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.

#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"
#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");
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
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");
}

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.

#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"
#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");
#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);
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");
}

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.

#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"
#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");
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
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");
}

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.

%

References