Fractals/Rational

< Fractals
Herman ring - image with c++ src code

Iteration of complex rational functions[1][2][3]

ExamplesEdit



Z3 over z5minus1.jpg
Z4 over z5minus1.jpg
Z5 over z5minus1.jpg
Z6 over z5minus1.jpg
Z7 over z5minus1.jpg


Z4 over z4minus1.jpg
Z4plus005 over z4minus1.jpg
Z4minus02i over z4minus1.jpg
Z4plus1 over z4minus1.jpg
Z4plus1 over z4minus005.jpg
Z4plus1 over z4.jpg
Extrema z4 over z4minus1.jpg
Extrema z4plus005 over z4minus1.jpg
Extrema z4minus02i over z4minus1.jpg
Extrema z4plus1 over z4minus1.jpg
Extrema z4plus1 over z4minus005.jpg
Extrema z4plus1 over z4.jpg


degree 6Edit

Julia set of rational function f(z)=z^2(3 − z^4 ) over 2.png

The Julia set of the degree 6 function f :[9]

f(z) = z^2\frac{3-z^4}{2}

There are 3 superattracting fixed points at :

  • z = 0
  • z = 1
  • z = ∞

All other critical points are in the backward orbit of 1.


How to compute iteration :

z:x+y*%i;
z1:z^2*(3-z^4)/2;
realpart(z1);
((x^2−y^2)*(−y^4+6*x^2*y^2−x^4+3)−2*x*y*(4*x*y^3−4*x^3*y))/2
imagpart(z1);
(2*x*y*(−y^4+6*x^2*y^2−x^4+3)+(x^2−y^2)*(4*x*y^3−4*x^3*y))/2 

Find fixed points using Maxima CAS :

z1:z^2*(3-z^4)/2;
s:solve(z1=z);
s:float(s);

result :

[z=−1.446857247913871,z=.7412709105660023,z=−1.357611535209976*%i−.1472068313260655,z=1.357611535209976*%i−.1472068313260655,z=1.0,z=0.0]

check multiplicities of the roots :

multiplicities;
[1,1,1,1,1,1]


 z1:z^2*(3-z^4)/2;
 s:solve(z1=z)$
 s:map(rhs,s)$
 f:z1;
 k:diff(f,z,1);
 define(d(z),k);
 m:map(d,s)$
 m:map(abs,m)$
 s:float(s);
 m:float(m);

Result : there are 6 fixed point 2 of them are supperattracting ( m=0 ), rest are repelling ( m>1 ):

 [−1.446857247913871,.7412709105660023,−1.357611535209976*%i−.1472068313260655,1.357611535209976*%i−.1472068313260655,1.0,0.0]
 [14.68114348748323,1.552374536603988,10.66447061028112,10.66447061028112,0.0,0.0]

Critical points :

[%i,−1.0,−1.0*%i,1.0,0.0]

ReferencesEdit