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Hi!

I can find a lot of beauty images about several fractals on the net, but very few about bifurcation fractal(s) and those are low quality ones.
Let this topic about this types of fractals, which are also very nice ones I think.
I used linear coloring with saturation (white color) and logarithmic x-axis, except if I write other.

(I use my self developed program for fractal generating, it was written in Pascal. It quite simple, a little hard to set zoom area and generating image with this, but I like it. :) )

First: The logistic map: x(i+1)=r*x(i)*(1-x(i)) ; where 0< x(1) <1 , 0< r <4
(http://en.wikipedia.org/wiki/Logistic_map (http://en.wikipedia.org/wiki/Logistic_map))

the red lines show the first few iterations
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_12_10_12.png)

the first few iterations, log. coloring
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_12_56_24.png)

x(i+1)=r*x(i)*(1-x(i)^2) ; where -1.2< x(1) <1.2 , 1< r <3
zoom
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_1_01_51.png)

if x(i)<0.5: x(i+1)=r*x(i), else: x(i+1)=r*(1-x(i)) ; where 0< x(1) <1 , 1< r <2
log coloring
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_1_05_38.png)

Delayed logistic map: x(i+2)=r*x(i+1)*(1-x(i)) ; where 0< x(1), x(2) <1 , 1.99< r <2.271
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_1_08_10.png)

log(log(x-axis))
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_1_10_26.png)

zoom, log(log(x-axis))
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_1_11_30.png)

zoom
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_1_12_32.png)

zoom, log x-axis (?)
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_00_38_177661858.png)

Delayed, averaged logistic map: x_temp=(x(i+1)+x(i))/2   x(i+2)=r*x_temp*(1-x_temp) ; where 0.83< x(1), x(2) <1.165 , 3.8< r <4.617
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_02_57_177752075.png)

zoom
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_02_56_17774225.png)

zoom
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_01_52_177721192.png)

zoom, log coloring,
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_01_53_17773637.png)

zoom
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_00_39_177671967.png)

zoom, log coloring
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_00_37_177641523.png)

zoom
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_00_38_17765445.png)

Delayed, geometric averaged logistic map: x_temp=sqrt(abs( x(i+1)*x(i) ))   x(i+2)=r*x_temp*(1-x_temp) ; where -1.215< x(1), x(2) <1.22 , 3.68< r <4.828
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_28_05_15_12_40_52.png)

with different saturation level in coloring
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_01_52_177711074.png)

first few iterations, log coloring
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_01_52_177701237.png)

zoom, first few iterations, log coloring
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_01_51_17769528.png)

zoom, first few iterations, log coloring
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_00_39_177681906.png)

The next one is the phase space of a simulated physical system: a dripping tap.
The model was simple:
The water drop was a mass point, which sits the end of a spring. The spring force corresponds to the force from the surface tension of water. When the elongation of the spring was bigger than a constant, the drop gets smaller (dripping). When dripping, the mass reduction of the drop and the reduction of elongation are depend on the actual velocity of the drop.

Note that this simulation do not gives you a point on the picture at every iteration, only if one drip is dropped. So calculation time is much longer than in the previous cases.

(http://highflowheating.co.uk/images/dripping-tap.png)

The bifurcations are not smooth, maybe bacause of the suddenly changes of the elongation of spring when drop. I didn't calculate with the oscillations of the spring... Edit: Or I should use smaller dt, or 4th order Runge-Kutta method instead of Euler one (see this later).
X-axis: parameter: flow rate of water
Y-axis: velocity of drops, 200,000 point/column (dt=5E-6 sec)
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_02_57_177761247.png)

The next is the Bifurcation diagram of double pendulum
(Description of the double pendulum: http://en.wikipedia.org/wiki/Double_pendulum (http://en.wikipedia.org/wiki/Double_pendulum)
Equations I calculated with are here on page 13 (optimize these before run!): http://online.redwoods.cc.ca.us/instruct/darnold/deproj/sp08/jaltic/presentation.pdf (http://online.redwoods.cc.ca.us/instruct/darnold/deproj/sp08/jaltic/presentation.pdf))
(http://upload.wikimedia.org/wikipedia/commons/thumb/7/78/Double-Pendulum.svg/294px-Double-Pendulum.svg.png)

Details of my setup (in SI system): L1=L2=1, m1=m2=1, theta1_initial=0, theta2_initial=0..pi, g=9.81, dt=1E-5, 3000 points/column
X-axis: theta2_initial (log. axis)
Y-axis: values of theta2 when m1 stops (w1=dtheta1/dt=0)
Calculation time: 17 hours @2.5GHz
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_57_177592349.png)

Zoom (dt=5E-6), log coloring (As you can see the pics are not symmetric...)
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_05_177582251.png)

I'm working on higher quality images... :)

there are some awesome works in this.
For some reason I never thought of this before, but what about a magnetic double pendulum?

