Difference between revisions of "File:Logiha300.jpg"
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+ | [[Complex map]] of the halfiterate or the [[Logistic operator]] $T(z)\!=\!s \,z\,(z\!-\!1)$ for |
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− | Importing image file |
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+ | $s\!=\!3$, upper map; |
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+ | $s\!=\!4$, intermadiate; |
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+ | $s\!=\!5$, at bottom. |
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+ | |||
+ | $u\!+\!\mathrm i v = \mathrm{LogisticOperator}_s^{1/2}(x\!+\!\mathrm i y)$ |
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+ | |||
+ | The function is real-holomorphic; the map are symmetric; so, for the two first cases, the only half of the map is shown. |
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+ | |||
+ | ==[[C++]] generator of the upper map== |
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+ | |||
+ | <poem><nomathjax><nowiki> |
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+ | #include <math.h> |
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+ | #include <stdio.h> |
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+ | #include <stdlib.h> |
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+ | #define DB double |
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+ | #define DO(x,y) for(x=0;x<y;x++) |
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+ | // using namespace std; |
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+ | #include <complex> |
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+ | typedef std::complex<double> z_type; |
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+ | #define Re(x) x.real() |
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+ | #define Im(x) x.imag() |
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+ | #define I z_type(0.,1.) |
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+ | #include "conto.cin" |
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+ | #include "efjh.cin" |
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+ | int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; |
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+ | int M=302,M1=M+1; |
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+ | int N=51,N1=N+1; |
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+ | DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. |
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+ | char v[M1*N1]; // v is working array |
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+ | //FILE *o;o=fopen("logi3e4a.eps","w");ado(o,304,204); |
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+ | FILE *o;o=fopen("logihalf3.eps","w");ado(o,304,104); |
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+ | fprintf(o,"152 2 translate\n 100 100 scale\n"); |
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+ | DO(m,M1) X[m]= -1.5+.02*(m-.5); |
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+ | DO(n,N1) Y[n]=0.+.02*(n-.5); |
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+ | |||
+ | for(m=-3;m<4;m++){if(m==0){M(.5*m,0)L(.5*m,1.04)} else{M(.5*m,0)L(.5*m,1)}} |
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+ | for(n=0;n<3;n++){ M( -1.5 ,.5*n)L(1.5,.5*n)} |
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+ | fprintf(o,".004 W 0 0 0 RGB S\n"); |
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+ | |||
+ | //maq(3.); |
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+ | maq(3.); |
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+ | DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} |
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+ | DO(m,M1){x=X[m]; //printf("%5.2f\n",x); |
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+ | DO(n,N1){y=Y[n]; z=z_type(x,y); |
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+ | // c=E(H(z))-1.; |
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+ | // c=F(1.+E(0.1*z)); |
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+ | // c=F(z); |
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+ | c=F(.5+E(z)); |
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+ | c=F(.5+E(c)); |
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+ | // c=boe(z); |
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+ | // c=.5*(1.-cos(exp((z+1.)/LQ))); |
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+ | // d=H(F(z-1.)); |
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+ | // p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.; |
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+ | // if(p>-99 && p<20) g[m*N1+n]=p; |
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+ | p=Re(c);q=Im(c); |
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+ | if(p>-99 && p<99) {g[m*N1+n]=p;} |
