Fractals/Rational

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

Herman ring - image with c++ src code


ExamplesEdit

Published online: June 15, 2010 DOI: 10.1007/s10114-010-6611-3 Http://www.ActaMath.com

GalleryEdit


Blaschke fractionEdit

 
Fatou sets for Blaschke fraction f(z) = rho * z^2 * (z-3) over (1-3z)

Analysis of critical points:

kill(all);
remvalue(all);
display2d:false;
ratprint : false; /* remove "rat :replaced " */



rho : -0.6170144002709304 +0.7869518599370003*%i;

define(f(z), rho * z^2 * (z-3)/(1-3*z));

/* first derivativa wrt z */
define( d(z), diff(f(z),z,1));


/* hipow does not expand expr, so hipow (expr, x) and hipow (expand (expr, x)) may yield different results */
n : hipow(num(expand(f(z))),z);
m : hipow(denom(expand(f(z))),z);

/* check if infinity is a fixed point */
limit(f(z),z,infinity);



/* finite critical points */

s:solve(d(z)=0)$
s : map(rhs,s)$
s : map('float,s)$
s : map('rectform,s)$

So there are 3 critical points :

  • 2 finite critical points : z=1.0 i z= 0.0
  • infinity

Dynamical plane consist of 3 basins

  • basin of attraction of fixed point z = infinity ( superattracting) with inf many componnets
  • basin of attraction of fixed point z = 0 ( superattracting) with inf many componnets
  • basin of parabolic period 3 cycle ( with z= 1 critical point)

Finite Blaschke productEdit

  is a finite Blaschke product of degree n.[11] It is: [12]

  • a rational function
  • an analytic function on the open unit disc such that f can be extended to a continuous function on the closed unit disc that maps the unit circle to itself
  • have no poles in the open unit disc
  • In particular, if ƒ satisfies the condition above and has no zeros inside the unit circle, then ƒ is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|ƒ(z)|)).
  • the Blaschke products B are rational perturbations of the doubling map of the circle R(z) = z^2 (equivalently given by θ → 2θ (mod 1)).
  • a finite Blaschke product may be uniquely described by the set of its critical points
  • rational map which fix a disc = which takes the closed unit disc D to itself
  • their iteration theory can be analyzed from the point of view of Fuchsian groups.
  • polynomials’ in the hyperbolic plane = hyperbolic polynomial
  • A finite Blaschke product, restricted to the unit circle, is a smooth covering map
  • the unit disk D, the unit circle ∂D and the complement of the closed unit disk C\D are all completely invariant sets for B


 

where

  •   is a unimodular constant. It is a point which lies on the unit circle:  
  •   is the multiplicity of the zero  
  •   is a finite sequence of n points in the open unit disc  


 


The building blocks[13] of Blaschke products are Mobius transformations of the form

 

where

  • ak ∈ D := {z ∈ C|, |z| < 1}
  • θk ∈ R.

A finite (infinite) Blaschke product has the form

 


Examples:

classificationEdit

There is a classification of finite Blaschke products in analogy with Möbius transformations.[14]

  • B is elliptic if the Denjoy-Wolff point z0 of B lies in D. |B' (z0)| < 1.
  • B is hyperbolic if the Denjoy-Wolff point z0 of B lies on ∂D and B'(z0) < 1,
  • B is parabolic if the Denjoy-Wolff point z0 of B lies on ∂D and B'(z0) = 1,


The Denjoy-Wolff point of B is a unique z0 ∈ D such that   for every z ∈ D

Julia setEdit

Let B be a finite Blaschke product of degree d > 1. Julia set   , the set on which the iterates   fail to be normal on any neighbourhood is either the unit circle   or a Cantor subset[15][16]

  • if B is elliptic, J(B) = ∂D,
  • if B is hyperbolic, J(B) is a Cantor subset of D,
  • if B is parabolic and z0 ∈ ∂D is the Denjoy-Wolff point of B,
    • J(B) = ∂D if B(z0) = 0
    • J(B) is a Cantor subset of ∂D if  


McMullen mapsEdit

singularly perturbed maps, also called McMullen maps[17]

  

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

The Bubble bath Julia setEdit

The quadratic rational function f:

 


The derivative wrt z is

 

The Julia set for f is called the Bubble Bath Julia set.[18] It is called the Bubble Bath for its visual similarity to a tub of bubbles.


