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Bifurcation map of a bouncing ball on opposing slopes.

Schematic of the system (not my work):
(http://demonstrations.wolfram.com/ChaoticDynamicsForBallBouncingBetweenTwoConstantSlopes/HTMLImages/index.en/popup_1.jpg)

This fractal is similar to bifurcation maps of double pendulum, but calculation is much faster, because this bouncing ball system is not feedbacked (no air resistance etc.), so we do not have to use successive approximation to calculate bouncings: every iteration gives us a point in the picture. This reduces calculation time from weeks to minites. :D

x-axis: initial velocity perpendicular with the slope
y-axis: velocities parallel to the slopes when bounce
Slopes angle: 60°, initial parallel velocity: 0.2
Equations are from a Hungarian book: Tél Tamás, Gruiz Márton: Kaotikus dinamika, Nemzeti Tankönyvkiadó, 2002.
(http://ttk.nyme.hu/mfi/Documents/Fizika-Kaotikus%20jelens%C3%A9gek/Kaotikus_jelens%C3%A9gek_I.pdf (http://ttk.nyme.hu/mfi/Documents/Fizika-Kaotikus%20jelens%C3%A9gek/Kaotikus_jelens%C3%A9gek_I.pdf))

Img #1
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_14_06_15_5_07_58.png)

this is cool, so, for clarification, what does this image tells us ?
 
and some questions:

- the bifurcation seems not to be visible, since bifurcation theory means splitting in 2, at which zoom levels can you spot the start and ends of the bifurcation
- can you provide zooms to the start of the bifurcation ?
- what do we see here? chaos or order ? i mean, is it chaotical behaviour of the bounces or does it follow certain lines ?

Thank you for this questions, it is important indeed.
This is the whole phase space of this system (cannot zoom out), so we can say this do not produces classical bifurcations. I call this picture bifurcation map because it is generated the same way as classical bifurcation maps. This system as is: not have periodic sections. In the other topic (http://www.fractalforums.com/new-theories-and-research/bifurcations-fractals-discovery/ (http://www.fractalforums.com/new-theories-and-research/bifurcations-fractals-discovery/)) I posted an explainig picture about that:
(http://nocache-nocookies.digitalgott.com/gallery/17/63_26_05_15_1_56_32_177522389.png)

So this system has only quasy periodic- and chaotic regions.

i see, it makes perfect sense to analyse existing dynamic systems like that

the interesting parts for these diagrams are the changing regions, can you render a zoom into a spike of the periodic line ?
or to the chaotic areas?

In fact I did that, I will found a good picture upload site for uotflank daily limit of this site. :D
It is a fractal, so you can see the same things in zooms too: only bubbles, namely quasy periodicity. In an other render shows not perpendicular but parallel velocity when bounce you can see interesting spikes...
I think the stucture of this fractal is similar with double pendumum's, but because of faster computation I can make deeper zooms. We will see if structure complexity is higher in zooms or not.

I played a little with this system and find that if initial velocity is perpendicular to slope that system has periodic behavior (with chaotic regions of course), else it is quasy peiodic (+ chaotic).

I think light patterns come from first slope and dark patterns from the other, but I will check it..

Img #2
(http://s3.postimg.org/ftuhcrmk3/Sorozat.png) (http://postimage.org/)
upload a picture (http://postimage.org/)

Video #1
https://youtu.be/-LrV-KsPTu8 (https://youtu.be/-LrV-KsPTu8)

First I zoom the periodic fractal.

This is from clean periodic region, 370x zoom, 5 hours of calculation time.

Img #3
Set:
(http://s3.postimg.org/9oggm3shf/LZ_01set.png) (http://postimage.org/)

Zoom:
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_16_06_15_10_19_22.png)

And one from the edge of periodic and chaotic regions. 500x zoom, 1.5 hours calculation.

Img #4
Set:
(http://s27.postimg.org/lh4azhhxv/LZ_02set.png) (http://postimage.org/)

Zoom:
(http://s24.postimg.org/8mj89xnj9/LZ_02zoom_500x_1_5_ra.png) (http://postimg.org/image/ar3lb0p5t/full/)

Light green lines in the chaotic region are periodic/soft chaotic regions (?).

As I see this system's fractals are much more simpler than double pendulum's ones, but this is a good model system this type of bifurcation maps because it requires 1E4 times less computation.

yay, the underlying method is far simpler, and the "chaotic" or "dense" regions dont exhibit the beautiful sub branching
but the quasiperiodic regions are what we encounter when watching a ball bounce this kind of thing ;) and assume it stays at a certain position longer, or we seem to be able to exactly predict the movements of the ball ( more or less ;) )

thank you bkercso.
I love what you do!
It's very inspiring!
 :thumbsup1:


I know this is a bit off-topic, but I've found a huge interest in bifurcation diagrams lately.
I believe that bifurcation and especially the "periodic order windows" that occur out of chaos are hugely important on a larger scale in reality.