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+ | if(q>-99 && q<99) {f[m*N1+n]=q;} |
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+ | // if(q>-99 && q<99 && fabs(q)>1.e-11 ) {f[m*N1+n]=q;} |
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+ | }} |
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+ | |||
+ | fprintf(o,"1 setlinejoin 2 setlinecap\n"); |
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+ | p=2;q=.4; |
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+ | //#include"plof.cin" |
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+ | for(m=-2;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 .6 0 RGB S\n"); |
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+ | for(m=0;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".002 W .9 0 0 RGB S\n"); |
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+ | for(m=0;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 0 .9 RGB S\n"); |
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+ | |||
+ | for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".006 W .9 0 0 RGB S\n"); |
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+ | for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".006 W 0 0 .9 RGB S\n"); |
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+ | for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".006 W 0 0 0 RGB S\n"); |
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+ | |||
+ | conto(o,f,w,v,X,Y,M,N, (0. ),-99,99); fprintf(o,".008 W .5 0 .5 RGB S\n"); |
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+ | |||
+ | fprintf(o,"0 setlinejoin 0 setlinecap\n"); |
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+ | |||
+ | //M(-3.02,0)L(0,0) |
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+ | //M(1.-1./Q,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); |
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+ | M(0.7,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); |
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+ | //for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)} |
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+ | for(n=0;n<12;n++) { M(0.7+.07*(n+.3),0) |
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+ | L(0.7+.07*(n+.7),0)} |
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+ | fprintf(o,"0.014 W 0 0 0 RGB S\n"); |
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+ | |||
+ | fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); |
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+ | system("epstopdf logihalf3.eps"); // for linux |
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+ | system( "open logihalf3.pdf"); // for mac |
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+ | getchar(); system("killall Preview"); |
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+ | } |
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+ | </nowiki></nomathjax></poem> |
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+ | ==[[C++]] generator of the second map== |
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+ | |||
+ | <poem><nomathjax><nowiki> |
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+ | #include <math.h> |
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+ | #include <stdio.h> |
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+ | #include <stdlib.h> |
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+ | #define DB double |
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+ | #define DO(x,y) for(x=0;x<y;x++) |
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+ | // using namespace std; |
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+ | #include <complex> |
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+ | typedef std::complex<double> z_type; |
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+ | #define Re(x) x.real() |
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+ | #define Im(x) x.imag() |
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+ | #define I z_type(0.,1.) |
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+ | #include "conto.cin" |
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+ | #include "efjh.cin" |
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+ | int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; |
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+ | int M=302,M1=M+1; |
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+ | int N=51,N1=N+1; |
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+ | DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. |
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+ | char v[M1*N1]; // v is working array |
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+ | //FILE *o;o=fopen("logi3e4a.eps","w");ado(o,304,204); |
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+ | FILE *o;o=fopen("logihalf4.eps","w");ado(o,304,104); |
