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 .

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 using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) define DB double
 * 4) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "ado.cin"
 * 5) 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"); 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");
 * 1) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);

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}