Difference between revisions of "File:Logi2d4t1500.jpg"

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{{oq|Logi2d4t1500.jpg|Original file ‎(2,698 × 2,656 pixels, file size: 801 KB, MIME type: image/jpeg) }}
−
Importing image file
  
  +
[[Complex map]] of the inverse of [[logistic sequence]] with parameter \(s\!=\!4\) :
  +
  +
\[
  +
u+\mathrm i v= \mathrm{ArcLogisticSequence}_4(x\!+\!\mathrm i y)
  +
\]
  +
  +
  +
This map appears as Fig.7.8 at page 79 of book
  +
«[[Superfunctions]]»<ref>
  +
https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas,algorithms,tables,graphics - 2020/7/28
  +
</ref><ref>https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas, algorithms, tables, graphics. Publisher: [[Lambert Academic Publishing]].
  +
</ref>
  +
<br>
  +
in order to show that the [[Holomorphic extension of the logistic sequence]]
  +
<ref>
  +
https://link.springer.com/article/10.3103/S0027134910020049 <br>
  +
http://mizugadro.mydns.jp/PAPERS/2010logistie.pdf
  +
D.Kouznetsov. [[Holomorphic extension of the logistic sequence]]. [[Moscow University Physics Bulletin]], 2010, No.2, p.91-98. (Russian version: p.24-31)
  +
</ref>
  +
is not so chaotic.
  +
==[[C++]] generator of map==
  +
<pre>
  +
#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 std::complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
#include "efjh.cin"
  +
/*
  +
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=201,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("logi2d4.eps","w");ado(o,124,124);
  +
fprintf(o,"62 62 translate\n 20 20 scale\n");
  +
DO(m,M1) X[m]=-3.+.03*(m-.5);
  +
DO(n,N1) Y[n]=-3.+.03*(n-.5);
  +
  +
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,".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=E(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 1 setlinecap\n");
  +
//p=.8;q=.4;
  +
p=2.;q=.5;
  +
//#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,".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 setlinejoin 0 setlinecap\n");
  +
  +
M(-3.02,0)L(0,0)
  +
//M(1.-1./Q,0)L(3,0)
  +
M(1.,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+.2* n,0)
  +
L(1+.2*(n+.4),0)}
  +
fprintf(o,"0.04 W 0 0 0 RGB S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf logi2d4.eps"); // for linux
  +
system( "open logi2d4.pdf"); // for mac
  +
getchar(); system("killall Preview");
  +
}
  +
</pre>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<pre>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\paperwidth 130pt
  +
\paperheight 128pt
  +
\topmargin -102pt
  +
\oddsidemargin -91pt
  +
\newcommand \sx {\scalebox}
  +
\newcommand \ing \includegraphics
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\begin{document}
  +
  +
\newcommand \axes {
  +
\normalsize
  +
\put( 5,126){\sx{.6}{$y$}}
  +
\put( 5,106.4){\sx{.6}{$2$}}
  +
\put( 5, 86.4){\sx{.6}{$1$}}
  +
\put( 5, 66.4){\sx{.6}{$0$}}
  +
\put( 0, 46.4){\sx{.6}{$-1$}}
  +
\put( 0, 26.4){\sx{.6}{$-2$}}
  +
\put( 24, 2){\sx{.6}{$-2$}}
  +
\put( 44, 2){\sx{.6}{$-1$}}
  +
\put( 68.4, 2){\sx{.6}{$0$}}
  +
\put( 89.2, 2){\sx{.6}{$1$}}
  +
\put(109.2, 2){\sx{.6}{$2$}}
  +
\put(126, 2){\sx{.6}{$x$}}
  +
}
  +
%\begin{picture}(122,122) \put( 4, 4){\ing{logi2c4a}}
  +
\begin{picture}(122,122) \put( 8, 6){\ing{logi2d4}}
  +
  +
\put( 52, 88){\rot{0}\sx{.99}{$u\!=\!0$}\ero}
  +
\put( 17,98){\rot{-36}\sx{.99}{$v\!=\!2$}\ero}
  +
\put( 92, 90.2){\rot{-2}\sx{.99}{$v\!=\!1$}\ero}
  +
\put( 91, 41){\rot{0}\sx{.99}{$v\!=\!-1$}\ero}
  +
\put( 16, 30){\rot{36}\sx{.99}{$v\!=\!-2$}\ero}
  +
{\put(12,66){\bf cut} }
  +
{\put(109,66){\bf cut} }
  +
\axes
  +
\end{picture}
  +
\end{document}
  +
</pre>
  +
  +
==References==
  +
  +
{{ref}}
  +
  +
{{fer}}
  +
==Keywords==
  +
  +
«[[Holomorphic extension of the Logistic sequence]]»,
  +
«[[LogisitcOperator]]»,
  +
<b>«[[LogisticSequence]]»</b>,
  +
«[[Table of superfunctions]]»,
  +
«[[Transfer equation]]»,
  +
«[[Superfunction]]»,
  +
«[[Superfunctions]]»,
  +
  +
[[Category:ArcLogisticSequence]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
  +
[[Category:Complex map]]
  +
[[Category:Elutin Pavel Vyacheslavovich]]
  +
[[Category:Latex]]
  +
[[Category:Logistic operator]]
  +
[[Category:Logistic sequence]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]

