File:Logi2c3T1000.png

Complex map of function LogisticSequence with critic value of parameter,

$ u\!+\!\mathrm i v= \mathrm{LogisticSequence}_4(x\!+\!\mathrm i y)$

The generators of the map are copypasted below

efjh.cin

// The code below should be stored as efjh.cin

DB LQ;
DB QQ,Q2,Q3,Q4,Q5,Q6,Q7;
DB C1,C2,C3,C4,C5,C6,C7;
DB S1,S2,S3,S4,S5,S6,S7,S8;

void maq(DB q){
Q=q;
QQ=Q*Q;
Q2=QQ;
Q3=QQ*Q;
Q4=QQ*QQ;
Q5=Q4*Q;
Q6=Q3*Q3;
Q7=Q6*Q;
LQ=log(Q);
C2=1.                                                           / (1.-Q) ;
C3=2.                                                           /((1.-Q)*(1.-Q2) );
C4=   (5.+Q)                                                    /((1.-Q)*(1.-Q2)*(1.-Q3) );
C5=2.*(7.+Q*(3.+Q*2.))                                          /((1.-Q)*(1.-Q2)*(1.-Q3)*(1.-Q4) );
C6=2.*(21.+Q*(14.+Q*(14.+Q*( 8.+Q*3.))))                        /((1.-Q)*(1.-Q2)*(1.-Q3)*(1.-Q4)*(1.-Q5));
C7=4.*(33.+Q*(30.+Q*(37.+Q*(32.+Q*(27.+Q*(12.+Q*(8.+Q)))))))    /((1.-Q)*(1.-Q2)*(1.-Q3)*(1.-Q4)*(1.-Q5)*(1.-Q6));

S1=1.;
S2=1.                                                           / (Q-1.);
S3=2.*Q                                                         /((Q-1.)*(Q2-1.));
S4=   Q *(1.+5.*Q*Q)                                            /((Q-1.)*(Q2-1.)*(Q3-1.) );
S5=2.*Q3*(3.+Q*(2.+QQ*7.))                                      /((Q-1.)*(Q2-1.)*(Q3-1.)*(Q4-1.) );
S6=2.*Q3*(1.+Q*(3.+QQ*(14.+Q*(14.+Q*(7.+QQ*21)))))              /((Q-1.)*(Q2-1.)*(Q3-1.)*(Q4-1.)*(Q5-1.));
S7=4.*Q4*(1.+Q*(5.+Q*(15.+Q*(12.+Q*(8.+Q*(30.+Q*(37.+Q*(27.+Q*(12.+QQ*(33.))))))))))
                                                               /((Q-1.)*(Q2-1.)*(Q3-1.)*(Q4-1.)*(Q5-1.)*(Q6-1.));
S8=Q4*(1.+QQ*(5.+Q*(60.+Q*(69.+Q*(89.+Q*(162.+Q*(415.+Q*(495.+Q*(531.+Q*(267.+Q*(630.+Q*(705.+Q*(621.+Q*(396+Q*(165.+Q*429)
             )     )      )      )      )       )       )       )       )       )       )       )       )
       )                                                       /((Q-1.)*(Q2-1.)*(Q3-1.)*(Q4-1.)*(Q5-1.)*(Q6-1.)*(Q7-1.));
;
printf("maq: Q=%9.6f\n",Q);}

z_type J(z_type z){ return .5-sqrt(.25-z/Q); }
z_type H(z_type z){ return Q*z*(1.-z); }

z_type f(z_type z){z_type e;e=exp(LQ*z);return e*(1.+e*(C2+e*(C3+e*(C4+e*(C5+e*(C6+e*C7))))));  }
z_type F(z_type z){     if(Re(z)>-6.) return H(F(z-1.));   return f(z);}
z_type e(z_type z){                     return z*(1.+z*(S2+z*(S3+z*(S4+z*(S5+z*(S6+z*(S7+z* S8)))))));}
z_type E(z_type z){     if(abs(z)>.1) return  E(J(z))+1.;
                       return log(e(z))/LQ;}

// Copyleft 2009,2012 by Dmitrii Kouznetsov

Generator of curves

// File efjh.cin above, and files ado.cin and conto.cin should be loaded to the working directory in order to compile the C++ code below

