File:Cr6spiral.png

Summary[edit]

c = 0.3749678400000000+0.2168721400000000*I 	    
multiplier m(c) = 0.4993345401687128+0.8663355369994206*I
Internal radius r(m) = 0.9999361207965108 
period = 1
internal angle t(m) = 0.1667830755386747 in turns  	 
t(m) - 1/6 = 0.0001164088720081 = 0.0006979657356246 * t(m)

 

This plot was created with Gnuplot.

Licensing[edit]

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Summary[edit]

// program by marcm200
coefficients read from input file spiral6.txt
	degree 2 coefficient = ( +1.0000000000000000 +0.0000000000000000*i) 
	degree 0 coefficient = ( -0.3749678400000000 +0.2168721400000000*i) 

Input polynomial p(z)=(1+0i)*z^2+(-0.3749678399999999967+0.21687213999999999103i)

1 critical points found

	cp#0: 0,0 . It's critical orbit is bounded and enters cycle #0 length=1 and it's stability = |multiplier|=0.66182 =attractive 
	internal angle = 0.43301199486547292672
cycle = {
-0.30202715286803538675,0.13520249234983439113 ; }

// progrm m-describe by Claude
the input point was +3.7496784e-01 + +2.1687213999999999e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
  with nucleus at +0e+00 + +0e+00 i
  the component has size 1.00000e+00 and is pointing west
  the atom domain has size 0.00000e+00
  the atom domain coordinates of the input point are -nan + -nan i
  the atom domain coordinates in polar form are nan to the east
  the nucleus is 4.33168e-01 to the west-south-west of the input point
  the input point is interior to this component at
  radius 9.99936e-01 and angle 0.166783075538674747 (in turns)
  the multiplier is +4.99335e-01 + +8.66336e-01 i
  a point in the attractor is +2.49664e-01 + +4.33167e-01 i
  external angles of this component are:
  .(0)
  .(1)

Maxima CAS src code[edit]

/*  
this is batch file for Maxima  5.13.0
http://maxima.sourceforge.net/
tested in wxMaxima 0.7.1
using draw package ( interface to gnuplot ) to draw on the screen


draws  critical orbit = orbit of critical point
 

Adam Majewski fraktal.republika.pl
with help of 
Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates
*/


/*  */

c:%i*0.21687214+0.37496784;


/* define function ( map) for dynamical system z(n+1)=f(zn,c)  */
f(z,c):=expand (z*z + c); /* expand speed up  computations and fix the stack overflow problem. Robert Dodier */

/* maximal number of iterations */
iMax:100000; /* to big couses bind stack overflow */
EscapeRadius:10;

/* define z-plane ( dynamical ) */
zxMin:-0.8;
zxMax:0.2;
zyMin:-0.2;
zyMax:0.8;

/* resolution is proportional to number of details and time of drawing */
iXmax:1000;
iYmax:1000;


/* compute critical point */
zcr:rhs(solve(diff(f(z,c),z,1)));
/* save critical point to 2 lists */
xcr:makelist (realpart(zcr), i, 1, 1); /* list of re(z) */
ycr:makelist (imagpart(zcr), i, 1, 1); /* list of im(z) */	

/* compute attracting fixed point */
fixed:solve([z=f(z,c)], [z]);
zfa:if cabs(2*rhs(fixed[1]))<1 then rhs(fixed[1]) else rhs(fixed[2]);
/* save attracting fixed point to 2 lists */
xfa:makelist (realpart(zfa), i, 1, 1); /* list of re(z) */
yfa:makelist (imagpart(zfa), i, 1, 1); /* list of im(z) */	


/* ------------------- compute forward orbit of critical point in 6 subsets ----------*/
z:zcr; /* first point  */
for i:1 thru 6 step 1 do
block
(
  z:f(z,c),
  if i=1 then orbit1:[z]
	elseif i=2 then orbit2:[z]
	elseif i=3 then orbit3:[z]
	elseif i=4 then orbit4:[z]
	elseif i=5 then orbit5:[z]
	elseif i=6 then orbit6:[z]
  );

for i:6 thru iMax step 1 do
block
(
  z:f(z,c),
  
  if abs(z)>EscapeRadius 
	then exit(i) 
	else (
		j:mod(i,6),
		if j=0 then orbit1:endcons(z,orbit1)
		elseif j=1 then orbit2:endcons(z,orbit2)
		elseif j=2 then orbit3:endcons(z,orbit3)
		elseif j=3 then orbit4:endcons(z,orbit4)
		elseif j=4 then orbit5:endcons(z,orbit5)
		elseif j=5 then orbit6:endcons(z,orbit6)
		
		)
  );







/* draw reversed orbit of beta  using draw package */
load(draw);
draw2d(
    title= concat(" dynamical plane for f(z,c):=z*z+",string(c)),
	user_preamble = "set key outside top",
	file_name = "cr6spiral",
    terminal  = 'png,
	pic_width  = iXmax,
    pic_height = iYmax,
	/* yrange = [zyMin,zyMax],
    xrange = [zxMin,zxMax],
	*/
	xlabel     = "Z.re ",
    ylabel     = "Z.im",
    point_type    = filled_circle,
	points_joined = false,
	
	point_size    = 0.8,
	key = "critical point",
	color		  =magenta,
	points(xcr,ycr),
	
	point_size    = 0.2,
	
	color		  =red,
	key = "1-spiral of critical orbit",
	points(map (realpart,orbit1),map (imagpart,orbit1)),
	
	color		  =orange,
	key = "2-spiral of critical orbit",
	points(map (realpart,orbit2),map (imagpart,orbit2)),
	
	color		  =yellow,
	key = "3-spiral of critical orbit",
	points(map (realpart,orbit3),map (imagpart,orbit3)),
	
	color		  =green,
	key = "4-spiral of critical orbit",
	points(map (realpart,orbit4),map (imagpart,orbit4)),
	
	color		  =cyan,
	key = "5-spiral of critical orbit",
	points(map (realpart,orbit5),map (imagpart,orbit5)),
	
	color		  =blue,
	key = "6-spiral of critical orbit",
	points(map (realpart,orbit6),map (imagpart,orbit6)),
	
	point_size    = 0.8,
	key = "attracting fixed point",
	color		  =black,
	points(xfa,yfa)
 );