Thank you.
Good question. Maybe this helps: http://www.math.tamu.edu/~mpilant/math614/StudentFinalProjects/Musick_Final.pdf (http://www.math.tamu.edu/~mpilant/math614/StudentFinalProjects/Musick_Final.pdf)  :D
But if you use real magnetic field model, not uniform, the system difficulty get more-more higher. Easier to do finite element simulation for these systems than calculate exact equations I think.

that double pendulum seems to be awfully nice chaotic :)

Yes! I love the shapes of the left part of this image:
(http://kepfeltoltes.hu/121022/Bitmap06_dt_5E-6_www.kepfeltoltes.hu_.png)

Nice pics. Bifurcation maps hadn't been meant to be pretty, they mostly are used to show that the mandelbrot set is meaningfull;)

What would be a delayed Mandelbrot set?...

It might be interesting to consider the complex maps.

Z(i+1) = C * Z(i) * ( 1 - Z(i)) where C is a complex version of r.

I suspect there will be restrictions on the phase of C for bounded maps.

Very interesting topic! For some reason I simply never asked myself what would a bifurcation diagram look like with another function than the logistic map. Looking forward to the next images.

This gives a 4D fractal: C(re), C(im), Z(re), Z(im). Interesting...

I tried taking some projections of the complex logistic map and it looked pretty boring. I should learn to test first post second.

Have you images to share? Maybe boring area did you see, or in poor quality...

If I recall correctly, I saw some images of the complex logistic map and it essentially showed Antibuddhabrot-like extensions. z-axis was mapped in the same way as the typical logistic map, which caused those extensions to be multiplied along several heights. It looked fairly pretty.
Also, if you look at the right planes of the Buddhabrot set, a logistic map is clearly visible.
Like this:
(http://farm5.static.flickr.com/4031/4428516360_80476119d0_o.jpg)

Wow, it's wonderful!

OFF:
And what about delayed Mandelbrot? It would be:
Z(1)=C
Z(2)=Z(1)^2+C
Z(i+2)=Z(i)^2+Z(i-1), or Z(i+2)=Z(i-1)^2+Z(i)...
Anybody? :)

The primary limitation appears to be for most phases of C (the complex value formally known as r) the iterations become unbounded. The first picture is the real map between 0 to 4 and is provided for reference. The second shows the same range for a phase shift of C of 0.52.

Ok, but you see a 4D fractal on a 2D plane. Did you choose the right section?
And I suggest to use more iterations, log coloring, oversampling and higher resolution for higher quality images.

Offtopic answer: didn't work too well, in terms of what the resulting image looks like when displayed as an escape time fractal. A bit mysterious, though, because the result has a triple symmetry, despite the numbers being squared.

I changed the definition slightly to
Z(0) := 0,
Z(1) := C,
z(n) := Z(n-1)^2 + Z(n-2),
which is equivalent, except that it starts at zero. This would have opened up the possibility of rendering Mandelbrot (starting value 0, C pixel dependent) and Julia (starting value pixel dependent, C invariant) variations.

Great!
Have you pics for post? :) I'm curious...

I found a Henon bifurcation map: http://commons.wikimedia.org/wiki/File:Henon_bifurcation_map_b%3D0.3.png (http://commons.wikimedia.org/wiki/File:Henon_bifurcation_map_b%3D0.3.png)
This is high resolution: 6,5 Mpix

(http://www.vectorizer.org/laggedMandel.png)

Interesting section.
Cube also has tripple symmetry...

I continue the simulation of double pendulum and create bifurcation map. Changed from Euler method to 4th order Runge-Kutta method, which taken approx. 50 times acceleration (!). (The method described with source code in this article: https://freddie.witherden.org/tools/doublependulum/report.pdf (https://freddie.witherden.org/tools/doublependulum/report.pdf), and on wikipedia: http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods (http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods)).

Setup was as at the first image, except the x-axis is linear.
Calculation time was the same (17 hours @ 2.5GHz), but the quality is much better.
Image is 1200x1800 pix, 4800 points/column, dt=1.3E-4.
I took some measurements and I found that with RK4 method dt is proportional with maxiter^(-0.4) at constant image quality (and with maxiter^(-1) with Euler method). Conclusion: higher resolution image not requires as more calculation time as with Euler method.
I measured the relative change of Hamiltonian (=sum of mechanical energy of the pendulum) after maxiter, because I experienced image quality depends on this parameter. This relates the precision of the simulation: theoretically the Hamiltonian are const. in time.
The relative change of energy after 4800 oscillations (not iterations!) of upper pendulum was: abs(dH/H0)=1E-17 at left side (small displacements) and 6E-14 at right side (large displacements).

