Fractals/Rational

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

Herman ring - image with c++ src code


ExamplesEdit

GalleryEdit


McMullen mapsEdit

singularly perturbed maps, also called McMullen maps[10]

  

degree 2Edit

Function:  


maxima

Maxima 5.41.0 http://maxima.sourceforge.net
using Lisp GNU Common Lisp (GCL) GCL 2.6.12
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) display2d:false;

(%o1) false
(%i2) f:z^2/(z^2-1);

(%o2) z^2/(z^2-1)
(%i3) dz:diff(f,z,1);

(%o3) (2*z)/(z^2-1)-(2*z^3)/(z^2-1)^2
(%i4) s:solve(f=z);

(%o4) [z = -(sqrt(5)-1)/2,z = (sqrt(5)+1)/2,z = 0]
(%i5) s:map('float,s);

(%o5) [z = -0.6180339887498949,z = 1.618033988749895,z = 0.0]
(%i6) 

So fixed points  :

  • z = -0.6180339887498949
  • z = 1.618033988749895
  • z = 0.0

degree 2 by Michael BeckerEdit

1 period 4 basinEdit

 
Julia set for f(z)=(z2+a) over (z2+b) a=-0.2+0.7i , b=0.917


degree 3 by Michael BeckerEdit

1 period 2 basinEdit

 
Julia set f(z)=1 over z3+z*(-3-3*I)

 

2 critical points : { -0.4550898605622273*I -1.098684113467809, 0.4550898605622273*I+1.098684113467809}; Both critical points tend to the periodic cycle.

There is only one attractive period cycle : period 2 cycle = {0, infinity}.

Whole plane ( sphere) is a basin of attraction of period 2 cycle ( which is divided into 2 components ). Julia set is a boundary.

2 period 2 basinsEdit


Function

    
 

where

  • a = 2.099609375
  • b = 0.349609375


Derivative:

 d(z):=-(3*z^2+2.099609375)/(z^3+2.099609375*z+0.349609375)^2

Critical points:

  [-0.8365822085525526*%i,0.8365822085525526*%i]
  

One can check it also using Wolfram Alpha

 solve (3*z^2+2.099609375)/(z^3+2.099609375*z+0.349609375)^2=0
 

the result:

 z = ± (5/16)* i* sqrt(43/6))
 

These are 2 finite critical points.



Infinity is a critical point too, as the 1st derivative's denominator degree is strictly greater than the numerator's. In numerical computations one can use the critical value (an image of critical point)

  
 
 

There are two period 2 cycle:

  • { +0.4101296722285255 +0.5079485669960778*I , +0.4101296722285255 -0.5079485669960778*I };
  • { +1.6890328811664648 +0.0000000000000000*I , +0.1147519899962205 +0.0000000000000000*I };

Both finite critical points fall into first cycle. Infinity ( or it's image zero) falls into the second cycle ( on the horizontal axis)


Infinity is not a fixed point

remvalue(all);
display2d:false;
define(f(z), 1/(z^3+ 2.099609375*z +  0.349609375));
(%i5)limit(f(z),z,infinity);
(%o5) 0
(%i6) limit(f(z),z,0);
(%o6) 2.860335195530726

degree 5Edit

by L. Javier Hernandez ParicioEdit

the rational map   has six fixed points:

  • ∞ ( repelling)
  • −0,809017 − 0,587785i
  • −0,809017 +0,587785i
  • 0,309017 − 0,951057i
  • 0,309017 + 0,951057i
  • 1

the basin of an end point associated to a fixed point (6= ∞) of f is the same that the attraction basin of the Newton-Raphson numerical method when it is applied to find the roots of the equation  [11]

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 :[12]

 

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]

degree 9 by Michael BeckerEdit

 
Julia set for f(z)=z2 over (z9-z+0.025)

ReferencesEdit

  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. Symmetries for Julia sets of rational maps by Gustavo Rodrigues Ferreira
  11. Exterior discrete semiflows: Basins of end points by L. Javier Hernandez Paricio, Miguel Maranon Grandes ,M. Teresa Rivas Rodrıguez
  12. ON THURSTON’S PULLBACK MAP by XAVIER BUFF, ADAM EPSTEIN, SARAH KOCH, AND KEVIN PILGRIM

See alsoEdit