I totally agree with the ideas of Robert Paterson:
http://smartpei.typepad.com/robert_patersons_weblog/2009/04/the-fractal-history-of-mankind-part-2-our-time.html

ray, kurzweil, singularity is near - nice read!

and actually I believe that: (see attached image)

cKleinhuis:
Well, now I think there is no contoured sub branching in chaotic regions. I made deep zooms on it and it's dimness I cannot attributed to rough time resolution because there is no time resolution here.
I see shaped patterns in it only when I take less iterations.

Chillheimer:
Thank you. :)

There is no doubt that our global word is unsustainable. Not only because of oil and wasting, but the picture is quite complex and on many areas there are majore problems, which has own fractal-like fine structure. Well, only the history we can see clear (if we)... :)

Img #5

Set:
(http://s8.postimg.org/5dtcqbqlh/LZ_05set.png) (http://postimg.org/image/i57iwu0dd/full/)


Zoom 78,000x (made by 15,000,000,000 bounces, during 35 minutes):
(http://s8.postimg.org/qoqwul8px/LZ_05zoom_3000x_0_3_ra.png) (http://postimg.org/image/6hdh2ab8h/full/)

I recalculated Img #4 with higher point/pixel density and upload to forum's gallery.

(http://nocache-nocookies.digitalgott.com/gallery/17/4917_19_06_15_11_44_54.png)

come on, lets do this in 3d.
and as a movie!°!

oh wait, I already did..  :o ;D  :dink:

https://www.youtube.com/watch?v=DxkAUz6VW6g


I rendered this short movie a few weeks ago and hadn't released it. somewhere in in amazing box, combined with amazing box SSE 2
your last pic reminded me of it. nice "coincidence" that those structures look so similar.. ;)

This video is private
?!

stupid me... set it to public. thx for ppointing that out.

edit: to make this post less useless, I'll add this little gem that I stumbled upon in the youtube recomendations:
https://www.youtube.com/watch?v=RnGPpjvugZo
 O0

hm, this guy has got a few other nice videos on his channel https://www.youtube.com/user/ch3trash0/videos

Your render @00:25 looks like a big skyscraper. :crazyeyes:

I give the equations of bouncing ball on twin slope (double slope), because the .pdf I linked first is not the book itself, only a review.
I wrote it in pascal language, so I use something similar now too. :tongue1:

Variables:
a: angle of slopes (alpha, 0..pi/2 [rad])
u: velocity component of the ball parallel with slope (0..1)
z: square of velocity component of the ball perpendicular to slope (approx. 0..0.3)
(mass of ball = gravity = 1)

I optimised it for faster running:

Quasy periodic zooms follow. :joy:

Img #6 Seems like a digital microchip...
alpha=60°, u(initial)=0.3, changed z, plotted z

Set:
(http://s11.postimg.org/sm87eh16r/LZ_09set.png) (http://postimg.org/image/7pbz9t35r/full/)

Zoom:
130,000x zoom, 20 min calc. time, average 7 points/pixel (you cannot see the underlying structure if iterate more)
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_25_06_15_11_06_15.png)


Img #7
alpha=60°, changed z, plotted u
I also changed u(initial), because I noticed that the degree of quasy periodicity depends on the angle of initial velocity to slope, so I kept this angle constant.
u(initial)=sqrt(z)*0.23  (remember: z is velocity^2 like quantity)

Set:
(http://s7.postimg.org/m633xb45n/LZ_10set.png) (http://postimg.org/image/nxw2s7nif/full/)

Zoom: strange that it shows blurred patterns. I think it is countured (quasy periodic) and blurred (chaotic) regions densely embedded in each other...
14000x zoom, 40 min calc. time
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_21_06_15_1_25_53.png)

Img #8

Set:
(http://s9.postimg.org/6dmrqkvi7/LZ_14set.png) (http://postimg.org/image/5bcl81cor/full/)


Zoom: 1.3E7x zoom, 13 points/pixel, 8 hours
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_01_07_15_3_04_51.png)

Img #9

Set:
(http://s23.postimg.org/6vdvn94cb/LZ_03set.png) (http://postimg.org/image/pnpqqu0qf/full/)

Zoom: Some filaments. Changed z, plotted u. Average 170 points/pixel, 1E4x zoom, 24 hours calculation.
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_22_06_15_9_53_19.png)

hey bkercso, thx for sharing the equations.
i wonder: is there something "more userfriendly" ? I'm no programmer and have no idea what to do with that code to get a picture. but I'd love to zoom around in some bifurcation diagrams..