I think the vertical widenings are from the first few iterations. I left the first 1% of points, but I took 20 times less iterations as for the other bifurcation fractals I posted. This one still requires a lot of processor time: one oscillation (means one point in the fractal) requires hundreds or thousands iterations. This image are made from 8.64 million oscillations.

Img #3 with log coloring
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_04_177551379.png)

with linear coloring with saturation
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_03_177541979.png)

that delayed Mset might be a distant relative of certain Julia-sets, Cantor-dust and/or the plot of p-adic numbers:
http://en.wikipedia.org/wiki/P-adic_number - image on the right.

It's a very close relative to the Manowar fractal in Fractint.

manowar
c = z1(0) = z(0) = pixel;  z(n+1) = z(n)^2 + z1(n) + c;  z1(n+1) = z(n);
Parameters: real & imaginary perturbations of z(0)

manowarj
z1(0) = z(0) = pixel;  z(n+1) = z(n)^2 + z1(n) + c;  z1(n+1) = z(n);
Parameters: real & imaginary parts of c

Here's a test animation I did a while back of the Julia set, tracing a spiral path for the parameters.

http://vimeo.com/moogaloop.swf?clip_id=49998325&amp;server=vimeo.com&amp;fullscreen=1

Ryan

New bifurcation map of double pendulum:
m1=10, m2=1, L1=L2=1, theta1_initial=0, theta2_initial=0..179.9 deg
Values: theta2 when m1 stops.
Calculation time: 18 hours @2.5GHz

Img #4 log coloring with optimized contrast
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_27_05_15_3_57_40.png)

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
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_05_177572189.png)

lin coloring with saturation
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_04_177561330.png)

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
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_56_32_177531735.png)

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.
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_55_21_177481445.png)

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 (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... :)

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 (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)

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
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_55_21_177471096.png)

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.

What a pity it's not moving... :D

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:

http://www.youtube.com/watch?v=0jiOSPUITdU

I didn't hear this before, but I like it! Like wind noise...
I found this circuit before: http://www.youtube.com/watch?v=bxQr8ql0Hz8 (http://www.youtube.com/watch?v=bxQr8ql0Hz8)
It generates the sound of Lorentz-attractor, from 3:40.

Here is a video of an analog generated bifurcation fractal, looks like the double pendulum's one:
http://www.youtube.com/watch?feature=player_embedded&v=FS8zNQmaC4c (http://www.youtube.com/watch?feature=player_embedded&v=FS8zNQmaC4c)

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...

And a phase modulated bif. map video:
http://www.youtube.com/watch?NR=1&v=i6GSBbb0dPk&feature=endscreen (http://www.youtube.com/watch?NR=1&v=i6GSBbb0dPk&feature=endscreen)

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

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 (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.
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_55_20_1774627.png)

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
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_55_20_177451636.png)

Well, that's why they are hard.

I've done a little experiment with Img #6:
Every time when I set initial conditions (remember: theta2_initial changes along x-axis)
- I calculated H0 (initial energy),
- set theta2_initial=Const*theta2_initial
- and calculate omega2 for same H0 (omega2 is the angular speed of lower pendulum)

Img#6 (original)
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 omega1=0
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_56_32_177531735.png)

Axis are linear and like as @Img #6.
Img #11 Const=0.5
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_02_57_17778298.png)

Img #12 Const=0.15
You can see in this picture that you can put high energy into the system without chaotic (and periodic?) motion if theta2_initial is small.
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_2_02_57_177772181.png)

I recalculated Img #6's calculation and plotted with different rendering criterion:

m1=2, m2=1, L1=L2=1, theta1_initial=0, dHavg_target=1E-11, dt_min=1.5E-4
x-axis: theta2_initial=0..179 deg (log allocation)
y-axis: beta2 when beta1=0 (betas are angular accelerations)
Average 12 points/pixel. I left the first 10% of points.

dHavg_target=1E-11 is high a little bit even at this low points/pixel rate: image quality in the two white ellipses are poor, blurred.
Calculation time: 14 hours @3GHz

Img #13
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_55_20_177441154.png)

I found that it was possible to make an image exactly like the 1st one posted on the thread using a sound wave. I have a program that plots a wave on the graph as it is played 44,000 times a every second,

I discovered an image like this , exactly like the 1st one on the thread, with an audio program which was plotting a graph of the sound 44,000 times a second, I had the loop in the sound, delayed by one sound sample, which sent a highly distorted and filtered high-frequency version of the wave back into itself, I think perhaps controlling the frequency modulation or something, anyway when I played with different settings it plotted a graph exactly like the 1st one here! the beginning of the wave was a regular low-frequency line, and asked the line progressed increasing numbers of high frequency feedback chaos lines forked off of each other in other words the wave was changing exponentially more in the middle of the graph and becoming regular at the end again.fun!

