Difference between revisions of "File:LogisticSecK2.jpg"

From TORI
Jump to navigation Jump to search
(Importing image file)
 
 
Line 1: Line 1:
  +
[[Logistic Sequence]] of real argument.
−
Importing image file
  
  +
  +
$y=F(x)=\mathrm{LogisticSequence}_s(x)$ versus $x$ for various values of parameter $s$.
  +
  +
where $F$ 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]]
  +
<ref name="logi">
  +
http://mizugadro.mydns.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)
  +
</ref>.
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:Explicit plot]]
  +
[[Category:Logistic operator]]
  +
[[Category:Logistic sequence]]
  +
  +
==[[C++]] generator of curves of the First picture==
  +
  +
<poem><nomathjax><nowiki>
  +
#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");
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[C++]] generator of curves of the Second picture==
  +
  +
<poem><nomathjax><nowiki>
  +
#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");
  +
}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\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}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
The free use is allowed, attribute the source.

Latest revision as of 08:42, 1 December 2018

Logistic Sequence of real argument.

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

where $F$ 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].

References

  1. ↑ http://mizugadro.mydns.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)

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.

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current06:13, 1 December 2018Thumbnail for version as of 06:13, 1 December 20186,807 × 3,044 (996 KB)Maintenance script (talk | contribs)Importing image file

You cannot overwrite this file.

The following page uses this file:

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=File:LogisticSecK2.jpg&oldid=19996"