Difference between revisions of "File:LogisticSecK2.jpg"

Logistic Sequence of real argument.

\(y=F(x)=\mathrm{LogisticSequence}_s(x)\) versus \(x`) for various values of parameter \(s\).

Here \(F\) is simplest superfunction of the Logistic operator, id est, solution of the transfer equation

\[ F(z\!+\!1) = s F(z)(1-F(z)) \]

for the logistic operator \(\mathrm{LogisticOperator}_s(z)=sz(1-z)\) as transfer function.

This solution exponentially approaches zero at minus infinity.

The holomorphic extension of the logistic sequence is described in 2010 in the Moscow University Physics Bulletin ^[1].

This plot is used as Fig.7.3 at page 73 of book «Superfunctions»^[2][3]
in order to show that the holomorphic extension of the Logistic sequence (Logistic Sequence ^[4] or logistic map sequence ^[5]) is pretty smooth at regular; the only oscillations become more frequent at the increase of the real part of the argument.

C++ generator of curves of the First picture

#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 "ado.cin"
#include "efjh.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
FILE *o;o=fopen("logi6.eps","w");ado(o,164,24);
fprintf(o,"62 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);

fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(m=-3;m<6;m++){if(m==0){M(m,-.04)L(m,1.06)} else{M(m,0)L(m,1)}}
for(n=0;n<2;n++){       M(  -3,n)L(5,n)}
fprintf(o,".008 W 0 0 0 RGB S\n");

maq(3.4);
DO(m,1004) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(m==0)M(x,y) else L(x,y);}
fprintf(o,".01 W 0 .7 0 RGB S\n");

maq(3.);
DO(m,1004) { x=-3.+8.*sqrt(.001*m); y=Re(F(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");

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

maq(3.8);
DO(m,1004) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(y>-2) { if(m==0)M(x,y) else L(x,y);} }
fprintf(o,".015 W 0 0 1 RGB [.001 .025] 0 setdash S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi6.eps");
        system(    "open logi6.pdf");
        getchar(); system("killall Preview");
}

C++ generator of curves of the Second picture

#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 "ado.cin"
#include "efjh.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
FILE *o;o=fopen("logi5.eps","w");ado(o,164,44);
fprintf(o,"62 22 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);

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

for(m=-3;m<6;m++){if(m==0){M(m,-1.06)L(m,1.06)} else{M(m,-1)L(m,1)}}
for(n=-1;n<2;n++){      M(  -3,n)L(5,n)}
fprintf(o,".008 W 0 0 0 RGB S\n");

maq(4.);
DO(m,1001) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(m==0)M(x,y) else L(x,y);}
fprintf(o,".01 W 0 .7 0 RGB S\n");

maq(3.9);
DO(m,1001) { x=-3.+8.*sqrt(.001*m); y=Re(F(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");

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

maq(4.1);
DO(m,1001) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(y>-2) { if(m==0)M(x,y) else L(x,y);} }
fprintf(o,".015 W 0 0 1 RGB [.001 .02] 0 setdash S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi5.eps");
        system(    "open logi5.pdf");
        getchar(); system("killall Preview");
}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 492pt
\paperheight 220pt
\topmargin -100pt
\oddsidemargin -72pt
\newcommand \sx {\scalebox}
\newcommand \ing \includegraphics
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\parindent 0pt
\pagestyle{empty}
\begin{document}
\sx{3}{\begin{picture}(166,22)
\put( 1,1){\includegraphics{logi6}}
%\put( 64,48){\sx{.6}{$F(x)$}}
\put( 50,16){\sx{.4}{$F(x)$}}
\put( 0,22){\sx{.3}{$1$}}
\put( 0, 3){\sx{.3}{$0$}}
%\put( -1, 3){\sx{.4}{$-1$}}
\put( 20, 0){\sx{.3}{$-2$}}
\put( 40, 0){\sx{.3}{$-1$}}
\put( 62.5, 0){\sx{.3}{$0$}}
\put( 82.5, 0){\sx{.3}{$1$}}
\put(102.5, 0){\sx{.3}{$2$}}
\put(122.5, 0){\sx{.3}{$3$}}
\put(142.5, 0){\sx{.3}{$4$}}
\put(162, 0){\sx{.3}{$x$}}
\put(91,15.5){\sx{.3}{$s\!=\!3$}}
\put(91,11){\sx{.3}{$s\!=\!3.4$}}
\put(93, 5){\sx{.3}{$s\!=\!3.8$}}
\end{picture}}
\vskip 9pt

\sx{3}{\begin{picture}(166,44)
\put( 1,1){\includegraphics{logi5}}
%\put( 64,48){\sx{.6}{$F(x)$}}
\put( 50,36){\sx{.4}{$F(x)$}}
\put(  0,41.5){\sx{.3}{$1$}}
\put(  0,22){\sx{.3}{$0$}}
\put( -1, 2){\sx{.3}{$-\!1$}}
\put( 20, 0){\sx{.3}{$-2$}}
\put( 40, 0){\sx{.3}{$-1$}}
\put( 62.5, 0){\sx{.3}{$0$}}
\put( 82.5, 0){\sx{.3}{$1$}}
\put(102.5, 0){\sx{.3}{$2$}}
\put(122.5, 0){\sx{.3}{$3$}}
\put(142.5, 0){\sx{.3}{$4$}}
\put(162, 0){\sx{.3}{$x$}}
\put(91.3,29){\sx{.3}{$s\!=\!3.9$}}
%\put(91,21){\sx{.3}{$s\!=\!4$}}
\put(92.5, 19.5){\sx{.3}{$s\!=\!4.1$}}
\end{picture}}
\end{document}

The free use is allowed, attribute the source.

References

Keywords

«Holomorphic extension of the Collatz subsequence», «LogisitcOperator», «LogisticSequence», «Table of superfunctions», «Transfer equation», «Superfunction», «Superfunctions»,