File:Logic4T.jpg

Explicit plot of iteration of the logistic operator

$T(z)=4 \,z \, (1\!-\!z)$

Lines $y=T^n(x)$ are drawn in the $x$,$y$ plane for various values of number $n$ of iteration.

With this value parameter $c\!=\!4$, the superfunction $F$ and the Abel function $G$ can be expressed in terms of elementary functions; this case can be used for the testing of the numerical implementation of the holomorphic extension of the logistic sequence, id est, superfunction $F.

Generators of the image are copipasted below. Construction of the iterates is described in 2010 in the Moscow University Physics Bulletin .

[C++]] generator of curves
// Do not forget to load also ado.cin and egjh.cin

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 4.*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("logic4.eps","w");ado(o,130,130); fprintf(o,"2 2 translate\n 100 100 scale\n"); M(0,0)L(1.25,0)L(1.25,1.25)L(0,1.25) fprintf(o,"C .001 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(4.);
 * 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(1.25,1.25)fprintf(o,".006 W 1 .3 1 RGB S\n"); M(0,0) DO(m,1521){x=.001*(m+.99); y=LO(x)    ; if(y>-.01 && 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+.99); y=LO(LO(x)) ; if(y>-.01 && 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+.99);c=F(.1*k+E(x)); y=Re(c);t=Im(c);if(y>=-.01 && 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+.99);c=F(-1.+E(x)); y=Re(c);t=Im(c);if(y>-.01 && 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+.99);c=F(-2.+E(x)); y=Re(c);t=Im(c);if(y>-.01 && 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+.99);c=F(-.1*k+E(x)); y=Re(c);t=Im(c);if(y>-.01 && 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 logic4.eps"); system(   "open logic4.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{logic4}} \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( 11,60){\sx{.7}{\rot{83}$n\!=\!2$\ero}} \put( 24,61){\sx{.7}{\rot{69}$n\!=\!1$\ero}} \put( 42,80){\sx{.64}{\rot{49}$n\!=\!0.7$\ero}} \put( 52,80){\sx{.64}{\rot{47}$n\!=\!0.5$\ero}} \put( 68,78,4){\sx{.64}{\rot{46}$n\!=\!0.2$\ero}}

\put( 76,74){\sx{.7}{\rot{45}$n\!=\!0$\ero}} \put(80,65.4){\sx{.6}{\rot{45}$n\!=\!-0.2$\ero}} \put(83.4,52){\sx{.6}{\rot{44}$n\!=\!-0.5$\ero}} \put(83,29.4){\sx{.7}{\rot{36}$n\!=\!-1$\ero}} \put(82,10.8){\sx{.7}{\rot{13}$n\!=\!-2$\ero}} \end{picture}} \end{document}