Hmm... upon looking back at the origins of this thread, I saw again the colored versions of the images and when I saw your inclusions of the early iterations as red, I was reminded of the "spectal" Buddhabrot, this forum has spawned a while back.
Do you think you could try to do one of those? Like, add up the per-iteration results not directly as exposures but rather as varying wavelengths of light.

The spectral buddhabrot had two really nice features: It really looked like all the different buddhabrots you get for different iteration-dephts across the used interval and it had the apprearance of a thin oil-film on water or a soap-bubble.
Potentially, besides being really pretty, it might also give you some unexpected insight into the structure of the noisy/blurry parts.
(http://www.fractalforums.com/gallery/3/838_27_07_10_1_02_35.jpeg)
(by cbuchner1 found here (http://www.fractalforums.com/images-showcase-%28rate-my-fractal%29/new-buddhabrot-gallery/msg46997/#msg46997), however that wasn't the first time this technique got used. I'm not sure who did the first render as I am unable to rediscover the corresponding thread)

Pictures please! :)

I comprehend what you talked about just now. I'm thinking about this, thank you. It might be a very good idea!

Blurry parts are interesting. I'm playing with Poincaré-sections from chaotic regions and for example @dHavg=1E-13 these are contoured while bifurcation diagram aren't. But I have to do more iterations and higher quality- and more images...
Maybe these are not blurry but too complicated for this horizontal resolution?

I recalculated Img #3:

m1=m2=1, L1=L2=1, theta1_initial=0, dHavg_target=1E-13, dt_min=3E-6
x-axis: theta2_initial=0..179.9 deg
y-axis: theta2 when omega1=0
Average 60 points/pixel (!), 2 Mpixels. I left the first 5% of points.
Calculation time: 8 days @12*2.67GHz (!)

Img #14
The vertical widenings around the second trisect point of x-axis are not from first iterations as I assumed before, there is a chaotic region with slow quasi-periodic to chaotic transition.
And yes, the 947th, 959th, 971th and 1800th pixel columns are missed, all white. Coding error: at this point I had to restart the calculation to continue (machiene turned off).
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_53_35_17742657.png)

Here it is in twice vertical resolution (average 30 points/pixel), without optimized contrast.
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_53_35_17743507.png)

A can't stop generating images, altough they are quite similar to each other. :)
In this new one I plotted theta2/theta2_initial normalized values instead of theta2 values. This normalization fitted the fractal to image area, and you already can see the quasi-periodic patterns even at low initial displacements. The environments of the two biggest "black point" are also good observable.

m1=10, m2=1, L1=1, L2=10, theta1_initial=0, dHavg_target=1E-13, dt_min=3E-5
x-axis: theta2_initial=0..150 deg  (log. allocation)
y-axis: theta2 when omega1=0
Average 10 points/pixel, 1.5 Mpixels. I left the first 10% of points.
Calculation time: 40 hours @2.5GHz

Img #15
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_53_34_17741260.png)

Yeah, well maybe you'll reveal something new once you get to the wavelength mapping.
Still, you did reveal some beautiful patterns.

one question i have

what does the density mean ??!
i mean the density is stronger at the upper and lower ends of the streaks, in between it is more light, does it mean that such events are more likely to happen !? at the exact upper and lower ends of a branch ?

the denser, the more often that place gets "hit" by what ever process is used for any given bifurcation diagram.
It must be that extreme values are hit more often than in-between ones.
I'm not sure if that's an accurate explanation but if you think of a typical sine-wave-trajectoiry-density of a simple harmonic oscillator, you'll find a similar increase in density towards the endges, due to change in velocity.
It'll slow down at the edges, thus creating more apparent density when averaged over all times, while it'll speed up at the point of stable equilibrium where velocity is maximized, resulting in reduced time-averaged density.

Since these bifurcation diagrams basically plot trajectories over time, they approach that same time-average.

hmm, thinking of the book hitchickers guide to galaxy, is this an indicator for the possibilty of events ?!
i mean, the more dense, the more possible an event is ?! considering the chaotic behaviour would this
be a hint for the possibilities of an event ?!

I'm thinking about for what I can base wavelength changeing, for what physical quantity.

Not exactly, these diagrams don't based on trajectories - unfortunately. If would be then I would get several points during one oscillation of the pendulum. But I get only 1 or 2, because I get a point only when the movement meet to the rendering criterion (for example when the angular speed of the upper pendulum is zero). Where the image is darker there are more points. Higher probability? We can say that but this is a deterministic system not a random one.
Because of that the geometry- and color patterns are really very interesting for me. The common thing in most of biurcation diagrams of double pendulum that there is a double dark band. At some systems they are close to each other (Img #3), at other systems they are moving away from each other with higher theta2_initial (Img #4).
I would say that the dark bands and dark points are attraktors of the given system.