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+ | fprintf(o,"152 2 translate\n 100 100 scale\n"); |
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+ | DO(m,M1) X[m]= -1.5+.02*(m-.5); |
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+ | DO(n,N1) Y[n]=0.+.02*(n-.5); |
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+ | |||
+ | for(m=-3;m<4;m++){if(m==0){M(.5*m,0)L(.5*m,1.04)} else{M(.5*m,0)L(.5*m,1)}} |
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+ | for(n=0;n<3;n++){ M( -1.5 ,.5*n)L(1.5,.5*n)} |
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+ | fprintf(o,".004 W 0 0 0 RGB S\n"); |
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+ | |||
+ | //maq(3.); |
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+ | maq(4.); |
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+ | DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} |
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+ | DO(m,M1){x=X[m]; //printf("%5.2f\n",x); |
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+ | DO(n,N1){y=Y[n]; z=z_type(x,y); |
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+ | // c=E(H(z))-1.; |
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+ | // c=F(1.+E(0.1*z)); |
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+ | // c=F(z); |
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+ | c=F(.5+E(z)); |
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+ | c=F(.5+E(c)); |
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+ | // c=boe(z); |
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+ | // c=.5*(1.-cos(exp((z+1.)/LQ))); |
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+ | // d=H(F(z-1.)); |
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+ | // p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.; |
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+ | // if(p>-99 && p<20) g[m*N1+n]=p; |
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+ | p=Re(c);q=Im(c); |
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+ | if(p>-99 && p<99) {g[m*N1+n]=p;} |
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+ | if(q>-99 && q<99) {f[m*N1+n]=q;} |
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+ | // if(q>-99 && q<99 && fabs(q)>1.e-11 ) {f[m*N1+n]=q;} |
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+ | }} |
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+ | |||
+ | fprintf(o,"1 setlinejoin 2 setlinecap\n"); |
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+ | p=2;q=.4; |
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+ | //#include"plof.cin" |
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+ | for(m=-2;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 .6 0 RGB S\n"); |
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+ | for(m=0;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".002 W .9 0 0 RGB S\n"); |
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+ | for(m=0;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 0 .9 RGB S\n"); |
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+ | |||
+ | for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".006 W .9 0 0 RGB S\n"); |
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+ | for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".006 W 0 0 .9 RGB S\n"); |
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+ | for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".006 W 0 0 0 RGB S\n"); |
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+ | |||
+ | conto(o,f,w,v,X,Y,M,N, (0. ),-99,99); fprintf(o,".008 W .5 0 .5 RGB S\n"); |
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+ | |||
+ | fprintf(o,"0 setlinejoin 0 setlinecap\n"); |
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+ | |||
+ | //M(-3.02,0)L(0,0) |
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+ | //M(1.-1./Q,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); |
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+ | M(.96,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); |
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+ | //for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)} |
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+ | for(n=0;n<10;n++) { M(.96+.07*(n+.3),0) |
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+ | L(.96+.07*(n+.7),0)} |
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+ | fprintf(o,"0.014 W 0 0 0 RGB S\n"); |
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+ | |||
+ | fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); |
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+ | system("epstopdf logihalf4.eps"); // for linux |
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+ | system( "open logihalf4.pdf"); // for mac |
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+ | getchar(); system("killall Preview"); |
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+ | } |
