File:Logic3T.jpg
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
- ↑ 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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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 06:13, 1 December 2018 | | 2,195 × 2,208 (1.16 MB) | Maintenance script (talk | contribs) | Importing image file |
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