I thought of mapping the wavelength simply to time, e.g. to iteration depth.
An alternate form might be to map it to orbit length if you keep track of that.

I notice that symmetry is broken in some graphs for higher energies. I would assume that the physics is still symmetric. Is this caused by the initial conditions? For example is it a matter of spinning clockwise vs counter clockwise?

We can see the broken symmetric in higher enegies, yes. (Maybe it breaks also in lower energies but because of the limited resolution we cannot see this (?). ) As you wrote, the only asymmetricity we put into the system is the initial conditions, and this is enough, so the motion is not symmetric. Values on the vertical axis are +- values, the midpoint of the axis is the zero point.

Can you integrate the total accumulated angle of each pendulum (How many times they spin around)? This would provide one test for the broken symmetry. 

I already integrated it in my head: zero. :) Even at max energy theta1_initial=0, theta2_initial=179.9°. So the pendulums don't spin around.
The asymmetricity what we see is something this: at X energy quasi-periodic points are around theta2=Y1, Y2 and -Y1+dy1, -Y2+dy2 (so when omega1=0 then theta2= around Ys).
As at lower energy there are hundreds of quasi-periodic points maybe dys are less, if they are exist.

Img #6 (again)
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_56_32_177522389.png)

I assume by maximum energy you mean both pendulums pointing up. If the system starts with that energy, then if the fixed pendulum is pointed up and the second pendulum is pointed down there will be enough energy for the primary pendulum to roll over.

That's maximum potential energy. However, if you put them *exactly* upright, you're actually in an unstable equilibrium and they would just stay like that.

No, the upper pendulum starts at 0° (=theta1_initial) and the other pendulum starts 0..179.9° (=theta2_initial). If I enable to roll over (put enough energy for this) the motion will be too chaotic and I won't get nice patterns.
Anyway the motion is asymmetric, like this:
http://article.wn.com/view/2009/12/09/Double_pendulum_video/ (http://article.wn.com/view/2009/12/09/Double_pendulum_video/)

I've done a fast calculation with Img #6's setup for check your idea. I increased pixel values 't(from previous event)/dt_min/10' instead of '1' at every event (when motion meets the rendering criterion). The result almost the same then Img #6. It's interesting...
Coloring is the same, points/pixel rate is almost the same (=4 vs 6). But it has less color depth, I think because of the following: which pixel are hitted more often there get less 't/dt_min/10' because 'more often' --> 'less time from previous event'. But we can see the patters so 'more often' wins over 'less time from previous event' (consider we don't know the sequence of points).

Img #16
Alternative bifurcation diagram
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_56_32_17751185.png)

too bad it is too small ;)

Yes, this is a pilot run (took half an hour of my life :) ).

Img #17
Same but increased pixel with '10000*theta1 (with sign)' instead of '1'.
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_56_32_17750743.png)

Img #18
Increased pixel with 100*abs(theta1).
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_56_32_177491325.png)

The patterns are very stable! ( Probably the lower color depth at Img #16,#17 caused by high values of pixels; because of this I used 100*abs(theta1) last time instead of 10000*abs(theta1). In white area the pixel value =0, so the interval between maxvalue-minvalue was small vs. maxvalue-0 what I colored with grayscale.).

My first explanation for the stability of patterns: correlation between theta2 vs time/theta1 is poor. If I see the Poincaré-section which I generated I am strengthened in this. So doesn't matter if I increase pixel value with '1' or with a 'random value'.

Here is the theta2 vs theta1 Poincaré-section of Img #6's setup @theta2_initial=160° (rendering criterion was the same: omega1=0):
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_58_177621030.png)

BUT: theta2_initial=160° is a chaotic region. In a quasi-periodic region this would be 2 curves, which means a higher correlation. So my explanation is not an evidence for me yet...
2 curves @theta2_initial=128°:
(http://kepfeltoltes.hu/121203/Bitmap_www.kepfeltoltes.hu_.png)

so, dont really got what you changed, are the lighter areas lesser visited ?!

btw, have you tried rendering a 1 dimensional mandelbrot bifurcation diagram !? i mean you can execute the mandelbrot function simply on a real number base z^2+c ...

so basically, the color mixing makes it all grey?

Lighter areas has less value. I registered under pixel 100*abs(theta1) at every visit, not the number of visit events.

On which real number it is working? if z<1 then z^2<z else z^2>=z.
If I take complex z with fixed Im(z)=0 my z(i) points will has Im(z(i))<>0.


If I understood correctly you based coloring on pixel values, and based pixel values on trajectory. I based coloring on pixel value, based pixel value on other parameter of system. And I used greyscale color palette. Is it the same?