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+ | </nowiki></nomathjax></poem> |
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+ | |||
+ | ==[[C++]] generator of the third map== |
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+ | |||
+ | <poem><nomathjax><nowiki> |
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+ | #include <math.h> |
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+ | #include <stdio.h> |
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+ | #include <stdlib.h> |
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+ | #define DB double |
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+ | #define DO(x,y) for(x=0;x<y;x++) |
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+ | // using namespace std; |
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+ | #include <complex> |
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+ | typedef std::complex<double> z_type; |
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+ | #define Re(x) x.real() |
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+ | #define Im(x) x.imag() |
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+ | #define I z_type(0.,1.) |
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+ | #include "conto.cin" |
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+ | #include "efjh.cin" |
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+ | int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; |
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+ | int M=302,M1=M+1; |
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+ | int N=101,N1=N+1; |
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+ | DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. |
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+ | char v[M1*N1]; // v is working array |
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+ | //FILE *o;o=fopen("logi3e4a.eps","w");ado(o,304,204); |
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+ | FILE *o;o=fopen("logihal5.eps","w");ado(o,304,204); |
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+ | fprintf(o,"152 102 translate\n 100 100 scale\n"); |
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+ | DO(m,M1) X[m]= -1.5+.02*(m-.5); |
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+ | DO(n,N1) Y[n]=-1.+.02*(n-.5); |
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+ | |||
+ | for(m=-3;m<4;m++){if(m==0){M(.5*m,-1.04)L(.5*m,1.04)} else{M(.5*m,-1)L(.5*m,1)}} |
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+ | for(n=-2;n<3;n++){ M( -1.5 ,.5*n)L(1.5,.5*n)} |
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+ | fprintf(o,".004 W 0 0 0 RGB S\n"); |
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+ | |||
+ | //maq(3.); |
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+ | maq(5.); |
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+ | DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} |
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+ | DO(m,M1){x=X[m]; //printf("%5.2f\n",x); |
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+ | DO(n,N1){y=Y[n]; z=z_type(x,y); |
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+ | // c=E(H(z))-1.; |
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+ | // c=F(1.+E(0.1*z)); |
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+ | // c=F(z); |
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+ | c=F(.5+E(z)); |
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+ | c=F(.5+E(c)); |
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+ | // c=boe(z); |
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+ | // c=.5*(1.-cos(exp((z+1.)/LQ))); |
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+ | // d=H(F(z-1.)); |
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+ | // p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.; |
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+ | // if(p>-99 && p<20) g[m*N1+n]=p; |
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+ | p=Re(c);q=Im(c); |
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+ | if(p>-99 && p<99) {g[m*N1+n]=p;} |
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+ | if(q>-99 && q<99) {f[m*N1+n]=q;} |
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+ | // if(q>-99 && q<99 && fabs(q)>1.e-11 ) {f[m*N1+n]=q;} |
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+ | }} |
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+ | |||
+ | fprintf(o,"1 setlinejoin 2 setlinecap\n"); |
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+ | p=2;q=.4; |
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+ | //#include"plof.cin" |
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+ | for(m=-2;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 .6 0 RGB S\n"); |
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+ | for(m=0;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".002 W .9 0 0 RGB S\n"); |
