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Fractal Math, Chaos Theory & Research => (new) Theories & Research => Topic started by: bkercso on May 21, 2016, 01:41:11 AM



Title: Bifurcations of Coupled systems
Post by: bkercso on May 21, 2016, 01:41:11 AM
I examined the well known coupled logistic map and the coupled standard map and got some nice pics.
Coupled means two same systems started from different initial conditions, and a coupling between them, which is symmetric in these cases:

The coupled logistic map:
Code:
x_temp:=parameter*x*(1-x);
x2_temp:=parameter*x2*(1-x2);
x:=x_temp+coupling*(x2_temp-x_temp);
x2:=x2_temp+coupling*(x_temp-x2_temp);

[parameter=0..4; x,x2=0..1; coupling=0..0.5]


The coupled standard map:
Code:
pi2:=2*pi();

p_temp:=p+k*sin(q);
while p_temp>pi2 do p_temp:=p_temp-pi2;
while p_temp<-pi2 do p_temp:=p_temp+pi2;
q_temp:=q+p_temp;
while q_temp>pi2 do q_temp:=q_temp-pi2;
while q_temp<-pi2 do q_temp:=q_temp+pi2;

p2_temp:=p2+k*sin(q2);
while p2_temp>pi2 do p2_temp:=p2_temp-pi2;
while p2_temp<-pi2 do p2_temp:=p2_temp+pi2;
q2_temp:=q2+p2_temp;
while q2_temp>pi2 do q2_temp:=q2_temp-pi2;
while q2_temp<-pi2 do q2_temp:=q2_temp+pi2;

p_diff:=p2_temp-p_temp;
while p_diff>pi2 do p_diff:=p_diff-pi2;
while p_diff<-pi2 do p_diff:=p_diff+pi2;
p2_diff:=p_temp-p2_temp;
while p2_diff>pi2 do p2_diff:=p2_diff-pi2;
while p2_diff<-pi2 do p2_diff:=p2_diff+pi2;
q_diff:=q2_temp-q_temp;
while q_diff>pi2 do q_diff:=q_diff-pi2;
while q_diff<-pi2 do q_diff:=q_diff+pi2;
q2_diff:=q_temp-q2_temp;
while q2_diff>pi2 do q2_diff:=q2_diff-pi2;
while q2_diff<-pi2 do q2_diff:=q2_diff+pi2;

p:=p_temp+coupling*p_diff;
p2:=p2_temp+coupling*p2_diff;
q:=q_temp+coupling*q_diff;
q2:=q2_temp+coupling*q2_diff;

while p>pi2 do p:=p-pi2;
while p<-pi2 do p:=p+pi2;
while q>pi2 do q:=q-pi2;
while q<-pi2 do q:=q+pi2;
while p2>pi2 do p2:=p2-pi2;
while p2<-pi2 do p2:=p2+pi2;
while q2>pi2 do q2:=q2-pi2;
while q2<-pi2 do q2:=q2+pi2;


The pics are generated with this program:
http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/ (http://www.fractalforums.com/windows-fractal-software/bifurcation-fractal-plotter-biffrapl/)

The coupled logistic map is a multiple attractor system (http://www.fractalforums.com/new-theories-and-research/multiple-attractor-bifurcation-fractals/ (http://www.fractalforums.com/new-theories-and-research/multiple-attractor-bifurcation-fractals/)), while in the case of the coupled standard map this doesn't make sense, as this system doesn't produce regular bifurcations, as seen before (http://www.fractalforums.com/new-theories-and-research/bifurcation-fractal-of-standard-map/ (http://www.fractalforums.com/new-theories-and-research/bifurcation-fractal-of-standard-map/)): this has no periodic intervals, just quasiperiodic and hard chaotic regions.
("Regular" systems, which produce "regular" bifurcations are opposite: only has periodic intervals, of which parameter-range is more and more narrow, while period length is longer and longer.)


