Difference between revisions of "File:Logic5T.jpg"

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[[Iteration]] of the [[Logistic operator]] $T(z)=5\, z\, (1\!-\!z)$,
−
Importing image file
  
  +
  +
$y\!=\!T^n(x)$
  +
  +
versus $x$ for various values of $n$ by
  +
<ref>
  +
http://www.springerlink.com/content/u712vtp4122544x4 , DOI 10.3103/S0027134910020049 <br>
  +
http://mizugadro.mydns.jp/PAPERS/2010logistie.pdf , reprint, English version<br>
  +
http://mizugadro.mydns.jp/PAPERS/2010logistir.pdf , reprint, Russian version<br>
  +
D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31)
  +
</ref>.
  +
  +
==[[C++]] generator of curves==
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// Need also [[ado.cin]] and [[efjh.cin]]
  +
  +
<poem><nomathjax><nowiki>
  +
#include <stdio.h>
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#include <stdlib.h>
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#define DB double
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#define DO(x,y) for(x=0;x<y;x++)
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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 "ado.cin"
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#include "efjh.cin"
  +
  +
DB LO(DB x){ return 5.*x*(1.-x);}
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
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FILE *o;o=fopen("logic5.eps","w");ado(o,130,130);
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fprintf(o,"2 2 translate\n 100 100 scale\n");
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#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
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#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
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M(0,0)L(1.25,0)L(1.25,1.25)L(0,1.25)
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fprintf(o,"C .003 W 0 0 0 RGB S\n");
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M(0,.25)L(1.25,.25) M(.25,0)L(.25,1.25)
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M(0,.50)L(1.25,.50) M(.50,0)L(.50,1.25)
  +
M(0,.75)L(1.25,.75) M(.75,0)L(.75,1.25)
  +
M(0,1.0)L(1.25,1.0) M(1.0,0)L(1.0,1.25)
  +
fprintf(o,".001 W 0 0 0 RGB S\n");
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fprintf(o,"1 setlinejoin 2 setlinecap\n");
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maq(5.);
  +
  +
M(0,0) L(1.25,1.25)fprintf(o,".006 W 1 .3 1 RGB S\n");
  +
M(0,0) DO(m,1521){x=.001*(m+.9); y=LO(x) ; if(y>=-.1 && y<=1.5)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
  +
M(0,0) DO(m,1521){x=.001*(m+.9); y=LO(LO(x)) ; if(y>=-.1 && y<=1.5)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
  +
  +
M(0,0) L(1.25,1.25) fprintf(o,".001 W 0 0 0 RGB S\n");
  +
for(k=1;k<21;k+=1){ M(0,0) DO(m,1521){x=.001*(m+.9);c=F(.1*k+E(x)); y=Re(c);t=Im(c);if(y>=-.1 && y<1.5 && fabs(t)<1.e-9)L(x,y) else break;} }
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fprintf(o,".001 W 0 0 .5 RGB S\n");
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  +
M(0,0) DO(m,1521){x=.001*(m+.9);c=F(-1.+E(x)); y=Re(c);t=Im(c);if(y>=-.1 && y<1.5 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".006 W 1 .5 0 RGB S\n");
  +
M(0,0) DO(m,1521){x=.001*(m+.9);c=F(-2.+E(x)); y=Re(c);t=Im(c);if(y>=-.1 && y<1.5 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".006 W 1 .5 0 RGB S\n");
  +
  +
for(k=1;k<21;k+=1){ M(0,0) DO(m,1521){x=.001*(m+.9);c=F(-.1*k+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1.5 && fabs(t)<1.e-9)L(x,y) else break;} }
  +
fprintf(o,".001 W .5 0 0 RGB S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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system("epstopdf logic5.eps");
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system( "open logic5.pdf");
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getchar(); system("killall Preview");
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}
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</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of curves==
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<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
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\usepackage{geometry}
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\paperwidth 1308pt
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\paperheight 1314pt
  +
\topmargin -100pt
  +
\oddsidemargin -74pt
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\textwidth 1540pt
  +
\textheight 1740pt
  +
\usepackage{graphicx}
  +
%\usepackage{overcite}
  +
%\usepackage{hyperref}
  +
%\usepackage{amssymb}
  +
%\usepackage{wrapfig}
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
%\setlength{\parskip}{2mm}
  +
%\setlength{\parindent}{0mm}
  +
\newcommand \ds {\displaystyle}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rme {\mathrm{e}}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \rmi {\mathrm{i}}
  +
\newcommand \eL[1] {\iL{#1} \end{eqnarray}}
  +
\newcommand \rf[1] {(\ref{#1})}
  +
\parindent 0pt
  +
\pagestyle{empty}
  +
\begin{document}
  +
\sx{10}{\begin{picture}(130,131)
  +
\put(3,4){\ing{logic5}}
  +
\put(0,128){\sx{.7}{$y$}}
  +
\put(0,103){\sx{.7}{$1$}}
  +
\put(0,54){\sx{.7}{$\frac{1}{2}$}}
  +
\put(0,3){\sx{.7}{$0$}}
  +
\put(25.2,1){\sx{.5}{$1/4$}}
  +
\put(50.4,1){\sx{.5}{$1/2$}}
  +
\put(75.6,1){\sx{.5}{$3/4$}}
  +
\put(104,.5){\sx{.6}{$1$}}
  +
\put(127,1){\sx{.6}{$x$}}
  +
\put( 9.9,62){\sx{.7}{\rot{86}$n\!=\!2$\ero}}
  +
\put( 19.5,62){\sx{.7}{\rot{73}$n\!=\!1$\ero}}
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\put( 53.3,93.5){\sx{.64}{\rot{50}$n\!=\!0.5$\ero}}
  +
\put( 60.3,92.5){\sx{.64}{\rot{49}$n\!=\!0.4$\ero}}
  +
\put( 67.6,92.5){\sx{.64}{\rot{47}$n\!=\!0.3$\ero}}
  +
\put( 73.9,90,4){\sx{.64}{\rot{46}$n\!=\!0.2$\ero}}
  +
\put( 81,88,4){\sx{.65}{\rot{45}$n\!=\!0.1$\ero}}
  +
\put( 88.6,87){\sx{.7}{\rot{44}$n\!=\!0$\ero}}
  +
\put( 92,81.5){\sx{.6}{\rot{43}$n\!=\!-0.1$\ero}}
  +
\put( 96,76.2){\sx{.6}{\rot{43}$n\!=\!-0.2$\ero}}
  +
\put(101,71.5){\sx{.6}{\rot{43}$n\!=\!-0.3$\ero}}
  +
\put(103.4,58){\sx{.6}{\rot{42}$n\!=\!-0.5$\ero}}
  +
\put(108,32.4){\sx{.7}{\rot{32}$n\!=\!-1$\ero}}
  +
\put(106.4,9.4){\sx{.7}{\rot{8}$n\!=\!-2$\ero}}
  +
\end{picture}}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Logistic operator]]
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[[Category:Iteration]]
  +
[[Category:C++]]
  +
[[Category:Explicit plot]]
  +
[[Category:Latex]]

