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 [1].
[C++]] generator of curves
// Do not forget to load also ado.cin and egjh.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 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");
#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 .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.);
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}
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)
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 06:13, 1 December 2018 | | 2,195 × 2,208 (1.66 MB) | Maintenance script (talk | contribs) | Importing image file |
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