Function f is defined for all z in the Riemann sphere   = it is defined on the whole Riemann sphere

 


  • The Fatou set of f is the basin of attraction of the 3-cycle consisting of the points 0, −1, and infinity. It is the only one attracting cycle and it is superattracting
  • The Julia set J(f) is the set of points whose orbits are not attracted to the above 3-cycle
  • the only critical points of f are:
    • z = 0 because it is the pole of order 3 of d(z), the zero of 1/d(z)
    • z = infinity because it is the zero of function d(z)


Maxima CAS code :

kill(all);
remvalue(all);
display2d:false;

define(f(z), (1 -z^2)/(z^2));

(%o3) f(z):=(1-z^2)/z^2
define( d(z), ratsimp(diff(f(z),z,1)));

(%o13) d(z):=-2/z^3
(%i14) limit(d(z),z,infinity);

(%o14) 0
(%i15) limit(d(z),z,0);

(%o15) infinity


(%i2) f(-1);
(%o2)                                  0
(%i3) limit(d(z),z,0);
(%o3)                             limit  d(z)
                                  z -> 0
(%i4) limit(f(z),z,0);
(%o4)                                 inf
(%i5) limit(f(z),z,inf);
(%o5)                                 - 1

Satbility of periodic cycle:

kill(all);
display2d:false;
ratprint : false; /* remove "rat :replaced " */


define(f(z), (1 -z^2)/(z^2));

F(z0):= block(
	[z],
	if is(z0 = 0) then 	z: limit(f(z),z,0)
	elseif is(z0 = infinity) then z: limit(f(z),z,infinity)
	elseif is(z0 = inf) then z: limit(f(z),z,inf)
	else z:f(z0),
	
	return(z)
)$

define( dz(z), ratsimp(diff(f(z),z,1)));

Dz(z0) := block(

	[m,z],
	if is(z0 = 0) then m: limit(dz(z),z,0)
	elseif is(z0 = infinity) then m: limit(dz(z),z,infinity)
	elseif is(z0 = inf) then m: limit(dz(z),z,inf)
	else m:dz(z0),
	
	return(m)

)$

GiveStability(z0, p):=block(
	[z,d],
	
	/* initial values */
	d : 1,
	z : z0,
	
	for i:1 thru p step 1 do (
	
		d : Dz(z)*d,	
		z: F(z)
		/*print("i = ", 0, "  d =",d, "  z = ", z)*/
	),
	
	return (cabs(d))
)$

GiveStability(-1,3);


See also :

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 3Edit

The components of the map   contain the attracting points that are the solutions to  . This is because the map is the one to use for finding solutions to the equation   by Newton–Raphson formula. The solutions must naturally be attracting fixed points.


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  [19]

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 :[20]

 

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. math.stackexchange question: relation-between-filled-julia-set-and-julia-set-of-a-rational-function?
  11. Finite Blaschke products: a survey by Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
  12. Dynamics of two-dimensional Blaschke products by ENRIQUE R. PUJALS and MICHAEL SHUB
  13. Color Visualization of Blaschke Product Mappings by Cristina Ballantine and Dorin Ghisa
  14. Epicycloids and Blaschke products by Chunlei Cao, Alastair Fletcher, Zhuan Ye
  15. THE EXPONENT OF CONVERGENCE OF A FINITE BLASCHKE PRODUCT by Gavin L. Jones. Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 245–254
  16. Blaschke products and parameter spaces by Katherine Plikuhn
  17. Symmetries for Julia sets of rational maps by Gustavo Rodrigues Ferreira
  18. A Thompson-Like Group for the Bubble Bath Julia Set by Jasper Weinrich-Burd, 2013
  19. Exterior discrete semiflows: Basins of end points by L. Javier Hernandez Paricio, Miguel Maranon Grandes ,M. Teresa Rivas Rodrıguez
  20. ON THURSTON’S PULLBACK MAP by XAVIER BUFF, ADAM EPSTEIN, SARAH KOCH, AND KEVIN PILGRIM

See alsoEdit