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