(This topic has the most replies in this category! :D )

Oh, well, the whole point of using a color pallette is to get rid of grey-scale colors. I'm not entirely sure if what you propose is exactly like I imagined it but if it is, then all you'd need to do is to use a color pallette, generated by converting wavelength responses to RGB values, or, as a first approximation, to use a rainbow color pallette...

I see (I think). So I would have to use e.g. HSV colorspace, Hue depends on other physical parameter (time, trajectory etc.) and Value depends on visit events of the pixel?
I wrote a HSVtoRGB converter into my rendering program before, maybe I develop my fractal generator program for store 2 numbers/pixel (for Hue and Value) and try this type of coloring...

something like that. Note how in the image I posted before (which is just reused from this forum) the Buddhabrot looks like made from a rainbow. I think it's colored by essentially linearly mapping iteration depth (e.g. time) to wavelength. I'd essentially like to see that but applied to your bifurcation map.

I see no difference between using color palette or grayscale palette. Moreover, at monochrome palette you know the sequence of colors: darker means higher. The novelty would be if I'd color pixels according to 2 parameters not only 1: darker means higher depth and colors means something special.
Buddhabrot shows the same patterns also in grayscale (?):
http://www.fractalfreak.com/BuddhaCompilation/BuddhaGallery.html#row6 (http://www.fractalfreak.com/BuddhaCompilation/BuddhaGallery.html#row6)

I found a site with a lot of physical simulation ideas (for creating bifurcation maps and other fractals :) ):
http://www.myphysicslab.com/ (http://www.myphysicslab.com/)

Ok, I'll post the image again:

(http://www.fractalforums.com/gallery/3/838_27_07_10_1_02_35.jpeg)

Note how the blue regions go out really far, the green regions are much more contained but the branches are sort of wide (both cases of low iteration depths), while for yellow and finally red values, the whole structure becomes subtler and subtler.
So I guess, based on those observations, this is done by inversely mapping wavelength to iteration-depth, or stated differently, it maps the corresponding colors' frequencies to iteration depth.

Since you only go through the visible spectrum once, it is similarly readible to a simple grey gradient but it targets a wider section of the human visual system, so the data is more distinquishable to us.
(cbuchner1, if you read this, can you confirm wether that's how you did it?)

kram1032: I red everything I can found about generating and coloring buddhabrots started from your link. I had no new idea to color my fractals... Bifurcation maps already colored as buddhabrots.
I try to color according to 'actual value - last value' of theta2, found a little bit different pattern but not significant:

Img #19 original coloring. dHavg=1E-8, average 3 points/pixel
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_58_177602475.png)

Img #20 special coloring detailed above. There are some difference in the middle columns of pixels.
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_58_58_17761376.png)

I rendered a 0.001 deg narrow band of chaotic region of Img #6. I think the reason is why we cannot see contoured patterns inside chaotic regions is they are too dense.

Img #21
m1=2, m2=1, L1=L2=1, theta1_initial=0, dHavg=1E-13
x-axis: theta2_initial=179.2 .. 179.201 deg
y-axis: theta2 when omega1=0
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_28_05_15_1_39_19.png)

yeah, that sure does look noisy as heck.
Ok, so no new insights here, I guess. Oh well...
As said, though, I wasn't able to find the original thread that lead to the first instance of a wavelength-based Buddhabrot...

This is a 1E-5 narrow band of chaotic region:

Img #22 Same as Img #21, but theta2_initial=179.200425 .. 179.200435
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_53_34_17740278.png)

nice. Now, patterns slowly become apparent.
It has some pretty arcs in it.

Yeah. The arcs are trange. I don't know yet what are these. I left the first 10% of points (and there are 3600 points/column), so is not from first iterations.
This is an approximation with dHavg=1E-13. I don't know how much this is similar with "reality", so what would be if I use higher precision.
I zoomed in this region...:
(http://kepfeltoltes.hu/121206/Kett_singa_nf12_179_2__179_201_fok__dHavg_1E-13_www_kepfeltoltes_hu__www.kepfeltoltes.hu_.png)

A 2.7E-9 deg wide section of upper chaotic region:
theta2_initial=179.200425383° ..
                   179.2004253857°

Img #23 2.9 Mpixels. I think my 20 digits- and dHavg=1E-15 precision is not enough for this.
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_53_34_17739984.png)

Img #24 Cowebs of Img #10.
theta2_initial=130.3° .. 133.3°, dHavg=1E-12, 20 points/pixel, 2.5 Mpixels, 2x oversampled
Simulated physical oscillation time of double pendulum: 1 year
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_50_32_177381928.png)

A zoom without contrast optimization
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_50_32_177371220.png)

Img #25 A zoom of Img #11.
theta2_initial=77.2° .. 79.8°, Const=0.5, 4 points/pixel, 3.2 Mpixels, dHavg=1E-13 (assume dHavg=1E-11 would be enough)
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_50_31_17736305.png)

Small displacement section

Img #26
theta2_initial=0..20°, 5 pojnts/pixel, 1.1 Mpixels, dHavg=1E-13, others as at Img #6 (but I undrawed fractal to fill image)
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_50_31_177352092.png)


Img #27 zoom of Img #26
theta2_initial=0..4°, 10 points/pixel, 2.1 Mpixels
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_50_30_17734561.png)

Oh gosh that's absolutely bufurcitabulous.

