Bifurcation fractal of Standard Map

[bkercso]

Standard map is a well studied chaotic system which is related to the movement of a charged particle in magnetic field. If we plot bifurcation diagram with it (I didn't find any in literature, only Poincare maps), it produces similar structures as double pendulum and bouncing ball on slopes I plotted before, and simple enough for fast calculation of high resolution fractals and videos of it.

Variables:
k: constant (0..10)
p,q: system variables (0..2*pi)

The iterated equations:

Code:

p(n+1)=p(n)+k*sin(q(n))
q(n+1)=q(n)+p(n+1)
keep p and q between 0..2*pi

I've made a video:
p(initial)=0. (If not, periodic points disappear.)
stepped q(initial) in time between 0..pi (write actual value to screen)
x-axis: stepped k between 0..kmax, where kmax=1.5556*q(initial)+0.3444
y-axis: plotted q between qmin..qmax, where qmin=  0.5926*q(initial)+0.0593,
                                                             qmax=-0.5926*q(initial)+6.3593

The video is 3:00 long, contains 4140 images (23 fps), calculation time was 1 day.

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

And a picture from it:
Img #1


About other bifurcation fractals see these topics:
http://www.fractalforums.com/new-theories-and-research/bifurcations-fractals-discovery/105/
and
http://www.fractalforums.com/new-theories-and-research/re-bifurcation-fractals-discovery/30/

Video is generated by a software which is a modification of BifFraPl. Topic of the latest:
http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/

[bkercso]

Some zooms:

Img #2

Set:



Zoom: changed k, plotted q, p(initial)=0, q(initial)=3.14


Img #3

Set:


Zoom: changed k, plotted q, p(initial)=0, q(initial)=0.7665

[Chillheimer]

I whish I had more time. I need three of me to check out all that is so fascinating! thx for sharing, I'll dive deeper into this and will annoy you with lots of questions, promised!

[bkercso]

Oh me... 
Just relax, good work needs time!

I'm working on a video about evolution of one pillar...

[bkercso]

Img #4

Set:


Zoom: p(initial)=0, q(initial)=3.0316, average 0.5 point/pixel

[bkercso]

A new video is made about evolution of a single pillar in standard map's bifurcation diagram while changing parameter q.

Stepped q(initial) in time between 0.73..1.43 (write actual value to screen)
p(initial)=0. (If not, periodic point become quasy periodic.)
x-axis: stepped k between kmin..kmax,
  where kmin=-0.6298*q(initial)+1.7367, kmax=3.92*q(initial)^3-13.67*q(initial)^2+14.7*q(initial)-3.475
y-axis: plotted q between qmin..qmax,
  where qmin=0.1282*q(initial)+1.4478, qmax=0.7492*q(initial)+2.0699

The video is 3:00 long, contains 4140 images (23 fps), calculation time was 1 day.

Video #2: Pillar evolution in Standard Map bifurcations HD
<a href="http://www.youtube.com/v/5qJ5xmFlOes&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/5qJ5xmFlOes&rel=1&fs=1&hd=1</a>