Print Page - Multiple attractor bifurcation fractals

Title: Multiple attractor bifurcation fractals
Post by: bkercso on May 20, 2016, 12:35:39 AM

Hi Folks,

After I realised that many chaotic systems has more than one chaotic attractor, I supplemented my little program Bifurcation Fractal Plotter with a trick in order to generate these fractals in full. The software also got some new chaotic equations.

http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/msg85310/#msg85310 (http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/msg85310/#msg85310)

The trick is simple: if we start the iteration with several initial value sets, the generator will explore all the trajectories! I wrote value sets not values, because chaotic systems with more than one variable has multiple attractors typically.
We can choose how many times to generate random initial values from the plotted range of the variables; 20..500 is recommended. If you use higher values, count densities will distribute more evenly between the attractors, but less iterations will be rejected as first few iterations: it could resulted background noise on some regions in the picture if the average iterations/pixel number is low.

Code:

Number of rejected iterations = Iterations@OneAttractor/2, but max. Maxiter*0.01

Maxiter = ImageWidth*AvgIterationsPerPixel
Iterations@OneAttractor = Maxiter/ChangeAttractor

It means that max. 1/3 of the total calculation time will be spent for rejected iterations (noise filtering), and it can be not enough with low Maxiter values. So eg. when scopeing for desired zoom area, don't afraid: background noise will disappear when calculate the high resolution image!

And an example why this change attractor matters:
IMG #1:
This one is generated with only one initial value set:
(http://s32.postimg.org/hrkpp9ocl/Bifurcation_C02_zoom1.png)

IMG #2:
... and with more than one initial value sets (this is a zoom of two coupled logistic maps, fractal -10 in my program):
(http://s32.postimg.org/w63avw5mt/Bifurcation_C02_zoom1_10.png)

Title: Re: Multiple attractor bifurcation fractals
Post by: bkercso on May 20, 2016, 01:10:09 AM

IMG #3:
The Henon map (fractal -11) (with hyperbolic coloring): for equations see Wikipedia...
(http://s32.postimg.org/40cph811h/Bifurcation_Henon_01_hyp.png)

IMG #4:
fractal -6 (with logarithmic coloring):

Code:

x_temp:=x;
x:=parameter*x*(1-x*previous);
if x_previous<0.5 then x:=1-x;
x_previous:=x_temp;

parameter=1.3..1.645 (on horizontal axis)
x=-1.2..1.2 (on vertical axis)

(http://s32.postimg.org/4l7te0b45/Bifurcation_A_fr_6.png)

Title: Re: Multiple attractor bifurcation fractals
Post by: bkercso on May 20, 2016, 01:15:32 AM

IMG #5:
fractal -7 (with logarithmic coloring):

Code:

x_temp:=x;
x:=parameter*x*(1-x*previous);
if x_previous<x then x:=1-x;
x_previous:=x_temp;

parameter=1.3..1.65 (on horizontal axis)
x=-1.2..1.5 (on vertical axis)

(http://s32.postimg.org/d4o17oldx/Bifurcation_A_fr_7.png)

IMG #6:
fractal -8 (with logarithmic coloring):

Code:

x_temp:=x;
x:=parameter*x*(1-x*previous);
if x_previous<x then x:=(x+x_previous)/2;
x_previous:=x_temp;

parameter=2.7..3.7 (on horizontal axis)
x=0..1 (on vertical axis)

(http://s32.postimg.org/heufej9l1/Bifurcation_A_fr_8.png)

Img #7:
A zoom of Img #6

(http://s32.postimg.org/kz8dy61j9/Bifurcation_A_fr_8_zoom01_hyp.png)

Title: Re: Multiple attractor bifurcation fractals
Post by: bkercso on May 20, 2016, 01:22:10 AM

IMG #8:
fractal -9 (with hyperbolic coloring):

Code:

x_temp:=x;
x:=parameter*x*(1-x*previous);
if x_previous<x then x:=(x+x_previous)/2
else                       x:=1-(x+x_previous)/2;
x_previous:=x_temp;

parameter=2..4.875 (on horizontal axis)
x=0.1..1.6 (on vertical axis)

(http://s32.postimg.org/yd195qix1/Bifurcation_A_fr_9_hyp.png)

Title: Re: Multiple attractor bifurcation fractals
Post by: bkercso on May 20, 2016, 01:29:40 AM

Img #9:
Zoom01 of Img #8 (fractal -9)

Set:
(http://s32.postimg.org/ybehxdnb9/Bifurcation_A_fr_9_zoom01_set.png)

Zoom:
(http://s32.postimg.org/h6aohddfp/Bifurcation_A_fr_9_zoom01_hyp.png)

Img #10:
Zoom02 of Img #8 (fractal -9)

Set:
(http://s32.postimg.org/c7s7zyfyd/Bifurcation_A_fr_9_zoom02_set.png)

Zoom:
(http://s32.postimg.org/519d6ffid/Bifurcation_A_fr_9_zoom02_hyp.png)

Img #11:
Zoom03 of Img #8 (fractal -9)

Set:
(http://s32.postimg.org/3mlkmxikl/Bifurcation_A_fr_9_zoom03_set.png)

Zoom:
(http://s32.postimg.org/z7uqhx5c5/Bifurcation_A_fr_9_zoom03_hyp.png)

Title: Re: Multiple attractor bifurcation fractals
Post by: Chillheimer on May 20, 2016, 04:36:24 PM

I love your work!
Good to see you're still active..
(sorry I still haven't found the time to dive into your program and the topic.. so little time, so many fascinating things in the world.. ;)

Title: Re: Multiple attractor bifurcation fractals
Post by: bkercso on May 20, 2016, 07:33:50 PM

Cheers! ;)
Somebody ask me about fractal generation, therefor I had new ideas... I had result for 2 new topic.

The software is much more better now than earlier! 88)

Title: Re: Multiple attractor bifurcation fractals
Post by: bkercso on May 23, 2016, 04:39:24 PM

I added some explanation to the 1st post, before the pictures, about how the program maps multiple attractors.

Welcome to Fractal Forums

Fractal Math, Chaos Theory & Research => (new) Theories & Research => Topic started by: bkercso on May 20, 2016, 12:35:39 AM