File:Logi2s5t.jpg

Complex map of the half iterate of the logistic operator

$T(z)=s\, z\,(1\!-\!z)$ for $s\!=\!5$

$u\!+\!\mathrm i v=T^{0.5}(x\!+\!\mathrm i y)$

$T^{0.5}(z)=F(0.5+G(z))$

C++ generator of map
//using namespace std; typedef std:: complex z_type; /* z_type arccos(z_type z){ return -I*log(z+I*sqrt(1.-z*z)); } z_type coe(z_type z){ return .5*(1.-cos(exp((z+1.)/LQ))); } z_type boe(z_type z){ return LQ*log(arccos(1.-2.*z))-1.; } z_type doe(z_type z){ return coe(1.+boe(z));; } int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=201,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("logi2s5.eps","w");ado(o,604,604); fprintf(o,"302 302 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-3.+.03*(m-.5); DO(n,N1) Y[n]=-3.+.015*(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"
 * 5) include "efjh.cin"

for(m=-3;m<4;m++){if(m==0){M(m,-3.04)L(m,3.04)} else{M(m,-3)L(m,3)}} for(n=-3;n<4;n++){        M(  -3  ,n)L(3,n)} fprintf(o,"2 setlinecap .008 W 0 0 0 RGB S\n");

maq(5.); 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=F(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=2.;q=.5; //#include"plof.cin" for(m=-2;m<2;m++) for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".005 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,".005 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,".005 W 0 0 .9 RGB S\n");

for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);  fprintf(o,".02 W .9 0 0 RGB S\n"); for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".02 W 0 0 .9 RGB S\n"); for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".02 W 0 0 0 RGB S\n");

conto(o,f,w,v,X,Y,M,N, (0. ),-p,p);  fprintf(o,".02 W .6 0 .6 RGB S\n");

fprintf(o,"0 setlinecap 0 setlinejoin 0 setlinecap\n");

//M(-3.02,0)L(0,0) //M(1.-1./Q,0)L(3,0) fprintf(o,"0.03 W 1 1 1 RGB S\n"); M(1.26,0)L(3,0) fprintf(o,"0.03 W 1 1 1 RGB S\n"); //for(n=0;n<16;n++) {M(-.2*n,0)L(-.2*(n+.4),0)} for(n=0;n<12;n++) {    M(1.26+.2*(n+.4),0) L(1.26+.2*(n+.8),0)} fprintf(o,"0.04 W 1 0 1 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logi2s5.eps"); // for linux system(   "open logi2s5.pdf"); // for mac getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphics} \usepackage{rotating} \paperwidth 640pt \paperheight 640pt \textwidth 700pt \textheight 700pt \topmargin -92pt \oddsidemargin -91pt \newcommand \sx {\scalebox} \newcommand \ing \includegraphics \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document}

\newcommand \axesS { \normalsize \put( 18,624){\sx{2.6}{$y$}} \put( 18,530){\sx{2.6}{$2$}} \put( 18,430){\sx{2.6}{$1$}} \put( 18,330){\sx{2.6}{$0$}} \put( -2,230){\sx{2.6}{$-1$}} \put( -2,130){\sx{2.6}{$-2$}} \put( -2, 30){\sx{2.6}{$-3$}} \put( 16, 10){\sx{2.6}{$-3$}} \put(116, 10){\sx{2.6}{$-2$}} \put(216, 10){\sx{2.6}{$-1$}} \put(332, 10){\sx{2.6}{$0$}} \put(432, 10){\sx{2.6}{$1$}} \put(532, 10){\sx{2.6}{$2$}} \put(622, 10){\sx{2.6}{$x$}} } \begin{picture}(640,628) %\put( 4, 4){\ing{logi2b3}} \put(36, 36){\ing{logi2s5}} \normalsize \put( 62, 424){\rot{ 7}\sx{4}{$v\!=\!4$}\ero} \put( 62, 326){\rot{ 0}\sx{4}{$v\!=\!0$}\ero} \put( 566, 328){\rot{ 0}\sx{4}{\bf cut}\ero} \put( 42, 232){\rot{-9}\sx{4}{$v\!=\!-4$}\ero} % \put( 140, 42){\rot{71}\sx{4}{$u\!=\!-4$}\ero} \put( 242, 44){\rot{66}\sx{4}{$u\!=\!0$}\ero} \put( 290, 40){\rot{59}\sx{4}{$u\!=\!2$}\ero} % \put( 464, 506){\rot{ 48}\sx{4}{$v\!=\!2$}\ero} \put( 504, 464){\rot{ 52}\sx{4}{$v\!=\!0$}\ero} \put( 556, 440){\rot{ 57}\sx{4}{$v\!=\!-2$}\ero} \axesS \end{picture} \end{document}