Fractals/Rational

From Wikibooks, open books for an open world
Jump to: navigation, search
Herman ring - image with c++ src code

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

Examples[edit]



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 6[edit]

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]

References[edit]

  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. ON THURSTON’S PULLBACK MAP by XAVIER BUFF, ADAM EPSTEIN, SARAH KOCH, AND KEVIN PILGRIM