File:Logi1a345T300.png

Explicit plots of various iterations of the Logistic operator with various values of parameter $s$.

$y=\mathrm{LogisticOperator}_s^{\,c}(x)$ for $s\!=\!3$ (left), $s\!=\!4$ (center), $s\!=\!5$ (right) at $c=1$ (black), $c=0.8$ (blue), $c=0.5$ (green), $c=0.2$ (red).

The non-integer iterates of the logisticOperator are calculated through the functions LogisticSequence and ArcLogisticSequence with

$ \mathrm{LogisticPoerator}_s^{\,c}(z)= \mathrm{LogisticSequence}_s\Big( c + \mathrm{ArcLogisticSequence}(z) \big)$

where logisticOperator is quadratic functin of special kind,

$\mathrm{LogisticOperator}_s(z)=sz(1\!-\!z)$.

Interpretation of the iterates of the logistic operator is simple for moderate values of the argument; at larger values, the branch should be specified, as the logistic operator is not a monotonous function.

Properties of the logistic sequence

C++ generator of curves for $s=3$
// FIels efjh.cin and ado.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"

main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("logi1a3.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 .003 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,".003 W 0 0 0 RGB S\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); maq(3.); 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(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.8+E(x))); L(x,y);}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 logi1a3.eps"); system(   "open logi1a3.eps"); getchar; system("killall Preview"); }
 * 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);

C++ generator of curves for $s=4$
// FIels efjh.cin and ado.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) 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.)
 * 4) include "ado.cin"
 * 5) include "efjh.cin"

main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("logi1a4.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 .003 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,".003 W 0 0 0 RGB S\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); 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(1,0) DO(m,101){x=1.-.0000999*m*m;y=Re(F(.8+E(x))); L(x,y);}fprintf(o,".006 W 0 0 .8 RGB S\n"); M(1,0) DO(m,101){x=1.-.0000999*m*m;y=Re(F(.5+E(x))); L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n"); M(1,0) DO(m,101){x=1.-.0000999*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 logi1a4.eps"); system(   "open logi1a4.pdf"); getchar; system("killall Preview"); }
 * 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);

C++ generator of curves for $s=5$
// FIels efjh.cin and ado.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) 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.)
 * 4) include "ado.cin"
 * 5) include "efjh.cin"

main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("logi1a5.eps","w");ado(o,140,140); 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 .003 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,".003 W 0 0 0 RGB S\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); maq(5); DB x0=1.25; DO(m,101){x=1.001-.00010001*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"); DO(m,101){x=1.17-.000116*m*m;y=Re(F(.8+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 .8 RGB S\n"); M(1.25,-.02) DO(m,101){x=x0-.0001249*m*m;y=Re(F(.5+E(x))); L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n"); M(1.25,-.01) DO(m,101){x=x0-.0001249*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 logi1a5.eps"); system(   "open logi1a5.pdf"); getchar; system("killall Preview"); }
 * 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);

Latex generator of labels
% Files logi1a3.pdf,logi1a4.pdf,logi1a5.pdf should be generated with the codes above in order to compile the Latex document below.

% %

\documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphics} % \usepackage{rotating} % \paperwidth 394pt % \paperheight 136pt % \topmargin -100pt % \oddsidemargin -75pt % \newcommand \sx {\scalebox} % \newcommand \ing \includegraphics % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \parindent 0pt % \pagestyle{empty} % \begin{document} % \newcommand \fiax { % \put(3,108){\sx{.9}{1}} % %\put(-9, 84){\sx{.9}{0.25}} % \put(3, 58){\sx{.9}{$\frac{1}{2}$}} % %\put(-9, 34){\sx{.9}{0.25}} % \put(3, 8){\sx{.9}{0}} % \put( 8,  1){\sx{.9}{0}} % \put( 55, 1){\sx{.9}{0.5}} % \put(108, 1){\sx{.9}{1}} % } % \begin{picture}(126,124) % \put(8,8){\ing{logi1a3}} \fiax \put(2,124){\sx{1}{$y\!=\!T^{c}(x)$}} % \put(31,63){\rot{51}\sx{.85}{$c\!=1$}\ero} % \put(33,53){\rot{49}\sx{.85}{$c\!=0.8$}\ero} % \put(39,48){\rot{48}\sx{.85}{$c\!=0.5$}\ero} % \put(44,40){\rot{46}\sx{.85}{$c\!=0.2$}\ero} % \put(118, 1){\sx{.9}{$x$}} % \put(44,20){\sx{1.6}{$s\!=\!3$}} % \end{picture} % \begin{picture}(126,124) \put(8,8){\ing{logi1a4}} \fiax \put(2,124){\sx{1}{$y\!=\!T^{c}(x)$}} % \put(32,84){\rot{57}\sx{.9}{$c\!=1$}\ero} % \put(39,70){\rot{54}\sx{.88}{$c\!=0.8$}\ero} % \put(46,65){\rot{51}\sx{.88}{$c\!=0.5$}\ero} % \put(54,54){\rot{49}\sx{.9}{$c\!=0.2$}\ero} % \put(118, 1){\sx{.9}{$x$}} % \put(44,20){\sx{1.6}{$s\!=\!4$}} % \end{picture} % \begin{picture}(140,127) \put(8,8){\ing{logi1a5}} \fiax %\put(0,124){\sx{1}{$y$}} % \put(34,107){\rot{61}\sx{.93}{$c\!=1$}\ero} % \put(45, 90){\rot{59}\sx{.92}{$c\!=0.8$}\ero} % \put(53, 77){\rot{55}\sx{.92}{$c\!=0.5$}\ero} % \put(58, 60){\rot{50}\sx{.92}{$c\!=0.2$}\ero} % \put(128, 1){\sx{.9}{$x$}} % \put(44,20){\sx{1.6}{$s\!=\!5$}} % \end{picture} % \end{document} %

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