 points/pixel, but now only 5. I'm developing the program/setting method for higher quality images.
Periodicity test doesn't help. Very interesting that the double pendulum's bifurcation map has not periodic intervals, only points (maybe).
As with the 4th order Runge-Kutta method the error of the calculation is proportional with dt^5, less dt also didn't help me as you see. The error of this calculation 10^5=1E5 times less than at Img #3...

Img #5 log coloring with optimized contrast


lin coloring with saturation

[bkercso]

m1=2, m2=1, L1=L2=1, theta1_initial=0, dt=1.9E-4
x-axis: theta2_initial=0..179 deg
y-axis: theta2 when m1 stops
Average 6 points/pixel. I left the first 10% of points.

Img #6 log coloring with optimized contrast

[bkercso]

m1=m2=1, L1=L2=1, theta1_initial=0, dt=1.5E-4
x-axis: theta2_initial=0..179 deg
y-axis: time from previous event when horizontal speed of m2 =0
Average 4 points/pixel. I left the first 10% of points.

Img #7 It isn't a zoom.

[bkercso]

I found one bifurcation map about double pendulum on the net: http://www.google.hu/imgres?q=bifurcation+map+double-pendulum&hl=hu&sa=X&biw=1280&bih=819&tbm=isch&prmd=imvns&tbnid=GMrVd6hBIjNGzM:&imgrefurl=http://www.sciencedirect.com/science/article/pii/S0094576510003528&docid=4DwFfPnLNOgpYM&imgurl=http://ars.sciencedirect.com/content/image/1-s2.0-S0094576510003528-gr2.jpg&w=512&h=749&ei=OMKXUKzTI6Hl4QTOi4AQ&zoom=1&iact=hc&vpx=649&vpy=217&dur=3744&hovh=272&hovw=186&tx=81&ty=109&sig=101475947789214596179&page=3&tbnh=162&tbnw=111&start=50&ndsp=30&ved=1t:429,r:14,s:50,i:279

I won't pay $31 for it...

[bkercso]

This article also contains some investigation about double pendulum with bifurcation maps:
http://www.google.hu/url?sa=t&rct=j&q=double+pendulum+bifurcation&source=web&cd=2&ved=0CCYQFjAB&url=https%3A%2F%2Fbitbucket.org%2Foangelo%2Fdouble-pendulum%2Fsrc%2Fd672eab7ed2d%2FArticles%2Fsdarticle.pdf&ei=mTuaUJSzIIX3sgbJgYHYCg&usg=AFQjCNGps4okpudWXNTqQV04aivpA6kW4A

[bkercso]

m1=m2=1, L1=L2=1, theta1_initial=0, dt=1E-4
x-axis: theta2_initial=0..179.9 deg
y-axis: theta2 when angle-acceleration of m1=0
Average 20 points/pixel. I left the first 10% of points.

Img #8

[kram1032]

that's clearly a space sqid spider, a space sqider. It has a squid head, squid tentacles, spider legs and the exhaust of a rocket.

[bkercso]

What a pity it's not moving...

[matsoljare]

I can imagine these formulas would be pretty useful for audio fractals as well. Have you heard my audio interpretation of the logistic map? It's the first minute of this vid:

<a href="http://www.youtube.com/v/0jiOSPUITdU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/0jiOSPUITdU&rel=1&fs=1&hd=1</a>

[bkercso]

I didn't hear this before, but I like it! Like wind noise...
I found this circuit before: <a href="http://www.youtube.com/v/bxQr8ql0Hz8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/bxQr8ql0Hz8&rel=1&fs=1&hd=1</a>
It generates the sound of Lorentz-attractor, from 3:40.

[bkercso]

Here is a video of an analog generated bifurcation fractal, looks like the double pendulum's one:
<a href="http://www.youtube.com/v/FS8zNQmaC4c&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/FS8zNQmaC4c&rel=1&fs=1&hd=1</a>

Comment of this video:
"For more information see noch-mehr-davon.de - Advanced lab course at the University of Göttingen"

Unfortunately I didn't find more info about it...

[bkercso]

And a phase modulated bif. map video:
<a href="http://www.youtube.com/v/i6GSBbb0dPk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/i6GSBbb0dPk&rel=1&fs=1&hd=1</a>

More about phase modulated bifurcation map (equations on page 4):
http://csc.ucdavis.edu/~chaos/courses/nlp/Projects2008/RyanJames/paper.pdf

[bkercso]

The first zoom of double pendulum's bifurcation map is done!
I developed an adaptive dt algorithm, which provide uniform calculation accuracy independently from energy of system (initial displacement) and from if region is "smooth" or "hard chaotic". And accelerates about 10-50% (dependent from pendulum geometry and energy).
The method is the follow:

(if dH<1E-20 then dH=1E-20)
if dH>dHmax then undo last iteration and dt(i+1)=dt_min
else dt(i+1)=( dt(i)*dt(i-1)*(dHavg_target/dH)^(1/8) )^0.5
if dt(i+1)<dt_min then dt(i+1)=dt_min
if dt(i+1)>dt_max then dt(i+1)=dt_max ,

where:
dH=abs( (H-H0)/H0 )
H: actual energy of the system (Hamiltonian)
H0: initial energy of the system
dt_initial=dt_min=1E-4 .. 1E-6 (depends on pendulum geometry and desired image quality)
dt_max=50*dt_min
dHavg_target=1E-11 .. 1E-15 (depends on pendulum geometry and desired image quality)

Edited:
I found the article based on which my calculation:
http://www.emte.siculorum.ro/~makozoltan/Szigorlat/s00.pdf
It's wrote in Hungarian, adaptive stepsize algorithm starts on page 66.

Img #9 (zoom into Img #4)
m1=10, m2=1, L1=L2=1, theta1_initial=0, theta2_initial=0..179.9 deg, dt_min=2E-5, dHavg_target=1E-13
Values: theta2 when angular speed of m1 is 0.
average 5.5 points/pixel, 600 x 6600 pixels, calculation time: 2 weeks @3GHz
I left the first 10% of points.

[bkercso]

Here is a higher quality version of Img #6. X-axis is logarithmic.
Average 100 points/pixel (!), 1000 x 1500 pixels, dt_min=3E-6, dHavg_target=1E-13, calculation time: 2 weeks @3.3GHz
The hard chaotic regions are still blurred...

Img#10

[kram1032]

Well, that's why they are hard.