File:Parabolic critical orbit of rational function ( Blaschke fraction).png

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/* 
[0.7121885831301268*%i-0.7019881922504848,
        0.6038629905954274*%i+0.7970881310050664,
        (-0.8738242051822692*%i)-0.4862419751909308,
        0.7121885831301268*%i-0.701988192250485]

       


*/

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



/* f(z) is used as a global function
   I do not know how to put it as a argument */

GiveOrbit(z0,OrbitLength):=
block(
 [z,Orbit],
 z:z0,
 Orbit:[z], 
 for i:1 thru OrbitLength step 1 do
        ( z:expand(f(z)),
         if cabs(z) > 2 then 
         	( print(z0), print("orbit of is escaping"),  
         	  return(Orbit)),
          Orbit:endcons(z ,Orbit)
          ),
 print(z0), print("orbit is not escaping"),  
 return(Orbit) 

)$



/* f(z) is used as a global function
   I do not know how to put it as a argument */

GiveOneArmOrbit(z0,OrbitLength):=
block(
 [z,Orbit],
 z:z0,
 Orbit:[z], 
 for i:1 thru OrbitLength step 1 do
        ( z:expand(f(z)),
          
         if cabs(z) > 2 then 
         	( print(z0), print("orbit of is escaping"),  
         	  return(Orbit)),
          Orbit:endcons(z ,Orbit)
          ),
 print(z0), print("orbit is not escaping"),  
 return(Orbit) 

)$

/* converts angle in radians in range -Pi to Pi
   to turns */
GiveTurn(a):=
( 
 
  if a<0 then a:a+2*%pi, /* from range (-Pi,Pi) to range (0,2Pi) */
   float(a/(2*%pi)) /* from radians to turns */
)$

/* give turn of complex number z */
cturn(z):=GiveTurn(carg(z))$





/* give Draw List from one point*/
/* 
converts complex number z = x*y*%i 
to the list in a draw format:  
[x,y] 
*/
d(z):=[float(realpart(z)), float(imagpart(z))]$
ToPoint(z):= points([d(z)])$ /* give Draw List from one point*/
ToPoints(myList):= points(map(d,myList))$



compile(all);




radius : 1.0;
t:1/3;
rho : -0.6170144002709304 +0.7869518599370003*%i;
f(z):= float(rectform((rho * z^2 * (z-3)/(1-3*z))))$



dz: diff(f(z),z,1);







iLength:10000;
Orbit: GiveOneArmOrbit(1.0,iLength)$



/* period 3 points */
e: f(f(f(z))) = z$

load (to_poly_solve);
s: to_poly_solve (e,z);

s: args(s); /*  https://stackoverflow.com/questions/12834709/create-a-union-into-a-list-in-maxima */
s:flatten(s);
s:map(rhs,s);

r:[];
for z in s do if (abs(abs(z) -1) < 0.1) then  r:cons(z,r);


cycle1:[];
z:r[1];
cycle1: cons(z, cycle1);
for i:1 thru 3 step 1 do (

	z:float(rectform(f(z))),
	cycle1:cons(z, cycle1)
	
);







turns:map(cturn,cycle1);

Orbit : ToPoints(Orbit)$
r:ToPoints(r)$
z32: ToPoint(cycle1[2]);
z31: ToPoint(cycle1[1]);
z33: ToPoint(cycle1[3]);
zcr: ToPoint(1);
cycle1: ToPoints(cycle1)$

load(draw); /* ( interface to gnuplot ) by Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates */

draw2d(
    title = "Parabolic period 3 orbit for f(z)= rho * z^2 * (z-3)/(1-3*z))",
    terminal  = png,
     user_preamble = "set size square; set key left top;", /* 360/26=13.85  ; 360/(2*26)=6,923 */
    file_name = concat("~/Dokumenty/julia_blaszke/period3/maxima/cycle_check/1over3/", concat("cycles_j_",string(iLength))),
    dimensions  = [600, 600],   /* Since Maxima 5.23, pic_width and pic_height are deprecated. */
    
    yrange = [-1.2,1.2],
    xrange = [-1.2, 1.2],
    
    xlabel     = "z.re ",
    ylabel     = "z.im",
    
    
   line_width        = 1,
    nticks = 50,
    color             = gray,
    transparent       = true,
    ellipse(0,0,1,1,0,360), /* unit circle */
    
    point_type    = filled_circle,
    points_joined = true,
    
    point_size    = 1.2,
    key="periodic points",
    color             = red ,
    cycle1,
    points_joined = false,
    point_size    = 1.2,
    key="critical point",
    color             = blue ,
    zcr,    
    
    point_size    = 0.8,
    points_joined = false,
    key="critical orbit ",
    color             = black ,
    Orbit
    
  );