Difference between revisions of "File:Logi5ab400.jpg"

Superfunction \(f\) of the logistic operator

\[ T(z)= s\,z\,(1\!-\!z) \]

constructed with the regular iteration at the upper fixed point \(\ L\!=\!1-1/s\ \) for \(\ s\!=\!4 \)

Graphic at the top: explicit plot \(y\!=\!f(x)\)

Complex map at the bottom: \[ u\!+\!\mathrm i v=f(x\!+\!\mathrm i y) \]

This map appears as Fig.7.15 at page 87 of book «Superfunctions»^[1][2]
in order to show that for the Holomorphic extension of the logistic sequence ^[3], the superfunctions constructed with the regular iteration at different fixed points are very different, although they satisfy the same Transfer equation

\[ F(z\!+\!1) = T(F(z)) \]

C++ generator of the graphic at the top

//Files ado.cin, logiu.cin should be loaded to the working directory in order to compile the 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 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"
//#include "u.cin"
#include "logiu.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  int M=801,M1=M+1;
  int N=401,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("logi5a4x.eps","w");ado(o,164,84);
fprintf(o,"102 42 translate\n 20 20 scale\n");
DO(m,M1) X[m]=-5.+.01*(m-.5);
DO(n,N1) Y[n]=-2.+.01*(n-.5);

for(m=-5;m<4;m++){if(m==0){M(m,-2.04)L(m,2.04)} else{M(m,-2)L(m,2)}}
for(n=-2;n<3;n++){         M(  -5  ,n)L(3,n)}
fprintf(o,".008 W 0 0 0 RGB S\n");

maq(4.);
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=U(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=.8;q=.3;
//#include"plof.cin"
for(m=-3;m<3;m++) 
for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".004 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,".004 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,".004 W 0 0 .9 RGB S\n");

for(m=1;m<6;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);   fprintf(o,".016 W .9 0 0 RGB S\n");
for(m=1;m<6;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);   fprintf(o,".016 W 0 0 .9 RGB S\n");
for(m=-4;m<6;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p);   fprintf(o,".016 W 0 0 0 RGB S\n");

                  conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p);   fprintf(o,".016 W .6 0 .6 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi5a4x.eps"); // for linux
        system(    "open logi5a4x.pdf"); // for mac
        getchar(); system("killall Preview");
}

C++ generator of the map at the bottom

// Files ado.cin, conto.cin, logiu.cin should be loaded 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 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 "ado.cin"
//#include "efjh.cin"
#include "logiu.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
FILE *o;o=fopen("logi5b4x.eps","w");ado(o,164,24);
fprintf(o,"102 2 translate\n 20 20 scale\n");
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

for(m=-5;m<4;m++){if(m==0){M(m,-.04)L(m,1.04)} else{M(m,0)L(m,1)}}
for(n=0;n<2;n++){       M(  -5,n)L(3,n)}
fprintf(o,".006 W 0 0 0 RGB S\n");

M(-5,.75)L(3,.75)
fprintf(o,".004 W 0 0 0 RGB S\n");

fprintf(o,"1 setlinejoin 2 setlinecap\n");

maq(4.);
DO(m,1034) { x=-5.08+8.*sqrt(.001*m); y=Re(U(x)); if(m==0)M(x,y) else L(x,y);}
fprintf(o,".01 W 0 .5 0 RGB S\n");

/*
maq(3.99);
DO(m,1001) { x=-5.+10.*sqrt(.001*m); y=Re(U(x)); if(m==0)M(x,y) else L(x,y);}
fprintf(o,".015 W 1 0 0 RGB [.03 .04] 0 setdash S\n");

maq(4.01);
DO(m,1001) { x=-5.+10.*sqrt(.001*m); y=Re(U(x)); if(m==0)M(x,y) else L(x,y);}
fprintf(o,".015 W 0 0 1 RGB [.01 .03] 0 setdash S\n");
*/
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi5b4x.eps");
        system(    "open logi5b4x.pdf");
        getchar(); system("killall Preview");
}

Latex generator of the labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 426pt
\paperheight 284pt
\topmargin -108pt
\oddsidemargin -90pt
\newcommand \sx {\scalebox}
\newcommand \ing \includegraphics
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}

\sx{2.6}{\begin{picture}(140,23) 
\put( 1, 1){\ing{logi5b4x}}
\put( 1, 21){\sx{.3}{$1$}}
\put( 1, 2){\sx{.3}{$0$}}

\put( 20.2,  0){\sx{.3}{$-4$}}
\put( 40.2,  0){\sx{.3}{$-3$}}
\put( 60.2,  0){\sx{.3}{$-2$}}
\put( 80.2,  0){\sx{.3}{$-1$}}
\put(102.6,  0){\sx{.3}{$0$}}
\put( 122.7,  0){\sx{.3}{$1$}}
\put(142.7,  0){\sx{.3}{$2$}}
\put(162,  0.4){\sx{.3}{$x$}}
\end{picture}}

%\sx{3.04}{\begin{picture}(140,85) 
\sx{2.6}{\begin{picture}(140,85) 
\put( 1, 1){\ing{logi5a4x}}
\put(  1, 81){\sx{.3}{$y$}}
\put(  1, 62){\sx{.3}{$1$}}
\put(  1, 42){\sx{.3}{$0$}}
\put( -1, 22){\sx{.3}{$-\!1$}}
%\put( -1, 2){\sx{.3}{$-\!2$}}
\put( 20.2,  0){\sx{.3}{$-4$}}
\put( 40.2,  0){\sx{.3}{$-3$}}
\put( 60.2,  0){\sx{.3}{$-2$}}
\put( 80.2,  0){\sx{.3}{$-1$}}
\put(102.6,  0){\sx{.3}{$0$}}
\put( 122.7,  0){\sx{.3}{$1$}}
\put(142.7,  0){\sx{.3}{$2$}}
\put(162,  0.4){\sx{.3}{$x$}}
\put(5.5,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}}
\put(25.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}}
\put(45.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}}
\put(65.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}}
\put(85.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}}
\put(105.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}}
\put(8,42.3){\sx{.3}{$v\!=\!0$}}
\put(20,39){\sx{.3}{\rot{90}$u\!=\!0.7$\ero}}
\put(30.6,38.6){\sx{.3}{\rot{90}$u\!=\!0.7$\ero}}
\put(37.4,38.4){\sx{.3}{\rot{90}$u\!=\!0.8$\ero}}

\put(50,45.3){\sx{.3}{$v\!=\!-0.1$}}
\put(52,42.3){\sx{.3}{$v\!=\!0$}}
\put(52,39.3){\sx{.3}{$v\!=\!0.1$}}

%\put(68,44.3){\sx{.25}{$v\!=\!0.1$}}
\put(69,42.3){\sx{.3}{$v\!=\!0$}}
%\put(67,40.3){\sx{.25}{$v\!=\!-0.1$}}
\end{picture}}
\end{document}

References

Keywords

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