Title: Bifurcation fractal of Standard Map Post by: bkercso on July 11, 2015, 10:55:14 PM 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)) 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 https://youtu.be/qmbwnrgLhhg (https://youtu.be/qmbwnrgLhhg) And a picture from it: Img #1 (http://nocache-nocookies.digitalgott.com/gallery/18/4917_11_07_15_10_48_21.png) About other bifurcation fractals see these topics: http://www.fractalforums.com/new-theories-and-research/bifurcations-fractals-discovery/105/ (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/ (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/ (http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/) Title: Re: Bifurcation fractal of Standard Map Post by: bkercso on July 11, 2015, 11:51:06 PM Some zooms:
Img #2 Set: (http://s15.postimg.org/68buz3jej/SZ_01set.png) (http://postimg.org/image/f3cp9m86v/full/) Zoom: changed k, plotted q, p(initial)=0, q(initial)=3.14 (http://nocache-nocookies.digitalgott.com/gallery/18/4917_13_07_15_10_18_43.png) Img #3 Set: (http://s23.postimg.org/b8pg179a3/SZ_08set.png) (http://postimage.org/) Zoom: changed k, plotted q, p(initial)=0, q(initial)=0.7665 (http://s17.postimg.org/9o8hty4kf/SZ_08zoom_ps_12_80s_Sh_U.png) (http://postimg.org/image/xf7vc24rf/full/) Title: Re: Bifurcation fractal of Standard Map Post by: Chillheimer on July 12, 2015, 12:35:53 AM 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! ;)
Title: Re: Bifurcation fractal of Standard Map Post by: bkercso on July 12, 2015, 09:54:52 PM Oh me... :scared:
Just relax, good work needs time! I'm working on a video about evolution of one pillar... Title: Re: Bifurcation fractal of Standard Map Post by: bkercso on July 13, 2015, 11:09:16 AM Img #4
Set: (http://s11.postimg.org/476gvwdnn/SZ_08set.png) (http://postimage.org/) Zoom: p(initial)=0, q(initial)=3.0316, average 0.5 point/pixel (http://s27.postimg.org/qc5mh17eb/SZ_08zoom_ps_0_5_1perc.png) (http://postimg.org/image/5s0sijrn3/full/) Title: Re: Bifurcation fractal of Standard Map Post by: bkercso on July 15, 2015, 12:45:03 AM 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 https://youtu.be/5qJ5xmFlOes (https://youtu.be/5qJ5xmFlOes) |