Latest revision as of 08:42, 1 December 2018

Iteration of the Logistic operator $T(z)=5\, z\, (1\!-\!z)$,

$y\!=\!T^n(x)$

versus $x$ for various values of $n$ by [1].

C++ generator of curves

// Need also ado.cin and efjh.cin


#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 "ado.cin"
#include "efjh.cin"

DB LO(DB x){ return 5.*x*(1.-x);}

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
FILE *o;o=fopen("logic5.eps","w");ado(o,130,130);
fprintf(o,"2 2 translate\n 100 100 scale\n");
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
M(0,0)L(1.25,0)L(1.25,1.25)L(0,1.25)
fprintf(o,"C .003 W 0 0 0 RGB S\n");
M(0,.25)L(1.25,.25) M(.25,0)L(.25,1.25)
M(0,.50)L(1.25,.50) M(.50,0)L(.50,1.25)
M(0,.75)L(1.25,.75) M(.75,0)L(.75,1.25)
M(0,1.0)L(1.25,1.0) M(1.0,0)L(1.0,1.25)
fprintf(o,".001 W 0 0 0 RGB S\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
maq(5.);

M(0,0) L(1.25,1.25)fprintf(o,".006 W 1 .3 1 RGB S\n");
M(0,0) DO(m,1521){x=.001*(m+.9); y=LO(x) ; if(y>=-.1 && y<=1.5)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
M(0,0) DO(m,1521){x=.001*(m+.9); y=LO(LO(x)) ; if(y>=-.1 && y<=1.5)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");

