Difference between revisions of "File:Logi2b3t1000.jpg"

Complex map of the half iterate of the logistic operator

\( T(z)= s\, z\,(1\!-\!z) \ \) for \(\ s\!=\!3 \)

\[ u\!+\!\mathrm i v=T^{0.5}(x\!+\!\mathrm i y) \]

\[ T^{0.5}(z)=F(0.5+G(z)) \]

\(F\) and \(G\) are Superfunction and Abelfunction for the Transferfunction \(T\)

This map appears as Fig.7.10 at page 82 of book «Superfunctions»^[1][2]
in order to show that the Holomorphic extension of the logistic sequence ^[3] is not so chaotic.

C++ generator of map

#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 std::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"
/*
z_type arccos(z_type z){ return -I*log(z+I*sqrt(1.-z*z)); }
z_type coe(z_type z){ return .5*(1.-cos(exp((z+1.)/LQ))); }
z_type boe(z_type z){ return LQ*log(arccos(1.-2.*z))-1.; }
z_type doe(z_type z){ return coe(1.+boe(z));; }
*/
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  int M=101,M1=M+1;
  int N=201,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("logi2b3.eps","w");ado(o,124,124);
fprintf(o,"62 62 translate\n 20 20 scale\n");
DO(m,M1) X[m]=-3.+.06*(m-.5);
DO(n,N1) Y[n]=-3.+.03*(n-.5);

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=E(H(z))-1.;
//      c=F(1.+E(0.1*z));
//      c=F(z);
        c=F(.5+E(z));
//      c=boe(z);
//      c=.5*(1.-cos(exp((z+1.)/LQ)));
//      d=H(F(z-1.));
//      p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
//      if(p>-4.9 && p<20) g[m*N1+n]=p;
        p=Re(c);q=Im(c);        
        if(p>-4.9 && p<4.9)     {g[m*N1+n]=p;}
//      if(q>-4.9 && q<4.9)     {f[m*N1+n]=q;}
        if(q>-4.9 && q<4.9 && fabs(q)>1.e-11 )  {f[m*N1+n]=q;}
                        }}

fprintf(o,"1 setlinejoin 2 setlinecap\n"); 
//p=.8;q=.4;
p=2.;q=.5;
//#include"plof.cin"
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");

fprintf(o,"0 setlinejoin 0 setlinecap\n");

//M(-3.02,0)L(0,0)  
M(1.-1./Q,0)L(3,0)  fprintf(o,"0.03 W 1 1 1 RGB S\n");
//for(n=0;n<16;n++) {M(-.2*n,0)L(-.2*(n+.4),0)}
for(n=0;n<12;n++) {     M(1-1./Q+.2* n,0)
                        L(1-1./Q+.2*(n+.4),0)}
fprintf(o,"0.04 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi2b3.eps"); // for linux
        system(    "open logi2b3.pdf"); // for mac
        getchar(); system("killall Preview");
}

//

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 130pt
\paperheight 130pt
\topmargin -104pt
\oddsidemargin -91pt
\newcommand \sx {\scalebox}
\newcommand \ing \includegraphics
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}

\newcommand \axes {
\normalsize
\put(  5,126){\sx{.6}{$y$}}
\put(  5,106.4){\sx{.6}{$2$}}
\put(  5, 86.4){\sx{.6}{$1$}}
\put(  5, 66.4){\sx{.6}{$0$}}
\put( 0, 46.4){\sx{.6}{$-1$}}
\put( 0, 26.4){\sx{.6}{$-2$}}
\put( 24,  1){\sx{.6}{$-2$}}
\put( 44,  1){\sx{.6}{$-1$}}
\put( 68.4,  1){\sx{.6}{$0$}}
\put( 89.2,  1){\sx{.6}{$1$}}
\put(109.2,  1){\sx{.6}{$2$}}
\put(126,  1){\sx{.6}{$x$}}
}
\begin{picture}(122,124) 
%\put( 4, 4){\ing{logi2b3}}
\put( 8, 6){\ing{logi2b3}}
\normalsize
\put( 12, 91){\rot{ 6}\sx{.8}{$v\!=\!4$}\ero}
\put( 12, 66){\rot{ 0}\sx{.8}{$v\!=\!0$}\ero}
\put(  5, 40){\rot{-6}\sx{.8}{$v\!=\!-4$}\ero}
\put(112,106){\rot{ 50}\sx{.8}{$v\!=\!0$}\ero}
\put(110, 26){\rot{-52}\sx{.8}{$v\!=\!0$}\ero}
%
\put(24, 7){\rot{73}\sx{.8}{$u\!=\!-4$}\ero}
\put(37.8, 8){\rot{69}\sx{.8}{$u\!=\!-2$}\ero}
\put(50, 8){\rot{65}\sx{.8}{$u\!=\!0$}\ero}
\put(62, 8){\rot{57}\sx{.8}{$u\!=\!2$}\ero}
%
\axes
\end{picture}
\end{document}

References

Keywords

«Abelfunction», «Holomorphic extension of the Logistic sequence», «Iterate», «LogisitcOperator», «LogisticSequence», «Regular iteration», «Table of superfunctions», «Transfer equation», «Superfunction», «Superfunctions»,