# Fractals/Rational

Iteration of complex rational functions

# Examples

## degree 2

Function: $f(z)={\frac {z^{2}}{z^{2}-1}}$ maxima

Maxima 5.41.0 http://maxima.sourceforge.net
using Lisp GNU Common Lisp (GCL) GCL 2.6.12
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 $z:f(z)=z$ :

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

## degree 6

The Julia set of the degree 6 function f :

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