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
#include "efjh.cin"
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 int M=408,M1=M+1;
 int N=602,N1=N+1;
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
char v[M1*N1]; // v is working array
FILE *o;o=fopen("logi2c3a.eps","w");ado(o,126,124);
fprintf(o,"62 62 translate\n 20 20 scale\n");
DO(m,M1) X[m]=-3.01+.015*(m-.5);
DO(n,300) Y[n]=-3.+.01*n;
          Y[300]=-0.002;
          Y[301]= 0.002;
for(n=302;n<N1;n++) Y[n]=-3.+.01*(n-1);
fprintf(o,"1 setlinejoin 2 setlinecap\n"); 
for(m=-3;m<4;m++){if(m==0){M(m,-3.04)L(m,3.04)} else{M(m,-3)L(m,3)}}
for(n=-3;n<4;n++){         M(  -3  ,n)L(3,n)}
fprintf(o,".008 W 0 0 0 RGB S\n");
maq(3.);
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); 
       c=F(z);
       p=Re(c);q=Im(c);        
       if(p>-49 && p<49)       {g[m*N1+n]=p;}
       if(q>-49 && q<49)       {f[m*N1+n]=q;}
                       }}
fprintf(o,"1 setlinejoin 2 setlinecap\n"); 
p=2.;q=.5;
for(m=-2;m<2;m++) 
for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".005 W 0 .6 0 RGB S\n");
for(m=0;m<2;m++) 
for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".005 W .9 0 0 RGB S\n");
for(m=0;m<2;m++) 
for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".005 W 0 0 .9 RGB S\n");
for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);   fprintf(o,".02 W .9 0 0 RGB S\n");
for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);   fprintf(o,".02 W 0 0 .9 RGB S\n");
for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p);   fprintf(o,".02 W 0 0 0 RGB S\n");
                 conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p);   fprintf(o,".02 W .6 0 .6 RGB S\n");
printf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf logi2c3a.eps"); // for linux
       system(    "open logi2c3a.pdf"); // for mac
       getchar(); system("killall Preview");
}

// Copyleft 2009, 2012 by Dmitrii Kouznetsov

Latex generator of labels

% File logi2c3a.pdf should be generated with the code above in order to compile the Latex document below

% %</br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphics} %<br> \usepackage{rotating} %<br> \paperwidth 128pt %<br> \paperheight 127pt %<br> \topmargin -104pt %<br> \oddsidemargin -90pt %<br> \newcommand \sx {\scalebox} %<br> \newcommand \ing \includegraphics %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \begin{document} %<br> \newcommand \axes { %<br> \put( 2,124){\sx{.5}{$y$}} %<br> \put( 2,105){\sx{.5}{$2$}} %<br> \put( 2,85){\sx{.5}{$1$}} %<br> \put( 2, 65){\sx{.5}{$0$}} %<br> \put(-.8, 45){\sx{.5}{$-\!1$}} %<br> \put(-.8, 25){\sx{.5}{$-\!2$}} %<br> \put( 22, 1){\sx{.5}{$-2$}} %<br> \put( 42, 1){\sx{.5}{$-\!1$}} %<br> \put( 65, 1){\sx{.5}{$0$}} %<br> \put( 85, 1){\sx{.5}{$1$}} %<br> \put(105, 1){\sx{.5}{$2$}} %<br> \put(125, 1){\sx{.5}{$x$}} %<br> } %<br> \begin{picture}(122,122) \put( 4, 4){\ing{logi2c3a}} %<br> \put(7,121.6){\rot{ 0.}\sx{.6}{$v\!=\!0$}\ero} %<br> \put(7, 94){\rot{ 0.}\sx{.6}{$u\!=\!0$}\ero} %<br> \put(7, 64.5){\rot{ 0.}\sx{.6}{$v\!=\!0$}\ero} %<br> \put(7, 35){\rot{ 0.}\sx{.6}{$u\!=\!0$}\ero} %<br> \put(7, 7){\rot{ 0.}\sx{.6}{$v\!=\!0$}\ero} %<br> \put(26,72){\rot{50}\sx{.5}{$u\!=\!0.1$}\ero} %<br> \put(39,69){\rot{62}\sx{.5}{$u\!=\!0.2$}\ero} %<br> \put(52,70.4){\sx{.5}{$v\!=\!0.1$}} %<br> \put(49,58.8){\sx{.5}{$v\!=\!-0.1$}} %<br> \axes %<br> \end{picture} %<br> \end{document} %<br> % %

%

References

http://tori.ils.uec.ac.jp/PAPERS/2010logistie.pdf D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31)