Difference between revisions of "File:KellerPlotT.png"

Iterates of Keller function

Explicit plot of various iterations \(t\) the Keller function.

\( y=\mathrm{Keller}^t(x)=\mathrm{Shoka}\Big( t + \mathrm{ArcShoka}(x)\Big) \)

See more detailed map at right (for non-integer iterates)
To plot this graphic, the iterates of the Keller function are implemented through the Shoka function and the ArcShoka function.
The detailed map is used as Fig.5.11 at page 56 of book «Superfunctions»^[1][2].

C++ generator of curves]]

// File ado.cin shold be loaded in order to compile the C++ code below.

 #include <math.h> 
 #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"

 z_type Shoka(z_type  z)  { return z + log(exp(-z)+(M_E-1.)); }
 z_type ArcShoka(z_type z){ return z + log((1.-exp(-z))/(M_E-1.)) ;}  

 #define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 #define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
 main(){ int j,k,m,n; DB x,y, a;
 FILE *o;o=fopen("KellerPlot.eps","w");ado(o,408,412);
 fprintf(o,"4 4 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n");
 for(m=0;m<5;m++){ M(m,0)L(m,4)}
 for(n=0;n<5;n++){ M(0,n)L(4,n)}
 M(0,0)L(4,4)
 fprintf(o,".01 W 0 0 0 RGB S\n");
 DO(n,134){x=.005+.01*n;y=Re(Shoka(3.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n");
 DO(n,216){x=.005+.01*n;y=Re(Shoka(2.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n");
 DO(n,154){x=.005+.02*n;y=Re(Shoka(1.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n");
 DO(n,101){x=.005+.04*n;y=Re(Shoka(-1.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n");
 DO(n,101){x=.005+.04*n;y=Re(Shoka(-2.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n");
 DO(n,101){x=.005+.04*n;y=Re(Shoka(-3.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n");
 fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
     system("epstopdf KellerPlot.eps");
     system(    "open KellerPlot.pdf"); //these 2 commands may be specific for macintosh
 getchar(); system("killall Preview");// if run at another operational sysetm, may need to modify
 }

Latex generator of labels

% File KellerPlot.pdf should be generated with the code above in order to compile the Latex document below.

\documentclass[12pt]{article} %<br>
\usepackage{geometry} %<br>
\usepackage{graphicx} %<br>
\usepackage{rotating} %<br>
\paperwidth 419pt %<br>
\paperheight 426pt %<br>
\topmargin -103pt %<br>
\oddsidemargin -83pt %<br>
\textwidth 1200pt %<br>
\textheight 600pt %<br>
\pagestyle {empty} %<br>
\newcommand \sx {\scalebox} %<br>
\newcommand \rot {\begin{rotate}} %<br>
\newcommand \ero {\end{rotate}} %<br>
\newcommand \ing {\includegraphics} %<br>
\begin{document} %<br>
\sx{1}{ \begin{picture}(810,410) %<br>
\put(1,9){\ing{KellerPlot}} % <br>
\put(-12,401){\sx{2.8}{$y$}} % <br>
\put(-12,303){\sx{2.8}{$3$}} % <br>
\put(-12,203){\sx{2.8}{$2$}} % <br>
\put(-12,103){\sx{2.8}{$1$}} % <br>
\put(0,-9){\sx{2.5}{$0$}} % <br>
\put(100,-9){\sx{2.5}{$1$}} % <br>
\put(200,-9){\sx{2.5}{$2$}} % <br>
\put(300,-9){\sx{2.5}{$3$}} % <br>
\put(392,-7){\sx{2.6}{$x$}} % <br>
 %\put(560,214){\rot{37}\sx{4}{$y=\mathrm{Tania}(x)$}\ero} % <br>
\put( 88,354){\rot{53}\sx{2.8}{$t\!=\!3$}\ero} %<br>
\put(160,354){\rot{50}\sx{2.8}{$t\!=\!2$}\ero} %<br>
\put(246,354){\rot{48}\sx{2.8}{$t\!=\!1$}\ero} %<br>
\put(336,350){\rot{45}\sx{2.8}{$t\!=\!0$}\ero} %<br>
\put(340,218){\rot{44}\sx{2.8}{$t\!=\!-1$}\ero} %<br>
\put(344,136){\rot{41}\sx{2.7}{$t\!=\!-2$}\ero} %<br>
\put(338, 68){\rot{34}\sx{2.7}{$t\!=\!-3$}\ero} %<br>
\end{picture} %<br>
}  %<br>
\end{document}
%

Copyleft status

Copyleft 2012 by Dmitrii Kouznetsov. The image and the generators above may be used for free; attribute the source.

References

Keywords

«ArcShoka», «Iterate», «Keller function», «Shoka function», «Superfunctions», «Transfer function», «Transferfunction»,