File:Logi5ab400.jpg

Superfunction $F$ of the logistic operator

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

constructed with the regular iteration at the upoer fixed point $L\!=\!1-1/s$ for $s\!=\!4$.

Graphic at the top: explicit plot $y\!=\!f(x)$

Complex map at the bottom: $u\!+\!\mathrm i v=F(x\!+\!\mathrm i y)$

C++ generator of the graphic at the top
Files ado.cin, logiu.cin should be loaded to the working directory in order to compile the code below

//using namespace std; typedef std::complex z_type; //#include "efjh.cin" //#include "u.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=801,M1=M+1; int N=401,N1=N+1; DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. char v[M1*N1]; // v is working array FILE *o;o=fopen("logi5a4x.eps","w");ado(o,164,84); fprintf(o,"102 42 translate\n 20 20 scale\n"); DO(m,M1) X[m]=-5.+.01*(m-.5); DO(n,N1) Y[n]=-2.+.01*(n-.5);
 * 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 "conto.cin"
 * 1) include "logiu.cin"

for(m=-5;m<4;m++){if(m==0){M(m,-2.04)L(m,2.04)} else{M(m,-2)L(m,2)}} for(n=-2;n<3;n++){        M(  -5  ,n)L(3,n)} fprintf(o,".008 W 0 0 0 RGB S\n");

maq(4.); DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} DO(m,M1){x=X[m]; //printf("%5.2f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); //     c=E(H(z))-1.; //     c=F(1.+E(0.1*z)); c=U(z); //     c=F(.5+E(z)); //     c=boe(z); //     c=.5*(1.-cos(exp((z+1.)/LQ))); //     d=H(F(z-1.)); //     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.; //     if(p>-4.9 && p<20) g[m*N1+n]=p; p=Re(c);q=Im(c); if(p>-4.9 && p<4.9)    {g[m*N1+n]=p;} //     if(q>-4.9 && q<4.9)     {f[m*N1+n]=q;} if(q>-4.9 && q<4.9 && fabs(q)>1.e-11 ) {f[m*N1+n]=q;} }}

fprintf(o,"1 setlinejoin 2 setlinecap\n"); //p=.8;q=.4; p=.8;q=.3; //#include"plof.cin" for(m=-3;m<3;m++) for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".004 W 0 .6 0 RGB S\n"); for(m=0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".004 W .9 0 0 RGB S\n"); for(m=0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".004 W 0 0 .9 RGB S\n");

for(m=1;m<6;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);   fprintf(o,".016 W .9 0 0 RGB S\n"); for(m=1;m<6;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);   fprintf(o,".016 W 0 0 .9 RGB S\n"); for(m=-4;m<6;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".016 W 0 0 0 RGB S\n");

conto(o,f,w,v,X,Y,M,N, (0. ),-p,p);  fprintf(o,".016 W .6 0 .6 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logi5a4x.eps"); // for linux system(   "open logi5a4x.pdf"); // for mac getchar; system("killall Preview"); }

C++ generator of the map at the bottom
// Files ado.cin, conto.cin, logiu.cin should be loaded in order to compile the C++ code below.

//using namespace std; typedef std::complex z_type; //#include "conto.cin" //#include "efjh.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("logi5b4x.eps","w");ado(o,164,24); fprintf(o,"102 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"
 * 1) include "logiu.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);

for(m=-5;m<4;m++){if(m==0){M(m,-.04)L(m,1.04)} else{M(m,0)L(m,1)}} for(n=0;n<2;n++){      M(  -5,n)L(3,n)} fprintf(o,".006 W 0 0 0 RGB S\n");

M(-5,.75)L(3,.75) fprintf(o,".004 W 0 0 0 RGB S\n");

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

maq(4.); DO(m,1034) { x=-5.08+8.*sqrt(.001*m); y=Re(U(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,".01 W 0 .5 0 RGB S\n");

/* maq(3.99); DO(m,1001) { x=-5.+10.*sqrt(.001*m); y=Re(U(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");

maq(4.01); DO(m,1001) { x=-5.+10.*sqrt(.001*m); y=Re(U(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,".015 W 0 0 1 RGB [.01 .03] 0 setdash S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logi5b4x.eps"); system(   "open logi5b4x.pdf"); getchar; system("killall Preview"); }

Latex generator of the labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphics} \usepackage{rotating} \paperwidth 426pt \paperheight 284pt \topmargin -108pt \oddsidemargin -90pt \newcommand \sx {\scalebox} \newcommand \ing \includegraphics \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document}

\sx{2.6}{\begin{picture}(140,23) \put( 1, 1){\ing{logi5b4x}} \put( 1, 21){\sx{.3}{$1$}} \put( 1, 2){\sx{.3}{$0$}}

\put( 20.2, 0){\sx{.3}{$-4$}} \put( 40.2, 0){\sx{.3}{$-3$}} \put( 60.2, 0){\sx{.3}{$-2$}} \put( 80.2, 0){\sx{.3}{$-1$}} \put(102.6, 0){\sx{.3}{$0$}} \put( 122.7, 0){\sx{.3}{$1$}} \put(142.7, 0){\sx{.3}{$2$}} \put(162, 0.4){\sx{.3}{$x$}} \end{picture}}

%\sx{3.04}{\begin{picture}(140,85) \sx{2.6}{\begin{picture}(140,85) \put( 1, 1){\ing{logi5a4x}} \put( 1, 81){\sx{.3}{$y$}} \put( 1, 62){\sx{.3}{$1$}} \put( 1, 42){\sx{.3}{$0$}} \put( -1, 22){\sx{.3}{$-\!1$}} %\put( -1, 2){\sx{.3}{$-\!2$}} \put( 20.2, 0){\sx{.3}{$-4$}} \put( 40.2, 0){\sx{.3}{$-3$}} \put( 60.2, 0){\sx{.3}{$-2$}} \put( 80.2, 0){\sx{.3}{$-1$}} \put(102.6, 0){\sx{.3}{$0$}} \put( 122.7, 0){\sx{.3}{$1$}} \put(142.7, 0){\sx{.3}{$2$}} \put(162, 0.4){\sx{.3}{$x$}} \put(5.5,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}} \put(25.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}} \put(45.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}} \put(65.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}} \put(85.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}} \put(105.6,39.6){\sx{.3}{\rot{90}$v\!=\!0$\ero}} \put(8,42.3){\sx{.3}{$v\!=\!0$}} \put(20,39){\sx{.3}{\rot{90}$u\!=\!0.7$\ero}} \put(30.6,38.6){\sx{.3}{\rot{90}$u\!=\!0.7$\ero}} \put(37.4,38.4){\sx{.3}{\rot{90}$u\!=\!0.8$\ero}}

\put(50,45.3){\sx{.3}{$v\!=\!-0.1$}} \put(52,42.3){\sx{.3}{$v\!=\!0$}} \put(52,39.3){\sx{.3}{$v\!=\!0.1$}}

%\put(68,44.3){\sx{.25}{$v\!=\!0.1$}} \put(69,42.3){\sx{.3}{$v\!=\!0$}} %\put(67,40.3){\sx{.25}{$v\!=\!-0.1$}} \end{picture}} \end{document}