Summary

Long description

This diagram is related with discrete dynamical system based on complex quadratic polynomials ${\displaystyle f_{c}(z)\,}$ when c=0

What program does :

• computes points of external ray for angle t=0 . See instruction zz:GiveRay(t,c)
• for each point in zz computes potential using simple method and saves to list pair of real numbers : distance to boundary and potential
• for each point in zz computes potential using full method and saves to list pair of real numbers : distance to boundary and potential
• draws above 2 sets of points using distance on horizontal axis and potential on vertical axis

Maxima src code

 /* 
 Maxima CAS batch file 
 Distributed under the GNU Public License. 
 draws 2 diagrams of potential in basin of attraction
 in infinity for fc(z)=z*z+c
 tested on : 
 wxMaxima 0.7.6 http://wxmaxima.sourceforge.net
 Maxima 5.16.3 http://maxima.sourceforge.net
 Using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
 */

 /* 
 gives a list of points ( complex numbers) 
 of external ray for angle 0 of unit circle 
 ( or f0(z) )
 */
 GiveRay(t,c):=
 block(
 [r], 
 /* range for drawing  R=2^r ; as r tends to 0 R tends to 1 */
 rMin:1E-10, /* 1E-4;  rMin > 0  ; if rMin=0 then program has infinity loop !!!!! */
 rMax:1, 
 caution:0.9330329915368074, /* r:r*caution ; it gives smaller r */
 /*  */
 R_max:200,
 /* */
 zz:[], /* array for z points of ray in fc plane */ 
 /*  some w-points of external ray in f0 plane  */
 r:rMax,
 while 2^r<R_max do r:2*r, /* find point w on ray near infinity (R>=R_max) in f0 plane */
 R:2^r,
 w:rectform(ev(R*exp(2*%pi*%i*t))), 
 z:w, /* near infinity z=w */
 zz:cons(z,zz),
 unless r<rMin do
 (	/* new smaller R */
 r:r*caution,  
 R:2^r,
 /* */
 z:rectform(ev(R*exp(2*%pi*%i*t))),
 zz:cons(z,zz)
 ),
 return(zz)
 )$

 /* gives distancve between 2 complex points */
 GiveDistance(c1,c2):=sqrt((realpart(c2)-realpart(c1))^2 +(imagpart(c2)-imagpart(c1))^2)$
 /* log(x) in Maxima is a natural (base e) logarithm of x  */
 log2(x) := log(x) / log(2);   

 /* returns floating point number 
 full definition
 */   
 GiveFLogPhi(z0,c,e_r,i_max):=
 block(
 [z:z0,
 logphi:log(cabs(z)),
 fac:1/2,
 i:0],
 while i<i_max and cabs(z)<e_r do
 (z:z*z+c,
 logphi:logphi+fac*log(cabs(1+c/(z*z))),
 i:i+1
 ),
 if i=iMax then logphi:0,
 return(float(logphi))
 )$

 /* returns floating point number 
 simple definition
 */   

 GiveSLogPhi(z0,c,e_r,i_max):=
 block(
 [z:z0,
 logphi,
 fac:1/2,
 i:0],
 while i<i_max and cabs(z)<e_r do
 (z:z*z+c,
 fac:fac/2,
 i:i+1
 ),
 if i=iMax 
 then logphi:0
 else logphi:fac*log(cabs(z)),
 return(float(logphi))
 )$

 /* ----------------------- main ----------------------------------------------------*/
 start:elapsed_run_time ();
 /* root point of period 2 component connected with period 1 component */
 c:0; /* hyperbolic */
 t:0;
 ER:3;
 iMax:1000;
 /* compute ray points & save to zz lists */
 zz:GiveRay(t,c)$
 /* landing points for external ray */
 /* landing point of ray  */
 zh:%e^(%i*t*2*%pi); 
 /* compute potential  = log(Phi)*/
 xy_FLogPhi:[];
 for z in zz do xy_FLogPhi:cons([GiveDistance(z,zh),GiveFLogPhi(z,c,ER,iMax)],xy_FLogPhi);
 xy_SLogPhi:[];
 for z in zz do xy_SLogPhi:cons([GiveDistance(z,zh),GiveSLogPhi(z,c,ER,iMax)],xy_SLogPhi);

 /* draw it using draw package by */
 x_max:1.0;
 y_max:20;

 load(draw); 
 draw2d(	terminal      = 'screen,
 /* */
 user_preamble =  "set grid;set key right bottom
 ",
 points_joined =true,
 point_size    =  0.6,
 point_type = filled_circle,
 xlabel = "distance to boundary",
 ylabel = "potential",
 /* */
 color         = green,
 key = " FullLogPhi",
 points(xy_FLogPhi),
 /* */
 color         = blue,
 key = " SimpleLogPhi",
 points(xy_SLogPhi)
 )$

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