M(0,0) L(1.25,1.25) fprintf(o,".001 W 0 0 0 RGB S\n");
for(k=1;k<21;k+=1){ M(0,0) DO(m,1521){x=.001*(m+.9);c=F(.1*k+E(x)); y=Re(c);t=Im(c);if(y>=-.1 && y<1.5 && fabs(t)<1.e-9)L(x,y) else break;} }
fprintf(o,".001 W 0 0 .5 RGB S\n");

M(0,0) DO(m,1521){x=.001*(m+.9);c=F(-1.+E(x)); y=Re(c);t=Im(c);if(y>=-.1 && y<1.5 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".006 W 1 .5 0 RGB S\n");
M(0,0) DO(m,1521){x=.001*(m+.9);c=F(-2.+E(x)); y=Re(c);t=Im(c);if(y>=-.1 && y<1.5 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".006 W 1 .5 0 RGB S\n");

for(k=1;k<21;k+=1){ M(0,0) DO(m,1521){x=.001*(m+.9);c=F(-.1*k+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1.5 && fabs(t)<1.e-9)L(x,y) else break;} }
fprintf(o,".001 W .5 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf logic5.eps");
       system( "open logic5.pdf");
       getchar(); system("killall Preview");
}

Latex generator of curves


 \documentclass[12pt]{article}
 \usepackage{geometry}
 \paperwidth 1308pt
 \paperheight 1314pt
\topmargin -100pt
\oddsidemargin -74pt
\textwidth 1540pt
\textheight 1740pt
\usepackage{graphicx}
%\usepackage{overcite}
%\usepackage{hyperref}
%\usepackage{amssymb}
%\usepackage{wrapfig}
\usepackage{graphics}
\usepackage{rotating}
%\setlength{\parskip}{2mm}
%\setlength{\parindent}{0mm}
\newcommand \ds {\displaystyle}
\newcommand \sx {\scalebox}
\newcommand \rme {\mathrm{e}}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\newcommand \eL[1] {\iL{#1} \end{eqnarray}}
\newcommand \rf[1] {(\ref{#1})}
\parindent 0pt
\pagestyle{empty}
\begin{document}
\sx{10}{\begin{picture}(130,131)
\put(3,4){\ing{logic5}}
\put(0,128){\sx{.7}{$y$}}
\put(0,103){\sx{.7}{$1$}}
\put(0,54){\sx{.7}{$\frac{1}{2}$}}
\put(0,3){\sx{.7}{$0$}}
\put(25.2,1){\sx{.5}{$1/4$}}
\put(50.4,1){\sx{.5}{$1/2$}}
\put(75.6,1){\sx{.5}{$3/4$}}
\put(104,.5){\sx{.6}{$1$}}
\put(127,1){\sx{.6}{$x$}}
\put( 9.9,62){\sx{.7}{\rot{86}$n\!=\!2$\ero}}
\put( 19.5,62){\sx{.7}{\rot{73}$n\!=\!1$\ero}}
\put( 53.3,93.5){\sx{.64}{\rot{50}$n\!=\!0.5$\ero}}
\put( 60.3,92.5){\sx{.64}{\rot{49}$n\!=\!0.4$\ero}}
\put( 67.6,92.5){\sx{.64}{\rot{47}$n\!=\!0.3$\ero}}
\put( 73.9,90,4){\sx{.64}{\rot{46}$n\!=\!0.2$\ero}}
\put( 81,88,4){\sx{.65}{\rot{45}$n\!=\!0.1$\ero}}
\put( 88.6,87){\sx{.7}{\rot{44}$n\!=\!0$\ero}}
\put( 92,81.5){\sx{.6}{\rot{43}$n\!=\!-0.1$\ero}}
\put( 96,76.2){\sx{.6}{\rot{43}$n\!=\!-0.2$\ero}}
\put(101,71.5){\sx{.6}{\rot{43}$n\!=\!-0.3$\ero}}
\put(103.4,58){\sx{.6}{\rot{42}$n\!=\!-0.5$\ero}}
\put(108,32.4){\sx{.7}{\rot{32}$n\!=\!-1$\ero}}
\put(106.4,9.4){\sx{.7}{\rot{8}$n\!=\!-2$\ero}}
\end{picture}}
\end{document}

References

  1. ↑ http://www.springerlink.com/content/u712vtp4122544x4 , DOI 10.3103/S0027134910020049
    http://mizugadro.mydns.jp/PAPERS/2010logistie.pdf , reprint, English version
    http://mizugadro.mydns.jp/PAPERS/2010logistir.pdf , reprint, Russian version
    D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31)

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