Img #1: Coupled logistic maps, coupling=0.01
(http://s32.postimg.org/u9aj65lol/fr_10_01_coup_0_01_hyp.png)

Img #2: coupling=0.1
(http://s32.postimg.org/6zutnjjlx/fr_10_02_coup_0_1_hyp.png)

Img #3: coupling=0.105 (download and switch between this and the previous one!)
(http://s32.postimg.org/9a460l4p1/fr_10_03_coup_0_105_hyp.png)


Img #4: Coupled standard maps, series; coupling=0; 1E-6; 1E-5; 1E-4; 1E-3; 1E-2; 3E-2; 1E-1; 2.5E-1; 5E-1
(http://s32.postimg.org/ro7t1yex1/Bifurcation_fraktal15_coupling_0_00000.png)
(http://s32.postimg.org/e6oc5r2md/Bifurcation_fraktal15_coupling_0_000001.png)
(http://s32.postimg.org/b0c653bph/Bifurcation_fraktal15_coupling_0_00001.png)
(http://s32.postimg.org/vdgsn5gcl/Bifurcation_fraktal15_coupling_0_00010.png)
(http://s32.postimg.org/m566nhfr9/Bifurcation_fraktal15_coupling_0_00100.png)
(http://s32.postimg.org/an3d68a9x/Bifurcation_fraktal15_coupling_0_01000.png)
(http://s32.postimg.org/4cgj8s2kl/Bifurcation_fraktal15_coupling_0_03000.png)
(http://s32.postimg.org/q0oblkdfp/Bifurcation_fraktal15_coupling_0_10000.png)
(http://s32.postimg.org/t4l7d4q2t/Bifurcation_fraktal15_coupling_0_25000.png)
(http://s32.postimg.org/jgqlcc5qd/Bifurcation_fraktal15_coupling_0_50000.png)

Title: Re: Bifurcations of Coupled systems
Post by: bkercso on May 22, 2016, 04:08:50 PM
Its interesting that if we stepped the coupling with a fixed parameter, we get similar structures at coupled logistic maps. This is because the structures seems to move on the standard bifurcation plot as we increase the coupling constant (just compare Img #2 and #3).


Img #5: coupled logistic map, stepped the coupling (0..0.2) and fixed the parameter (3.8): where periodicity appears, the two maps are synchronized
(http://s32.postimg.org/lpax3r4c5/Bifurcation_fr_10_mixup01_hyp.png)


Img #6: coupled logistic map, stepped the coupling (0..0.14) and fixed the parameter (3.6):
(http://s32.postimg.org/9ke76jvsl/Bifurcation_fr_10_mixup02_hyp.png)

Img #7: coupled logistic map, stepped the coupling (0..0.14) and fixed the parameter (3.99):
(http://s32.postimg.org/5s0pe7wz9/Bifurcation_fr_10_mixup03.png)

Img #8: zoom of Img #7
Set: the big object belongs to one attractor, and the six little ones belong a 2nd attractor
(http://s32.postimg.org/5aoe0phv9/Bifurcation_fr_10_mixup03_zoom01_set.png)

Zoom: its grooved because of the small number of trials of change attractor (50), therefor program spend more iterations on some attractor that on others
(http://s32.postimg.org/btd3j9lmd/Bifurcation_fr_10_mixup03_zoom01_hyp.png)

Title: Re: Bifurcations of Coupled systems
Post by: bkercso on May 22, 2016, 10:56:08 PM
Img #9: a zoom of Img #8

Set:
(http://s32.postimg.org/4bfacen79/Bifurcation_fr_10_mixup03_zoom02_set.png)

Zoom: with average 100,000 iterations/pixel, 2 Mpix, 500 trials to change attractor at every value of coupling
(I'm thinking how can solve easily to color each attractor with different colors...?)
(http://s32.postimg.org/z8lmburxx/Bifurcation_fr_10_mixup03_zoom02.png)

Title: Re: Bifurcations of Coupled systems
Post by: bkercso on May 22, 2016, 11:27:19 PM
Let's see the other coupled system here, the coupled standard map! It behaves different. I fixed parameter to two values, 1st for a chaotic region and 2nd to a quasy periodic region:

Img #10: Choose parameter values to fix.
(http://s33.postimg.org/6drubhyrz/Bifurcation_fr15_mixup_sets.png)

