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:
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:
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/The coupled logistic map is a multiple attractor system (
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/): 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
Img #2: coupling=0.1
Img #3: coupling=0.105 (download and switch between this and the previous one!)
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