File:LogisticSecK2.jpg

Logistic Sequence of real argument.

$y=F(x)=\mathrm{LogisticSequence}_s(x)$ versus $x$ for various values of parameter $s$.

where $F$ is simplest superfunction of the Logistic operator, id est, solution of the transfer equation

$F(z\!+\!1) = s F(z)(1-F(z))$

for the logistic operator $\mathrm{LogisticOperator}_s(z)=sz(1-z)$ as transfer function.

This solution exponentially approaches zero at minus infinity.

The holomorphic extension of the logistic sequence is described in 2010 in the Moscow University Physics Bulletin .

C++ generator of curves of the First picture
using namespace std; typedef complex z_type; //#include "conto.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("logi6.eps","w");ado(o,164,24); fprintf(o,"62 2 translate\n 20 20 scale\n");
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) 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.)
 * 1) include "ado.cin"
 * 2) include "efjh.cin"
 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

fprintf(o,"1 setlinejoin 2 setlinecap\n"); for(m=-3;m<6;m++){if(m==0){M(m,-.04)L(m,1.06)} else{M(m,0)L(m,1)}} for(n=0;n<2;n++){      M(  -3,n)L(5,n)} fprintf(o,".008 W 0 0 0 RGB S\n");

maq(3.4); DO(m,1004) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,".01 W 0 .7 0 RGB S\n");

maq(3.); DO(m,1004) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,".015 W 1 0 0 RGB [.03 .04] 0 setdash S\n");

fprintf(o,"1 setlinejoin 1 setlinecap\n");

maq(3.8); DO(m,1004) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(y>-2) { if(m==0)M(x,y) else L(x,y);} } fprintf(o,".015 W 0 0 1 RGB [.001 .025] 0 setdash S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logi6.eps"); system(   "open logi6.pdf"); getchar; system("killall Preview"); }

C++ generator of curves of the Second picture
using namespace std; typedef complex z_type; //#include "conto.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("logi5.eps","w");ado(o,164,44); fprintf(o,"62 22 translate\n 20 20 scale\n");
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) 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.)
 * 1) include "ado.cin"
 * 2) include "efjh.cin"
 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

fprintf(o,"1 setlinejoin 2 setlinecap\n");

for(m=-3;m<6;m++){if(m==0){M(m,-1.06)L(m,1.06)} else{M(m,-1)L(m,1)}} for(n=-1;n<2;n++){     M(  -3,n)L(5,n)} fprintf(o,".008 W 0 0 0 RGB S\n");

maq(4.); DO(m,1001) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,".01 W 0 .7 0 RGB S\n");

maq(3.9); DO(m,1001) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,".015 W 1 0 0 RGB [.03 .04] 0 setdash S\n");

fprintf(o,"1 setlinejoin 1 setlinecap\n");

maq(4.1); DO(m,1001) { x=-3.+8.*sqrt(.001*m); y=Re(F(x)); if(y>-2) { if(m==0)M(x,y) else L(x,y);} } fprintf(o,".015 W 0 0 1 RGB [.001 .02] 0 setdash S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logi5.eps"); system(   "open logi5.pdf"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphics} \usepackage{rotating} \paperwidth 492pt \paperheight 220pt \topmargin -100pt \oddsidemargin -72pt \newcommand \sx {\scalebox} \newcommand \ing \includegraphics \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \parindent 0pt \pagestyle{empty} \begin{document} \sx{3}{\begin{picture}(166,22) \put( 1,1){\includegraphics{logi6}} %\put( 64,48){\sx{.6}{$F(x)$}} \put( 50,16){\sx{.4}{$F(x)$}} \put( 0,22){\sx{.3}{$1$}} \put( 0, 3){\sx{.3}{$0$}} %\put( -1, 3){\sx{.4}{$-1$}} \put( 20, 0){\sx{.3}{$-2$}} \put( 40, 0){\sx{.3}{$-1$}} \put( 62.5, 0){\sx{.3}{$0$}} \put( 82.5, 0){\sx{.3}{$1$}} \put(102.5, 0){\sx{.3}{$2$}} \put(122.5, 0){\sx{.3}{$3$}} \put(142.5, 0){\sx{.3}{$4$}} \put(162, 0){\sx{.3}{$x$}} \put(91,15.5){\sx{.3}{$s\!=\!3$}} \put(91,11){\sx{.3}{$s\!=\!3.4$}} \put(93, 5){\sx{.3}{$s\!=\!3.8$}} \end{picture}} \vskip 9pt

\sx{3}{\begin{picture}(166,44) \put( 1,1){\includegraphics{logi5}} %\put( 64,48){\sx{.6}{$F(x)$}} \put( 50,36){\sx{.4}{$F(x)$}} \put( 0,41.5){\sx{.3}{$1$}} \put( 0,22){\sx{.3}{$0$}} \put( -1, 2){\sx{.3}{$-\!1$}} \put( 20, 0){\sx{.3}{$-2$}} \put( 40, 0){\sx{.3}{$-1$}} \put( 62.5, 0){\sx{.3}{$0$}} \put( 82.5, 0){\sx{.3}{$1$}} \put(102.5, 0){\sx{.3}{$2$}} \put(122.5, 0){\sx{.3}{$3$}} \put(142.5, 0){\sx{.3}{$4$}} \put(162, 0){\sx{.3}{$x$}} \put(91.3,29){\sx{.3}{$s\!=\!3.9$}} %\put(91,21){\sx{.3}{$s\!=\!4$}} \put(92.5, 19.5){\sx{.3}{$s\!=\!4.1$}} \end{picture}}

\end{document}

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