I set up an sound oscillation circuit in the computer which made those fractals once, except my HDD from that year was hit by a fractal light from the sky which had a branch pattern and a high voltage, so i don't think i have kept the circuit for the fractal oscillator, anyways it was too chaotic for tonal sound. the sound of the lightning that hit the house was similar to when a car hits the wall of the room you are standing in, and then everything was fine for a couple of days and then my power supply started smoking.

i did find a piccy of it in one of the less excited modes,


it went from this kind of waveform as in picture, to exacltly the same waveform as in the giant pictures, through various destabilisation phases. The digital setup was on program called Reaktor and it was an oscillator going through a peakEQ filter that boosed the signal and sent it back into the oscillator frequency.

it was mostly a stable oscillator, unitil the filter sent back frequencies that made the oscillator totally go crazy.

This is the bifurcation map of a dripping tap (dripping faucet) model published by Thomas Schmidt and Marko Marhl in 1997 ("A simple mathematical model of a dripping tap", http://fy.chalmers.se/~f7xiz/TIF081C/drippingfaucet.pdf (http://fy.chalmers.se/~f7xiz/TIF081C/drippingfaucet.pdf)).
x-axis: water afflux (1E-7..1E-4) (linear scaling)
y-axis: speed of drops (0..1)

Simulation used Runge-Kutta numerical method with semi-adaptive stepsize: dt was 2E-5, and when dropped undo last step and switched to dt=dt/100. Did this 3 times at every drop, so I got million times precision around droppings.

Picture made from 20,000 drops/column, calculation time was 15 hours on 2,5 GHz processor core.

Img #28 Dripping faucet: velocity vs. afflux (whole map)
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_09_06_15_4_13_50.png)

wow, another thread I hadn't noticed before.. incredible pictures bkercso! I didn't know there is so much variation in simple bifurcation.. fascinating!
please keep em coming!

Thanks, Chillheimer! I feel strong interest in structure of chaotic phisical systems, so I plan play a little with dripping tap and a bouncing ball on double slope model in the future... :)

Long way to eternity :) (Zoom of Img #28.)

x-axis: water afflux, 4.2E-5..4.47E-5
y-axis: velocity of drops (~0.2..0.5)

Average 50 points/pixel, dt=2E-5, calculation time was 53 hours, log. coloring.

Img #29
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_12_06_15_10_36_52.png)

A new topic is tarted for bouncing ball's fractals:
http://www.fractalforums.com/new-theories-and-research/re-bifurcation-fractals-discovery/msg84980/#msg84980 (http://www.fractalforums.com/new-theories-and-research/re-bifurcation-fractals-discovery/msg84980/#msg84980)

I rendered a zoom of dripping tap's bifurcations.
x-axis: water afflux (4.43E-5..4.95E-5) (log scaling)
y-axis: time between drops (0.2..0.7)
Calculation time: 70 hours @2.5 GHz
Log coloring, average 10 points/pixel

Img #30
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_15_06_15_7_23_04.png)

I tired the nebula rendering on these (same method as in the buddhabrot nebulabrot renderings).  It doesn't help bring out any more details.  Here are some tests...

Logistic Map

(https://farm1.staticflickr.com/255/19135044465_1930cb39e5_c.jpg)

Henon Attractor

(https://farm1.staticflickr.com/362/18512516424_b5cb70ca44_b.jpg)

Delayed Logistic Map

(https://farm1.staticflickr.com/287/18947329730_5929f5e7c1_b.jpg)

It seems like as you plot the Y axis pixels for each iteration loop they cover a random spread and do not concentrate in areas for low iterations vs higher iterations so you end up with an even almost grayscale soup.

Using other shaded color palettes to map the iteration hit counts of pixels can help to tease out more details depending on the formula and palette.  These are some quick test render results from the latest version of Visions Of Chaos http://softology.com.au/voc.htm

Logistic Map

(https://farm1.staticflickr.com/334/18916750908_a2868e6e86_c.jpg)

Henon Attractor

(https://farm1.staticflickr.com/278/18913768240_079091bebc_c.jpg)

Delayed Logistic

(https://farm1.staticflickr.com/268/18918318039_7531c9eedc_c.jpg)

Bouncing Ball

(https://farm1.staticflickr.com/274/18948150000_3fde960e15_c.jpg)

Jason.