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+ | for(m=0;m<2;m++) |
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+ | for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 0 .9 RGB S\n"); |
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+ | |||
+ | for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".006 W .9 0 0 RGB S\n"); |
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+ | for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".006 W 0 0 .9 RGB S\n"); |
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+ | for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".006 W 0 0 0 RGB S\n"); |
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+ | |||
+ | conto(o,f,w,v,X,Y,M,N, (0. ),-99,99); fprintf(o,".008 W .5 0 .5 RGB S\n"); |
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+ | |||
+ | fprintf(o,"0 setlinejoin 0 setlinecap\n"); |
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+ | |||
+ | //M(-3.02,0)L(0,0) |
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+ | //M(1.-1./Q,0)L(3,0) |
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+ | M(1.25,0)L(3,0) |
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+ | fprintf(o,"0.01 W 1 1 1 RGB S\n"); |
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+ | //for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)} |
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+ | for(n=0;n<4;n++) { M(1.25+.06*(n+.3),0) |
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+ | L(1.25+.06*(n+.7),0)} |
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+ | fprintf(o,"0.014 W 0 0 0 RGB S\n"); |
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+ | |||
+ | fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); |
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+ | system("epstopdf logihal5.eps"); // for linux |
||
+ | system( "open logihal5.pdf"); // for mac |
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+ | getchar(); system("killall Preview"); |
||
+ | } |
||
+ | </nowiki></nomathjax></poem> |
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+ | |||
+ | ==[[Latex]] generator of the labels== |
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+ | |||
+ | <poem><nomathjax><nowiki> |
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+ | \documentclass[12pt,a4paper,oneside]{book} |
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+ | %\newcommand \EN[1] {{#1}} % make the English version |
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+ | \newcommand \EN[1] {{}} % suppress the English version |
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+ | \newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported) |
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+ | %\newcommand \RU[1] {{}} % suppress the Russian version |
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+ | %The Japanese version is not yet supported. While \JP is used to suppress several lines at once. |
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+ | \newcommand \JP[1] {{}} % ореsuppress some text |
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+ | %\usepackage[space]{cite }% If exist. |
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+ | \usepackage[utf8]{inputenc} |
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+ | \usepackage[T2A]{fontenc} |
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+ | \usepackage[russian]{babel} |
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+ | \usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx} |
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+ | |||
+ | \usepackage{rotating} |
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+ | \usepackage{hyperref} |
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+ | \usepackage{wrapfig} |
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+ | \usepackage{geometry} |
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+ | %\renewcommand\baselinestretch{.94} |
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+ | %\renewcommand\baselinestretch{.94} |
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+ | %\renewcommand\baselinestretch{1.15} |
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+ | %\renewcommand\baselinestretch{1.2} |
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+ | %\renewcommand\baselinestretch{1.2} |
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+ | |||
+ | \paperwidth150mm |
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+ | %\paperheight 257mm |
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+ | \paperheight 216mm |
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+ | %\textwidth 175mm |
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+ | \textwidth 150mm |
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+ | %\oddsidemargin 5mm |
