Herman ring - image with c++ src code

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

Examples

• Newton fractals
• commons:Category:Complex rational maps
• f(z)=z2/(z9-z+0,025) ^[4]
• f(z)=(z3-z)/(dz2+1) where d=-0,003+0,995i ^[5]
• f(z)=(z3-z)/(dz2+1) where d=1,001· e2Pi/30 ^[6]
• Multibrot sets by Xender^[7]
• ${\displaystyle f_{\lambda }(z)=z^{n}+{\frac {\lambda }{z^{d}}}}$^[8]
• Rational Julia Sets by Marc McClure
• f(z) = (z^n+c)/(c^n+z), for n = -2 ^[9]
• Jasper Weinrich Burd: A Thompson-Like Group for the Bubble Bath Julia Set

• Rational functions
• 
  ${\displaystyle {\frac {z}{z-1}}}$
• 
  ${\displaystyle {\frac {z^{2}}{z^{2}-1}}}$
• 
  ${\displaystyle {\frac {z^{3}}{z^{3}-1}}}$
• 
  ${\displaystyle {\frac {z^{4}}{z^{4}-1}}}$
• 
  ${\displaystyle {\frac {z^{5}}{z^{5}-1}}}$
• 
  ${\displaystyle {\frac {z^{7}}{z^{7}-1}}}$
• 
• 
• 
• 
• 
• 

degree 6

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

The Julia set of the degree 6 function f :^[10]

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

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]

References

1. ↑ Julia Sets of Complex. Polynomials and Their. Implementation on the Computer. by CM Stroh
2. ↑ Julia sets by Michael Becker.
3. ↑ DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS by R. HAGIHARA AND J. HAWKINS
4. ↑ f(z)=z2/(z9-z+0,025) by Esmeralda Rupp-Spangle
5. ↑ f(z)=(z3-z)/(dz2+1) where d=-0,003+0,995i by Esmeralda Rupp-Spangle
6. ↑ f(z)=(z3-z)/(dz2+1) where d=1,001· e2Pi/30 by Esmeralda Rupp-Spangle
7. ↑ Rhapsody in Numbers by Xender
8. ↑ Julia Sets for Rational Maps by PAUL BLANCHARD , CUZZOCREO, ROBERT L. DEVANEY, DANIEL M. LOOK, ELIZABETH D. RUSSELL
9. ↑ fractalforums : Fractal Math, Chaos Theory & Research > General Discussion > Do z-->z/c² or z-->z*c² create a fractal
10. ↑ ON THURSTON’S PULLBACK MAP by XAVIER BUFF, ADAM EPSTEIN, SARAH KOCH, AND KEVIN PILGRIM

• Rational maps: the structure of Julia sets from accessible Mandelbrot sets by Fitzgibbon, Elizabeth Laura
• Iteration of Rational Functions by Omar Antolín Camarena