Difference between revisions of "File:Logic3T.jpg"

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[[Explicit plot]] of iterates of the [[logistic operator]] with parameter equal to 3;
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Importing image file
  
  +
  +
$T(z)=3\, z\, (1\!-\!z)$
  +
  +
$y=T^n(x)$ is shown versus $x$ for various values of number $n$ of iterate.
  +
  +
The iteration of the logistic operator is described in 2010 at the [[Moscow University Physics Bulletin]]
  +
<ref>
  +
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/>
  +
  +
==[[C++]] generator of curves==
  +
// files [[ado.cin]] and [[efjh.cin]] should be loaded to the working directory in order to compile the [[C++]] code below
  +
<poem><nomathjax><nowiki>
  +
#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 3.*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("logic3.eps","w");ado(o,104,104);
  +
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);
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#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
  +
M(0,0)L(1,0)L(1,1)L(0,1)
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fprintf(o,"C .001 W 0 0 0 RGB S\n");
  +
M(0,.25)L(1,.25) M(.25,0)L(.25,1)
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M(0,.50)L(1,.50) M(.50,0)L(.50,1)
  +
M(0,.75)L(1,.75) M(.75,0)L(.75,1)
  +
fprintf(o,".001 W 0 0 0 RGB S\n");
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
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maq(3.);
  +
//maq(4.);
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// DO(m,101){x=1.-.0000999*m*m;y=Re(F(1.+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 0 RGB S\n");
  +
  +
M(0,0) L(1,1)fprintf(o,".006 W 1 .3 1 RGB S\n");
  +
//M(0,0) DO(m,1021){x=.001*(m+.99);c=F(1.+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
  +
//M(0,0) DO(m,1021){x=.001*(m+.99);c=F(1.+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
  +
  +
//M(0,0) DO(m,1021){x=.001*(m+.99); y=3*x*(1.-x) ;if(y>=0 && y<=1.)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
  +
M(0,0) DO(m,1021){x=.001*(m+.99); y=LO(x) ; if(y>=0 && y<=1.)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
  +
M(0,0) DO(m,1021){x=.001*(m+.99); y=LO(LO(x)) ; if(y>=0 && y<=1.)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
  +
  +
M(0,0) L(1,1) fprintf(o,".001 W 0 0 0 RGB S\n");
  +
for(k=1;k<21;k+=1){ M(0,0) DO(m,1021){x=.001*(m+.99);c=F(.1*k+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && 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,1021){x=.001*(m+.5);c=F(-0.8+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".003 W .8 0 0 RGB S\n");
  +
  +
M(0,0) DO(m,1021){x=.001*(m+.99);c=F(-1.+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && 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,1021){x=.001*(m+.99);c=F(-2.+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && 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,1021){x=.001*(m+.99);c=F(-.1*k+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && fabs(t)<1.e-9)L(x,y) else break;} }
  +
fprintf(o,".001 W .5 0 0 RGB S\n");
  +
  +
/*
  +
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.8+E(x))); if(y>0 && y<1) L(x,y); else break;}fprintf(o,".006 W 0 0 .8 RGB S\n");
  +
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.5+E(x))); L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n");
  +
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.2+E(x))); L(x,y);}fprintf(o,".006 W .8 0 0 RGB S\n");
  +
*/
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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system("epstopdf logic3.eps");
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system( "open logic3.pdf");
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getchar(); system("killall Preview");
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}
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</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of curves]]==
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\paperwidth 1058pt
  +
\paperheight 1064pt
  +
\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}(130,106)
  +
\put(3,4){\ing{logic3}}
  +
\put(0,103){\sx{.7}{$y$}}
  +
%\put(0,103){\sx{.7}{$1$}}
  +
\put(0,79.1){\sx{.7}{$\frac{3}{4}$}}
  +
\put(0,54){\sx{.7}{$\frac{1}{2}$}}
  +
\put(0,28.9){\sx{.7}{$\frac{1}{4}$}}
  +
\put(0.4,3){\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(102,1){\sx{.6}{$x$}}
  +
\put( 11,36){\sx{.7}{\rot{81}$n\!=\!2$\ero}}
  +
\put( 18,36){\sx{.7}{\rot{62}$n\!=\!1$\ero}}
  +
\put( 36.4,44){\sx{.52}{\rot{47}$n\!=\!0.3$\ero}}
  +
\put( 42,43.8){\sx{.52}{\rot{45}$n\!=\!0.1$\ero}}
  +
  +
\put( 85.6,84){\sx{.6}{\rot{45}$n\!=\!0$\ero}}
  +
\put(45,40.4){\sx{.52}{\rot{43}$n\!=\!-0.1$\ero}}
  +
\put(50,36){\sx{.52}{\rot{41}$n\!=\!-0.4$\ero}}
  +
\put(43,18.4){\sx{.66}{\rot{27}$n\!=\!-1$\ero}}
  +
\put(42,8.8){\sx{.66}{\rot{11}$n\!=\!-2$\ero}}
  +
\end{picture}}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Explicit plot]]
  +
[[Category:Logistic sequence]]
  +
[[Category:Logistic operator]]
  +
[[Category:iteration]]
  +
[[Category:C++]]
  +
[[Category:Latex]]
  +
[[Category:Book]]
  +
[[Category:Superfunction]]
  +
[[Category:Abel function]]

Latest revision as of 08:42, 1 December 2018

Explicit plot of iterates of the logistic operator with parameter equal to 3;

$T(z)=3\, z\, (1\!-\!z)$

$y=T^n(x)$ is shown versus $x$ for various values of number $n$ of iterate.