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+ | \oddsidemargin -72pt |
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+ | %\topmargin -17mm |
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+ | \topmargin -103pt |
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+ | \textheight 257mm |
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+ | \pagestyle{empty} |
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+ | \large |
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+ | |||
+ | \usepackage{color} |
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+ | \definecolor{rose}{rgb}{1,.9,1} |
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+ | \renewcommand\thesection{\arabic{section}} |
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+ | \renewcommand\figurename{Рис.} |
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+ | %\renewcommand\refname{References} |
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+ | %\renewcommand\figurename{Fig.} |
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+ | \newcommand \be {\begin{eqnarray}} |
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+ | \newcommand \ee {\end{eqnarray} } |
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+ | \newcommand \sx {\scalebox} |
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+ | \newcommand \rme {{\rm e}} %%makes the base of natural logarithms Roman font |
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+ | %\newcommand \rme {{e}} %%makes the base of natural logarithms Italics font; choose one of these |
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+ | \newcommand \rmi {{\rm i}} %%imaginary unity is always roman font |
||
+ | \newcommand \ds {\displaystyle} |
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+ | \newcommand \bN {\mathbb{N}} |
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+ | \newcommand \bC {\mathbb{C}} |
||
+ | \newcommand \bR {\mathbb{R}} |
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+ | \newcommand \cO {\mathcal{O}} |
||
+ | \newcommand \cF {\mathcal{F}} |
||
+ | \newcommand \rot {\begin{rotate}} |
||
+ | \newcommand \ero {\end{rotate}} |
||
+ | \newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!} |
||
+ | \newcommand \pS {{~}~{~}} |
||
+ | \newcommand \fac {\mathrm{Factorial}} |
||
+ | \newcommand {\rf}[1] {(\ref{#1})} |
||
+ | \newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} %make the labels visible |
||
+ | %\newcommand{\iL}[1] {\label{#1}} %make the labels invisible; choose one of these options |
||
+ | \newcommand \eL[1] {\iL{#1}\ee} |
||
+ | \newcommand \ing \includegraphics |
||
+ | \newcommand \tet {\mathrm{tet}} |
||
+ | \parindent 0pt |
||
+ | \begin{document} |
||
+ | \JP{ |
||
+ | \color{red} |
||
+ | \begin{picture}(425,729.88) |
||
+ | \put(0,0){\rule{1pt}{729.88pt}} |
||
+ | \put(425,0){\rule{1pt}{729.88pt}} |
||
+ | \put(0,0){\rule{426pt}{1pt}} |
||
+ | \put(0,728.88){\rule{426pt}{1pt}} |
||
+ | \end{picture} |
||
+ | %\vskip -740pt |
||
+ | \vskip -729pt |
||
+ | \color{black} |
||
+ | }\hskip 2pt |
||
+ | \sx{1.32}{\begin{picture}(320,120) |
||
+ | \put(16,16){\ing{logihalf3}} |
||
+ | \put(8,112){\sx{1.2}{$y$}} |
||
+ | \put(8, 64.4){\sx{1.17}{$\displaystyle \frac{1}{2}$}} |
||
+ | \put(8, 14){\sx{1.2}{$0$}} |
||
+ | \put(60,5){\sx{1.2}{$-1$}} |
||
+ | \put(166,5){\sx{1.2}{$0$}} |
||
+ | \put(266,5){\sx{1.2}{$1$}} |
||
+ | \put(312,5){\sx{1.24}{$x$}} |
||
+ | \put(26,49.2){\sx{1.2}{\rot{12} $v\!=\!4$ \ero}} |
||
+ | \put(29,32){\sx{1.2}{\rot{6} $v\!=\!2$ \ero}} |
||
+ | \put(30,15){\sx{1.2}{$v\!=\!0$}} |
||
+ | \put(222.4,32){\sx{1.2}{\rot{90}$v\!=\!0$\ero}} |
||
+ | \put(238,24){\sx{1.2}{\rot{58}\bf cut \ero}} |
||
+ | \put(288,15){\sx{1.2}{\rot{0}\bf cut \ero}} |
||
+ | \put(170,113){\sx{1.2}{\rot{-20}$u\!=\!3$\ero}} |
||
+ | \put(161,100){\sx{1.2}{\rot{-30}$u\!=\!2$\ero}} |
||
+ | \put(151,87){\sx{1.2}{\rot{-41}$u\!=\!1$\ero}} |
||
+ | \put(136.6,76){\sx{1.2}{\rot{-54}$u\!=\!0$\ero}} |
||
+ | \put(125,68){\sx{1.2}{\rot{-72}$u\!=\!-1$\ero}} |
||
+ | \put(112,64){\sx{1.2}{\rot{-79}$u\!=\!-2$\ero}} |
||
+ | \put(97,59){\sx{1.2}{\rot{-81}$u\!=\!-3$\ero}} |
||
+ | \put(84,56){\sx{1.2}{\rot{-83}$u\!=\!-4$\ero}} |
||
+ | \end{picture}} |
||
+ | \vskip 4pt |
||
+ | |||
+ | \hskip 2pt |
||
+ | \sx{1.32}{\begin{picture}(320,120) |
||
+ | \put(16,16){\ing{logihalf4}} |
||
+ | \put(8,112){\sx{1.2}{$y$}} |
||
+ | \put(8, 64.4){\sx{1.17}{$\displaystyle \frac{1}{2}$}} |
||
+ | \put(8, 14){\sx{1.2}{$0$}} |
||
+ | \put(60,5){\sx{1.2}{$-1$}} |
||
+ | \put(166,5){\sx{1.2}{$0$}} |
||
+ | \put(266,5){\sx{1.2}{$1$}} |
||
+ | \put(312,5){\sx{1.24}{$x$}} |
||
+ | \put(26,40){\sx{1.2}{\rot{9} $v\!=\!4$ \ero}} |
||
+ | \put(29,27.7){\sx{1.2}{\rot{5} $v\!=\!2$ \ero}} |
||
+ | \put(30,15){\sx{1.2}{$v\!=\!0$}} |
||
+ | \put(222.4,32){\sx{1.2}{\rot{90}$v\!=\!0$\ero}} |
||
+ | \put(251,24){\sx{1.2}{\rot{69}\bf cut \ero}} |
||