Latest revision as of 16:28, 21 August 2025


Complex map of the inverse of logistic sequence with parameter \(s\!=\!4\) :

\[ u+\mathrm i v= \mathrm{ArcLogisticSequence}_4(x\!+\!\mathrm i y) \]


This map appears as Fig.7.8 at page 79 of book «Superfunctions»[1][2]
in order to show that the Holomorphic extension of the logistic sequence [3] is not so chaotic.

C++ generator of map

#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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
#include "efjh.cin"
/*
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=201,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("logi2d4.eps","w");ado(o,124,124);
fprintf(o,"62 62 translate\n 20 20 scale\n");
DO(m,M1) X[m]=-3.+.03*(m-.5);
DO(n,N1) Y[n]=-3.+.03*(n-.5);

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,".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=E(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 1 setlinecap\n"); 
//p=.8;q=.4;
p=2.;q=.5;
//#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,".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 setlinejoin 0 setlinecap\n"); 

M(-3.02,0)L(0,0)  
//M(1.-1./Q,0)L(3,0)  
M(1.,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+.2* n,0)
                        L(1+.2*(n+.4),0)}
fprintf(o,"0.04 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi2d4.eps"); // for linux
        system(    "open logi2d4.pdf"); // for mac
        getchar(); system("killall Preview");
}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 130pt
\paperheight 128pt
\topmargin -102pt
\oddsidemargin -91pt
\newcommand \sx {\scalebox}
\newcommand \ing \includegraphics
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}

\newcommand \axes {
\normalsize
\put(  5,126){\sx{.6}{$y$}}
\put(  5,106.4){\sx{.6}{$2$}}
\put(  5, 86.4){\sx{.6}{$1$}}
\put(  5, 66.4){\sx{.6}{$0$}}
\put( 0, 46.4){\sx{.6}{$-1$}}
\put( 0, 26.4){\sx{.6}{$-2$}}
\put( 24,  2){\sx{.6}{$-2$}}
\put( 44,  2){\sx{.6}{$-1$}}
\put( 68.4,  2){\sx{.6}{$0$}}
\put( 89.2,  2){\sx{.6}{$1$}}
\put(109.2,  2){\sx{.6}{$2$}}
\put(126,  2){\sx{.6}{$x$}}
}
%\begin{picture}(122,122) \put( 4, 4){\ing{logi2c4a}}
\begin{picture}(122,122) \put( 8, 6){\ing{logi2d4}}

\put( 52, 88){\rot{0}\sx{.99}{$u\!=\!0$}\ero}
\put( 17,98){\rot{-36}\sx{.99}{$v\!=\!2$}\ero}
\put( 92, 90.2){\rot{-2}\sx{.99}{$v\!=\!1$}\ero}
\put( 91, 41){\rot{0}\sx{.99}{$v\!=\!-1$}\ero}
\put( 16, 30){\rot{36}\sx{.99}{$v\!=\!-2$}\ero}
{\put(12,66){\bf cut} }
{\put(109,66){\bf cut} }
\axes
\end{picture}
\end{document}

References

  1. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.
  3. ↑ https://link.springer.com/article/10.3103/S0027134910020049
    http://mizugadro.mydns.jp/PAPERS/2010logistie.pdf D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31)

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