n-furcation in the Mandelbrot set

• choose path on the parameter plane ( p/q internal ray)
• For parameters from 2d path on the parameter plane one can compute periodic points z (dynamic plan = 2D frame )
• draw 3D birurcation diagram from dynamical frames

Periodic points z

• for period 1 ( fixed point) using explicit method
• for higher periods using Newton method

complex double GiveFixed(complex double c){
/* 

Equation defining fixed points : z^2-z+c = 0
	z*2+c = z
	z^2-z+c = 0

coefficients of standard form ax^2+ bx + c  
 	a = 1 , b = -1 , c = c
 
The discriminant d is 

	d=b^2- 4ac 
	d = 1 - 4c
	
 
 alfa =  (1-sqrt(d))/2 
*/

	complex double d = 1-4*c;
	complex double z = (1-csqrt(d))/2.0;
	return z;

}

Bifurcation = 2-furcation

Path on the parameter plane ( along real slice of Mandelbrot set = real axis, all poointa are real, Imaginary part im(c) = 0)

• c = 0 ( interior of period 1 component, center of period 1 component)
• -3/4 > c > 0 ( interior of period 1 component, internal ray 1/2 )
• c = -3/4 ( common point of boundary of period 1 component and period 2 component, root point = bifurcation point)
• -1 > c > -3/4 (interior of period 2 component, internal ray 0 )

Trifurcation = 3-furcation

Path on the parameter plane

• c = 0 ( interior of period 1 component, center of period 1 component)
• from c=0 to c = -0.125000000000000 +0.649519052838329i ( interior of period 1 component, internal ray 1/3 )
• c = -0.125000000000000 +0.649519052838329i ( common point of boundary of period 1 component and period 3 component, root point = bifurcation point)
• from root to center ( interior, internal ray for angle =0)
• center c = -0.122561166876654 +0.744861766619744i (interior, center of period 3 component)

Here imaginary part is not 0. One can use

• the multiplier map to aproximate c
• github repo with c code
• wikibooks : first derivative wrt z

Period 3 points for c = 0

Newton method finds 8 points

• two period 3 cycles
• two fixed points: repellling z=1 and superattracting z=0

 periodic points are: 
 z = +1.000000000000000000; +0.000000000000000000 exact period = 1 stability = 2.000000000000000000
 z = +0.000000000000000000; +0.000000000000000000 exact period = 1 stability = 0.000000000000000000
 
 z = +0.623489801858733531; +0.781831482468029809 exact period = 3 stability = 8.000000000000000000
 z = -0.222520933956314404; +0.974927912181823607 exact period = 3 stability = 8.000000000000000000
 z = -0.900968867902419126; +0.433883739117558120 exact period = 3 stability = 8.000000000000000000
 z = -0.900968867902419126; -0.433883739117558120 exact period = 3 stability = 8.000000000000000000
 z = -0.222520933956314404; -0.974927912181823607 exact period = 3 stability = 8.000000000000000000
 z = +0.623489801858733531; -0.781831482468029809 exact period = 3 stability = 8.000000000000000000

Here are 2 repelling period 3 cycles and 2 fixed points ( period 1 cycles )

scr code

• m.gp - gnuplot bash file for drawing path3.png diagram
• n.c = c file for finding periodic points using Newton method
• m.c = c file for computing c and zf fixed points along internla rays
• m.txt = txt file with result of m.c program (# radius cx cy zxf zyf ). It will be used for creating graphic

git

echo "# " >> README.md
git init
git add README.md
git commit -m "first commit"
git branch -M main
git remote add origin [email protected]:adammaj1/mandelbrot-trifurcation.git
git push -u origin main

cd existing_folder
git add .
git commit -m "Initial commit"
git push -u origin main

subdirectory

mkdir images
git add *.png
git mv  *.png ./images
git commit -m "move"
git push -u origin main

then link the images:

![](./png/n.png "description") 

to overwrite

git mv -f 

Local repo

~/Dokumenty/mandelbrot-trifurcation/mandelbrot-trifurcation/