+ | \put(288,15){\sx{1.2}{\rot{0}\bf cut \ero}} |
||
+ | \put(170,114){\sx{1.2}{\rot{-19}$u\!=\!4$\ero}} |
||
+ | \put(162,105){\sx{1.2}{\rot{-30}$u\!=\!3$\ero}} |
||
+ | \put(155,95){\sx{1.2}{\rot{-37}$u\!=\!2$\ero}} |
||
+ | \put(146,85){\sx{1.2}{\rot{-43}$u\!=\!1$\ero}} |
||
+ | \put(137,77){\sx{1.2}{\rot{-54}$u\!=\!0$\ero}} |
||
+ | \put(127.4,72){\sx{1.2}{\rot{-69}$u\!=\!-1$\ero}} |
||
+ | \put(117.6,65){\sx{1.2}{\rot{-75}$u\!=\!-2$\ero}}% |
||
+ | \put(106.6,62){\sx{1.2}{\rot{-78}$u\!=\!-3$\ero}}% |
||
+ | \put(97.6,59){\sx{1.2}{\rot{-82}$u\!=\!-4$\ero}} |
||
+ | \end{picture}} |
||
+ | \vskip 4pt |
||
+ | |||
+ | \hskip 2pt |
||
+ | \sx{1.32}{\begin{picture}(320,220) |
||
+ | \put(16,16){\ing{logihal5}} |
||
+ | \put(8,212){\sx{1.2}{$y$}} |
||
+ | \put(8,164.5){\sx{1.17}{$\displaystyle \frac{1}{2}$}} |
||
+ | \put(8,114){\sx{1.2}{$0$}} |
||
+ | \put(-2,64.5){\sx{1.16}{$\displaystyle-\frac{1}{2}$}} |
||
+ | \put(-1,14){\sx{1.2}{$-1$}} |
||
+ | \put(60,5){\sx{1.2}{$-1$}} |
||
+ | \put(166,5){\sx{1.2}{$0$}} |
||
+ | \put(266,5){\sx{1.2}{$1$}} |
||
+ | \put(312,5){\sx{1.24}{$x$}} |
||
+ | \put(26,137){\sx{1.2}{\rot{9} $v\!=\!4$ \ero}} |
||
+ | \put(29,125){\sx{1.2}{\rot{4} $v\!=\!2$ \ero}} |
||
+ | \put(30,114.5){\sx{1.2}{$v\!=\!0$}} |
||
+ | \put(28,105){\sx{1.2}{\rot{-4} $v\!=\!-2$ \ero}} |
||
+ | \put(25,94){\sx{1.2}{\rot{-9} $v\!=\!-4$ \ero}} |
||
+ | % |
||
+ | \put(172,201.8){\sx{1.2}{\rot{-22} $u\!=\!4$ \ero}} |
||
+ | \put(167,192.8){\sx{1.2}{\rot{-29} $u\!=\!3$ \ero}} |
||
+ | \put(159.8,183.8){\sx{1.2}{\rot{-36} $u\!=\!2$ \ero}} |
||
+ | \put(152,176){\sx{1.2}{\rot{-47} $u\!=\!1$ \ero}} |
||
+ | % |
||
+ | \put(82,21){\sx{1.2}{\rot{60} $u\!=\!-4$ \ero}} |
||
+ | \put(95,18){\sx{1.2}{\rot{56} $u\!=\!-2$ \ero}} |
||
+ | \put(110,17){\sx{1.2}{\rot{48} $u\!=\!0$ \ero}} |
||
+ | \put(130,17){\sx{1.2}{\rot{40} $u\!=\!2$ \ero}} |
||
+ | \put(156,17){\sx{1.2}{\rot{30} $u\!=\!4$ \ero}} |
||
+ | % |
||
+ | \put(263,124){\sx{1.2}{\rot{74}\bf cut \ero}} |
||
+ | \put(298,115){\sx{1.2}{\rot{0}\bf cut \ero}} |
||
+ | \put(222.4,102){\sx{1.2}{\rot{90}$v\!=\!0$\ero}} |
||
+ | \end{picture}} |
||
+ | |||
+ | \end{document} |
||
+ | </nowiki></nomathjax></poem> |
||
+ | |||
+ | ==References== |
||
+ | <references/> |
||
+ | |||
+ | [[Category:C++]] |
||
+ | [[Category:Complex map]] |
||
+ | [[Category:Book]] |
||
+ | [[Category:BookMap]] |
||
+ | [[Category:Iterate]] |
||
+ | [[Category:Latex]] |
||
+ | [[Category:Logistic operator]] |
||
+ | [[Category:Superfunction]] |
Latest revision as of 08:42, 1 December 2018
Complex map of the halfiterate or the Logistic operator $T(z)\!=\!s \,z\,(z\!-\!1)$ for $s\!=\!3$, upper map; $s\!=\!4$, intermadiate; $s\!=\!5$, at bottom.
$u\!+\!\mathrm i v = \mathrm{LogisticOperator}_s^{1/2}(x\!+\!\mathrm i y)$
The function is real-holomorphic; the map are symmetric; so, for the two first cases, the only half of the map is shown.
Contents
C++ generator of the upper 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"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=302,M1=M+1;
int N=51,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("logi3e4a.eps","w");ado(o,304,204);
FILE *o;o=fopen("logihalf3.eps","w");ado(o,304,104);
fprintf(o,"152 2 translate\n 100 100 scale\n");
DO(m,M1) X[m]= -1.5+.02*(m-.5);
DO(n,N1) Y[n]=0.+.02*(n-.5);
for(m=-3;m<4;m++){if(m==0){M(.5*m,0)L(.5*m,1.04)} else{M(.5*m,0)L(.5*m,1)}}
for(n=0;n<3;n++){ M( -1.5 ,.5*n)L(1.5,.5*n)}
fprintf(o,".004 W 0 0 0 RGB S\n");
//maq(3.);
maq(3.);
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=F(.5+E(c));
// 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>-99 && p<20) g[m*N1+n]=p;
p=Re(c);q=Im(c);
if(p>-99 && p<99) {g[m*N1+n]=p;}
if(q>-99 && q<99) {f[m*N1+n]=q;}
// if(q>-99 && q<99 && fabs(q)>1.e-11 ) {f[m*N1+n]=q;}
}}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2;q=.4;
//#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,".002 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,".002 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,".002 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,".006 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,".006 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,".006 W 0 0 0 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (0. ),-99,99); fprintf(o,".008 W .5 0 .5 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) fprintf(o,"0.01 W 1 1 1 RGB S\n");
M(0.7,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n");
//for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)}
for(n=0;n<12;n++) { M(0.7+.07*(n+.3),0)
L(0.7+.07*(n+.7),0)}
fprintf(o,"0.014 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf logihalf3.eps"); // for linux
system( "open logihalf3.pdf"); // for mac
getchar(); system("killall Preview");
}
C++ generator of the second 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"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=302,M1=M+1;