Hi,
Uhh, my comments and some of my variables are in Hungarian, and you can use my code under freePascal compiler. In it's current form cannot compile an .exe from it because you can set parameters in declaration section.
I use this as follows: generate a fractal, program writes point density by pixel in a .txt. With an other program (wrote in Delphi) convert it to .bmp. After that open the picture with IrfanView, select an area of it and typeing the four number from window header to my program to zoom. Not so user friendly.

But if somebody feel like and have enought time to write a software for bifurcation fractals I give the whole code and needed support with pleasure.

Just I searched fractal software which can render bifurcation maps and I'm surprised that I did not find one.

Img #10: Chinese pillars with baobab like-roots.
We saw similar structures at double pendulum in the other topic. (The slow quasy periodic to chaotic transition mustache is also common in both fractals, but this is not seen in this zoom.) The common procedure in both case that an angle was changed along x-axis. Here I changed alpha from 59° to 61°. For fix z and u I used a map with changed z. u value selected from video I linked before.

Set:
(http://s14.postimg.org/q7dpq1chd/LZ_18set.png) (http://postimg.org/image/ga2owz4vh/full/)

Zoom:
(http://nocache-nocookies.digitalgott.com/gallery/17/4917_06_07_15_9_51_23.png)

Img #11: Blurred patterns of Img #7. I was right, these are densly packed quasy periodic and chaotic regions. So beauty... ;D

Set:
(http://s4.postimg.org/3qd6747kt/LZ_16set_C.png) (http://postimage.org/)

Zoom: 1.5E5x zoom, 8 points/pixel, 23 hours
(http://s21.postimg.org/ie6xgp0g7/LZ_16zoom_C_1_5_E4x_ps_8_23_ra.png) (http://postimg.org/image/6cbjmjr7n/full/)

And I'm working on make my softver usable before share it... :dink:

That's very generous! Thank you!
Now I really hope that someone takes up that challenge...

Yes, but did you read my last sentence below the pictures?

nope i missed that! those pics are such a nice distraction ;)

wohoo! :)
 :joy:

Ultra Fractal can, although it's not necessarily a simple thing to do.  Here's one of mine:

http://www.kerrymitchellart.com/gallery24/bifurcation.html

Thx for the info. I didn't find much more pics with google. :)
But yours is a nice render!

Thanks for the inspiration bkercso.  I got back into my bifurcation code which was nothing more than a very pixelated black on white logistic renderer.  After some much needed updates in output quality and new fromuals here are some results.

Logistic Map
(https://farm1.staticflickr.com/316/19130135882_33a0e4fd6e_c.jpg)

(https://farm1.staticflickr.com/466/19078189086_b1f740ec47_c.jpg)

Henon Attractor
(https://farm1.staticflickr.com/294/18480807273_50f453c352_c.jpg)

(https://farm1.staticflickr.com/467/18915338749_1c317d83fd_c.jpg)

Delayed Logistic Map
(https://farm1.staticflickr.com/332/19139284881_1ff97bc0b3_c.jpg)

(https://farm1.staticflickr.com/317/19078607316_4f5bc4105d_c.jpg)

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

(https://farm1.staticflickr.com/313/19130138582_4c9a3cf5ee_c.jpg)

(https://farm1.staticflickr.com/390/19130139162_a195a1c73c_c.jpg)

For full res images of the above see https://www.flickr.com/photos/39445835@N05/

I have included these new functions in the Bifurcation mode of Visions Of Chaos (http:/softology.com.au/voc.htm)

Jason.

Wow, that's it! Thank you for sharing pics and also for the software.
My - far simpler - software is almost done, now writing some more help messages. I will share it in the next few days. But translating the code to English will take more time.

Regarding scale independent behavior of fractals try to use log scaling on parameter axis for undraw interesting sections.

                 ***********************
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/)
                 ***********************

Just I've made a series which tries reveal the structure of chaotic region, do this with increasing iteration number picture to picture.

Plotted z, changed alpha.
z=0.022, u=0.231 (fixed velocity-slope angle), alpha=59..61.
Iterations: 0.01, 0.3, 1, 3, 10, 30, 100, 300, 1000, 3000, 10000, 30000

Img #12
(http://s18.postimg.org/94vh3mq8p/L_Blur_sorozat.png) (http://postimg.org/image/jrpa91ydx/full/)

Img #13
Patterns in hard chaotic regions:
The same image calculated with 8 digits and 20 digits variables. Conclusion: the blurred hard chaotic regions is own feature of this (these) system(s); the lack of contoured patterns do not come from calculation errors, not from finite precision of calculations!

(http://nocache-nocookies.digitalgott.com/gallery/17/4917_29_06_15_11_29_11.png)