# Fractals/Rational

Iteration of complex rational functions

# Examples

## McMullen maps

singularly perturbed maps, also called McMullen maps

 $R_{\lambda }(z)=z^{m}+\lambda +z^{d}$ ## 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 3 by Michael Becker

### 1 period 2 basin

$f(z)={\frac {1}{z^{3}+z*(-3-3*I)}}$

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 basins

Function

 $f(z)={\frac {1}{z^{3}+a*z+b}}$ 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)

 $f(\infty )=0$ 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 5

### by L. Javier Hernandez Paricio

the rational map $h(z)={\frac {1+4z^{5}}{5z^{4}}}$  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 $z^{5}-1=0$ 

## 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]