The iteration of the logistic operator is described in 2010 at 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

// files ado.cin and efjh.cin should be loaded to the working directory in order to compile the C++ code below


#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 3.*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("logic3.eps","w");ado(o,104,104);
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(1,0)L(1,1)L(0,1)
fprintf(o,"C .001 W 0 0 0 RGB S\n");
M(0,.25)L(1,.25) M(.25,0)L(.25,1)
M(0,.50)L(1,.50) M(.50,0)L(.50,1)
M(0,.75)L(1,.75) M(.75,0)L(.75,1)
fprintf(o,".001 W 0 0 0 RGB S\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
maq(3.);
//maq(4.);
// DO(m,101){x=1.-.0000999*m*m;y=Re(F(1.+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 0 RGB S\n");

M(0,0) L(1,1)fprintf(o,".006 W 1 .3 1 RGB S\n");
//M(0,0) DO(m,1021){x=.001*(m+.99);c=F(1.+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
//M(0,0) DO(m,1021){x=.001*(m+.99);c=F(1.+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");

//M(0,0) DO(m,1021){x=.001*(m+.99); y=3*x*(1.-x) ;if(y>=0 && y<=1.)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
M(0,0) DO(m,1021){x=.001*(m+.99); y=LO(x) ; if(y>=0 && y<=1.)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");
M(0,0) DO(m,1021){x=.001*(m+.99); y=LO(LO(x)) ; if(y>=0 && y<=1.)L(x,y) else break;}fprintf(o,".006 W 0 1 1 RGB S\n");

M(0,0) L(1,1) fprintf(o,".001 W 0 0 0 RGB S\n");
for(k=1;k<21;k+=1){ M(0,0) DO(m,1021){x=.001*(m+.99);c=F(.1*k+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && 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,1021){x=.001*(m+.5);c=F(-0.8+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && fabs(t)<1.e-9)L(x,y) else break;}fprintf(o,".003 W .8 0 0 RGB S\n");

M(0,0) DO(m,1021){x=.001*(m+.99);c=F(-1.+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && 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,1021){x=.001*(m+.99);c=F(-2.+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && 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,1021){x=.001*(m+.99);c=F(-.1*k+E(x)); y=Re(c);t=Im(c);if(y>0 && y<1 && fabs(t)<1.e-9)L(x,y) else break;} }
fprintf(o,".001 W .5 0 0 RGB S\n");

/*
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.8+E(x))); if(y>0 && y<1) L(x,y); else break;}fprintf(o,".006 W 0 0 .8 RGB S\n");
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.5+E(x))); L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n");
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.2+E(x))); L(x,y);}fprintf(o,".006 W .8 0 0 RGB S\n");
*/
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf logic3.eps");
       system( "open logic3.pdf");
       getchar(); system("killall Preview");
}

Latex generator of curves]]


 \documentclass[12pt]{article}
 \usepackage{geometry}
 \paperwidth 1058pt
 \paperheight 1064pt
\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}(130,106)
\put(3,4){\ing{logic3}}
\put(0,103){\sx{.7}{$y$}}
%\put(0,103){\sx{.7}{$1$}}
\put(0,79.1){\sx{.7}{$\frac{3}{4}$}}
\put(0,54){\sx{.7}{$\frac{1}{2}$}}
\put(0,28.9){\sx{.7}{$\frac{1}{4}$}}
\put(0.4,3){\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(102,1){\sx{.6}{$x$}}
\put( 11,36){\sx{.7}{\rot{81}$n\!=\!2$\ero}}
\put( 18,36){\sx{.7}{\rot{62}$n\!=\!1$\ero}}
\put( 36.4,44){\sx{.52}{\rot{47}$n\!=\!0.3$\ero}}
\put( 42,43.8){\sx{.52}{\rot{45}$n\!=\!0.1$\ero}}

\put( 85.6,84){\sx{.6}{\rot{45}$n\!=\!0$\ero}}
\put(45,40.4){\sx{.52}{\rot{43}$n\!=\!-0.1$\ero}}
\put(50,36){\sx{.52}{\rot{41}$n\!=\!-0.4$\ero}}
\put(43,18.4){\sx{.66}{\rot{27}$n\!=\!-1$\ero}}
\put(42,8.8){\sx{.66}{\rot{11}$n\!=\!-2$\ero}}
\end{picture}}
\end{document}

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