int N=51,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("logi3e4a.eps","w");ado(o,304,204);
FILE *o;o=fopen("logihalf4.eps","w");ado(o,304,104);
fprintf(o,"152 2 translate\n 100 100 scale\n");
DO(m,M1) X[m]= -1.5+.02*(m-.5);
DO(n,N1) Y[n]=0.+.02*(n-.5);
for(m=-3;m<4;m++){if(m==0){M(.5*m,0)L(.5*m,1.04)} else{M(.5*m,0)L(.5*m,1)}}
for(n=0;n<3;n++){ M( -1.5 ,.5*n)L(1.5,.5*n)}
fprintf(o,".004 W 0 0 0 RGB S\n");
//maq(3.);
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=F(z);
c=F(.5+E(z));
c=F(.5+E(c));
// 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>-99 && p<20) g[m*N1+n]=p;
p=Re(c);q=Im(c);
if(p>-99 && p<99) {g[m*N1+n]=p;}
if(q>-99 && q<99) {f[m*N1+n]=q;}
// if(q>-99 && q<99 && fabs(q)>1.e-11 ) {f[m*N1+n]=q;}
}}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2;q=.4;
//#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,".002 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,".002 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,".002 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,".006 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,".006 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,".006 W 0 0 0 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (0. ),-99,99); fprintf(o,".008 W .5 0 .5 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) fprintf(o,"0.01 W 1 1 1 RGB S\n");
M(.96,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n");
//for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)}
for(n=0;n<10;n++) { M(.96+.07*(n+.3),0)
L(.96+.07*(n+.7),0)}
fprintf(o,"0.014 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf logihalf4.eps"); // for linux
system( "open logihalf4.pdf"); // for mac
getchar(); system("killall Preview");
}
C++ generator of the third 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"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=302,M1=M+1;
int N=101,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("logi3e4a.eps","w");ado(o,304,204);
FILE *o;o=fopen("logihal5.eps","w");ado(o,304,204);
fprintf(o,"152 102 translate\n 100 100 scale\n");
DO(m,M1) X[m]= -1.5+.02*(m-.5);
DO(n,N1) Y[n]=-1.+.02*(n-.5);
for(m=-3;m<4;m++){if(m==0){M(.5*m,-1.04)L(.5*m,1.04)} else{M(.5*m,-1)L(.5*m,1)}}
for(n=-2;n<3;n++){ M( -1.5 ,.5*n)L(1.5,.5*n)}
fprintf(o,".004 W 0 0 0 RGB S\n");
//maq(3.);
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=F(.5+E(c));
// 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>-99 && p<20) g[m*N1+n]=p;
p=Re(c);q=Im(c);
if(p>-99 && p<99) {g[m*N1+n]=p;}
if(q>-99 && q<99) {f[m*N1+n]=q;}
// if(q>-99 && q<99 && fabs(q)>1.e-11 ) {f[m*N1+n]=q;}
}}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2;q=.4;
//#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,".002 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,".002 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,".002 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,".006 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,".006 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,".006 W 0 0 0 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (0. ),-99,99); fprintf(o,".008 W .5 0 .5 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(1.25,0)L(3,0)
fprintf(o,"0.01 W 1 1 1 RGB S\n");
//for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)}
for(n=0;n<4;n++) { M(1.25+.06*(n+.3),0)
L(1.25+.06*(n+.7),0)}
fprintf(o,"0.014 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf logihal5.eps"); // for linux
system( "open logihal5.pdf"); // for mac
getchar(); system("killall Preview");
}
Latex generator of the labels
\documentclass[12pt,a4paper,oneside]{book}
%\newcommand \EN[1] {{#1}} % make the English version
\newcommand \EN[1] {{}} % suppress the English version
\newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported)
%\newcommand \RU[1] {{}} % suppress the Russian version
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
\newcommand \JP[1] {{}} % ореsuppress some text
%\usepackage[space]{cite }% If exist.
\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[russian]{babel}
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}
\usepackage{rotating}
\usepackage{hyperref}
\usepackage{wrapfig}
\usepackage{geometry}
%\renewcommand\baselinestretch{.94}
%\renewcommand\baselinestretch{.94}
%\renewcommand\baselinestretch{1.15}
%\renewcommand\baselinestretch{1.2}
%\renewcommand\baselinestretch{1.2}
\paperwidth150mm
%\paperheight 257mm
\paperheight 216mm
%\textwidth 175mm
\textwidth 150mm
%\oddsidemargin 5mm
\oddsidemargin -72pt
%\topmargin -17mm
\topmargin -103pt
\textheight 257mm
\pagestyle{empty}
\large
\usepackage{color}
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References
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 06:13, 1 December 2018 | 1,771 × 2,551 (1.81 MB) | Maintenance script (talk | contribs) | Importing image file |
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