Img #11: Fixed parameter to "1", stepped coupling: coupling=0..0.5, x=-6.28..6.28 [k=1.2, p_initial=1.78, q_initial=3.14, plotted p]
Images also shows the zoom are of next img.
(http://s33.postimg.org/a2zhy3xvz/Bifurcation_fr15_mixup01_zoom02_set.png)

Img #12: zoom of selected area of Img #11, coupling=-5E-4..0.124, x=-1.34..2.43
(http://s33.postimg.org/jqdklfkgf/Bifurcation_fr15_mixup01_zoom02.png)

Img# 13: a zoom of the above selected area: we get the regular patterns of the "bifurcation" fractal of standard map
(http://s33.postimg.org/hlu6amt73/Bifurcation_fr15_mixup01_zoom01.png)

Img #14:
Set: (this is Img #12 without rotation)
(http://s33.postimg.org/p2w13uzpr/Bifurcation_fr15_mixup01_zoom03_set.png)

Zoom: noise: system is periodic, but the attractor changes rapidly with coupling
(http://s33.postimg.org/r135l895b/Bifurcation_fr15_mixup01_zoom03.png)

Title: Re: Bifurcations of Coupled systems
Post by: bkercso on May 22, 2016, 11:42:12 PM
Now parameter is fixed to "2" (as showed on Img #10): a quasy periodic region:

Img #15: coupling=0..0.5, x=-6.28..6.28 [k=1.2, p_initial=2.92 (this is showed on Img #10), q_initial=3.14, plotted p]
Started from a quasy periodic setup and applying some coupling, a fast transitions takes place to chaos. Then, after a certain amount of coupling quasy periodicity appears again - on that place (ie. between thap p values), where was without coupling -, but on a strange way: interrupted with narrow "noisy" region(s?). When system is chaotic, attractor is shifted to an other region.
(http://s33.postimg.org/lu2sc5b9r/Bifurcation_fr15_mixup02.png)

Img #16: But how fast is the first transition from q.p. to chaos? Top left corner of Img #15: we don't get sharp patterns...
(http://s33.postimg.org/fgqkbdcnj/Bifurcation_fr15_mixup02_zoom01.png)

Img #17: and a q.p. region from the right side of Img #15, without (?) narrow interruptions
Set:
(http://s33.postimg.org/gkg8rcx1r/Bifurcation_fr15_mixup02_zoom03_set.png)

Zoom:
(http://s33.postimg.org/8yq8z6j4f/Bifurcation_fr15_mixup02_zoom03.png)

Title: Re: Bifurcations of Coupled systems
Post by: bkercso on May 23, 2016, 12:40:30 PM
An interesting article about Earth's climate as a result of coupled nonlinear systems (which are ocean and atmosphere):
https://wattsupwiththat.com/2015/11/23/chaos-climate-part-2-chaos-stability/ (https://wattsupwiththat.com/2015/11/23/chaos-climate-part-2-chaos-stability/)

And some additional explanation to the previous images of coupled logistic maps:
We can see that system can shift from periodic to chaotic behavior or vica versa when change the parameter or the amount of coupling. Moreover, in some regions system has multiple attractors, which means that the system can behaves chaotic or periodic with a fixed parameter and coupling, depending on the initial conditions.

Title: Re: Bifurcations of Coupled systems
Post by: bkercso on May 23, 2016, 01:30:36 PM
Measure and simulation of coupled nonlinear oscillators (article: http://rsta.royalsocietypublishing.org/content/368/1911/343 (http://rsta.royalsocietypublishing.org/content/368/1911/343)); larger version: right click - view image:
(http://d29qn7q9z0j1p6.cloudfront.net/content/roypta/368/1911/343/F3.large.jpg?width=800&height=600&carousel=1)


"An interesting property of chaotic systems is that two systems, when properly coupled together, can synchronize with one another and evolve along the same chaotic orbit [...]. Many proposed applications of chaos, including secure communication systems, sensor networks and data assimilation and prediction, rely on this phenomenon of synchronization between chaotic oscillators"



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