Bifurcation Diagram Bouncing Ball

[Chillheimer]

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

oh wait, I already did..    

<a href="https://www.youtube.com/v/DxkAUz6VW6g&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/DxkAUz6VW6g&rel=1&fs=1&hd=1</a>


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..

[panzerboy]

This video is private
?!

[Chillheimer]

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

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

[bkercso]

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

[bkercso]

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.

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)

Code:

Iterate:

if u-2*sqrt(z)*tg(a)+sqrt(1-z)>=0 then u:=u-2*sqrt(z)*tg(a)  {stay on current slope}
else begin   {jump to other slope}
 z_prev:=z;
 z:=-z*(1+0.5*sin(4*a)*tg(a))-u^2*0.5*sin(4*a)*ctg(a)+u*sqrt(z)*sin(4*a)+2*cos(a)^2;
 u:=-u+sqrt(z_prev)*tg(a)-sqrt(z)*tg(a);
end;

I optimised it for faster running:

Code:

cosa2:=cos(a)^2;
sin4a:=sin(4*a);
tga:=tg(a);
ctga:=ctg(a);
jumped:=false;
temp0:=sqrt(1-z);

iteration begin
if not(jumped) then
begin
 sqrtz:=sqrt(z);
 temp1:=sqrtz*tga;
end;
jumped:=false;
temp2:=2*temp1;
temp3:=u-temp2;
if temp3>=-temp0 then u:=temp3  {stay on current slope}
else begin  {jump to other slope}
 temp4:=0.5*sin4a;
 temp5:=u*temp4;
 z:=-z*(1+temp4*tga)+temp5*(-u*ctga+2*sqrtz)+2*cosa2;
 jumped:=true;
 temp0:=sqrt(1-z);
 sqrtz:=sqrt(z);
 temp1_:=temp1;
 temp1:=sqrtz*tga;
 u:=-u+temp1_-temp1;
end;
iteration end;

[bkercso]

Quasy periodic zooms follow.

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

Set:


Zoom:
130,000x zoom, 20 min calc. time, average 7 points/pixel (you cannot see the underlying structure if iterate more)



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:


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

[bkercso]

Img #8

Set:



Zoom: 1.3E7x zoom, 13 points/pixel, 8 hours

[bkercso]

Img #9

Set:


Zoom: Some filaments. Changed z, plotted u. Average 170 points/pixel, 1E4x zoom, 24 hours calculation.

[Chillheimer]

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..

[bkercso]

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.

[bkercso]

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

[bkercso]

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:


Zoom:

[bkercso]

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

Set:


Zoom: 1.5E5x zoom, 8 points/pixel, 23 hours


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

[Chillheimer]

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.

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

[bkercso]

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