Could you share the code for the iteration loop?  I would love to experiment with it.

Your image is so much more detailed that the included sample diagram in the paper you linked to.  You should email the authors and give them a look at your results.

Jason.

Shure, of course! I did not integrated it into my big program, but I share it now. Periodicity test accelerates it, but I think adaptive stepsize would not, because water afflux is constant, so system is always changing with almost the same velocity.
For further info please see the first linked article.

Parameter- and value range (v and t devided by picturesize):

water afflux: 1E-6..1E-4
v: 0..1
t: 0.1..0.6
picturesize: 0.0003 for v, 0.003 for t

Initial values:

The iterated code (use 4rd order Rungke-Kutta method):

When l>lmax I undo last step and continue with dt=dt/10. Do this 6 times.

After when l>lmax again, dripping:

Emailing to authors is a good idea, but dripping faucet is one of the most discussed chaotic system. An other author pair (Ken Kiyono and Nobuko Fuchikami) publicated about this system approx. over 10 years. It is worth read theirs results:
http://arxiv.org/pdf/chao-dyn/9904012.pdf (http://arxiv.org/pdf/chao-dyn/9904012.pdf)

(http://s11.postimg.org/6rxsowg77/Fluid_dynamics_of_dripping_tap.png) (http://postimage.org/)

Your last pic about bouncing ball with this linear coloring and oversampling is really very detailed. Thx for it!

I recalculated Img #30 with average 35 points/pixel, it took 9 days. Virtual physical dripping time was approx. 1 year.

Img #31

(http://nocache-nocookies.digitalgott.com/gallery/17/4917_27_06_15_10_35_30.png)

                 ***********************
The topic of my software Bifurcation Fractal Plotter is started! :o
http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/ (http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/)
                 ***********************

Oops, I missed an important row from the code of dripping tap!  :embarrass:


I also updated the code above.

Lol for the high voltage fractal. ;) You were lucky to survive.... cheers :beer:

I've made some new zooms, why not. ;) It's in delayed, averaged logistic map (fractal=-2 in my program BifFraPl).

Img #32

Set:
(http://s22.postimg.org/ocpc4lvxd/FZ_2_01set.png) (http://postimg.org/image/ibrn7j9b1/full/)

Zoom: 3700x, 500 point/pixel, calc. time: 18 min. on 2 CPU-cores
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_04_07_15_1_26_42.png)


And a newer seen before (neither now ;D) hidden bifurcation diagram of it:
(Cannot unhide with more iterations, I tried it.)

Img #33
(http://s29.postimg.org/rjo7golt3/FZ_2_02zoom_1_E5x_ps_250.png) (http://postimg.org/image/4i7maxm5f/full/)

Img #34 Fine structure of delayed, geometric averaged logistic map.

Set:
(http://s8.postimg.org/7ulutoh11/FZ_3_02set.png) (http://postimage.org/)

Zoom: 1,5x oversampling, 300 points/pixel, 6 min. on 2 cores
(http://s16.postimg.org/anwc1jcp1/FZ_3_02zoom_ps_300_12perc_lin.png) (http://postimg.org/image/fmjug2ght/full/)

I calculated a zoom of roots of double pendulum (of Img #10). It took 1 week.
We can understand these structures if see the video about standard map:
http://www.fractalforums.com/new-theories-and-research/bifurcation-fractal-of-standard-map/ (http://www.fractalforums.com/new-theories-and-research/bifurcation-fractal-of-standard-map/)

Img #35

Set:
(http://s14.postimg.org/wu2b5vw4x/Kett_singa_nf07_zoom01_set.png) (http://postimage.org/)

Zoom (empty sections are cutted out; right click for higher resolution), dt=4E-5, dHavg=1E-12, 5 points/pixel:
(http://nocache-nocookies.digitalgott.com/gallery/18/4917_14_07_15_11_01_47.png)

I unhided the hidden bifurcation map of Img #33!
If you interested how I did it, please visit the Multiple attractor bifurcation fractals topic:

http://www.fractalforums.com/new-theories-and-research/multiple-attractor-bifurcation-fractals/msg92974/#msg92974 (http://www.fractalforums.com/new-theories-and-research/multiple-attractor-bifurcation-fractals/msg92974/#msg92974)

The reason of the hidden map is that this chaotic system changed between its chaotic attractors as I stepped the parameter along the horizontal axis.

Img #36: (sorry for low res... :embarrass:)
(http://s32.postimg.org/5gcdapkjp/FZ_2_02zoom_1_E5x_Multiple_attr.png)