File:Logi2c4T1000.png

Complex map of function LogisticSequence with critic value of parameter,
 * $ u\!+\!\mathrm i v= \mathrm{LogisticSequence}_4(x\!+\!\mathrm i y)$

The generators of the map are copypasted below

logiu
// The code below should be stored as logiu.cin

DB LQ; DB QQ,Q2,Q3,Q4,Q5,Q6,Q7; DB C1,C2,C3,C4,C5,C6,C7; DB S1,S2,S3,S4,S5,S6,S7,S8;

void maq(DB q){ Q=q; QQ=Q*Q; Q2=QQ; Q3=QQ*Q; Q4=QQ*QQ; Q5=Q4*Q; Q6=Q3*Q3; Q7=Q6*Q; LQ=log(Q); C2=1. / (1.-Q) ; C3=2. /((1.-Q)*(1.-Q2) ); C4=  (5.+Q)                                                    /((1.-Q)*(1.-Q2)*(1.-Q3) ); C5=2.*(7.+Q*(3.+Q*2.))                                         /((1.-Q)*(1.-Q2)*(1.-Q3)*(1.-Q4) ); C6=2.*(21.+Q*(14.+Q*(14.+Q*( 8.+Q*3.))))                       /((1.-Q)*(1.-Q2)*(1.-Q3)*(1.-Q4)*(1.-Q5)); C7=4.*(33.+Q*(30.+Q*(37.+Q*(32.+Q*(27.+Q*(12.+Q*(8.+Q)))))))   /((1.-Q)*(1.-Q2)*(1.-Q3)*(1.-Q4)*(1.-Q5)*(1.-Q6)); S1=1.; S2=1. / (Q-1.); S3=2.*Q                                                        /((Q-1.)*(Q2-1.)); S4=  Q *(1.+5.*Q*Q)                                            /((Q-1.)*(Q2-1.)*(Q3-1.) ); S5=2.*Q3*(3.+Q*(2.+QQ*7.))                                     /((Q-1.)*(Q2-1.)*(Q3-1.)*(Q4-1.) ); S6=2.*Q3*(1.+Q*(3.+QQ*(14.+Q*(14.+Q*(7.+QQ*21)))))             /((Q-1.)*(Q2-1.)*(Q3-1.)*(Q4-1.)*(Q5-1.)); S7=4.*Q4*(1.+Q*(5.+Q*(15.+Q*(12.+Q*(8.+Q*(30.+Q*(37.+Q*(27.+Q*(12.+QQ*(33.)))))))))) /((Q-1.)*(Q2-1.)*(Q3-1.)*(Q4-1.)*(Q5-1.)*(Q6-1.)); S8=Q4*(1.+QQ*(5.+Q*(60.+Q*(69.+Q*(89.+Q*(162.+Q*(415.+Q*(495.+Q*(531.+Q*(267.+Q*(630.+Q*(705.+Q*(621.+Q*(396+Q*(165.+Q*429) )    )      )      )      )       )       )       )       )       )       )       )       )        )                                                       /((Q-1.)*(Q2-1.)*(Q3-1.)*(Q4-1.)*(Q5-1.)*(Q6-1.)*(Q7-1.)); ; printf("maq: Q=%9.6f\n",Q);}

z_type J(z_type z){ return .5-sqrt(.25-z/Q); } z_type H(z_type z){ return Q*z*(1.-z); } z_type f(z_type z){z_type e;e=exp(LQ*z);return e*(1.+e*(C2+e*(C3+e*(C4+e*(C5+e*(C6+e*C7)))))); } z_type F(z_type z){    if(Re(z)>-6.) return H(F(z-1.));   return f(z);} z_type e(z_type z){                    return z*(1.+z*(S2+z*(S3+z*(S4+z*(S5+z*(S6+z*(S7+z* S8)))))));} z_type E(z_type z){    if(abs(z)>.1) return  E(J(z))+1.; return log(e(z))/LQ;} //#include "efjh.cin" z_type u(z_type z){ DB q1 =Q-1.; DB q2 =Q-2.; DB q3 =Q-3.; DB q2L=log(q2); DB U0=q1/Q; DB U1=1.; DB U2=-Q/(q1*q2); DB U3=-2.*QQ/(q1*q1*q2*q2*q3); DB U4=-2.*Q3*(Q-7.)/(q1*q1*q1*q2*q2*q2*(-21.+Q*(22.+Q*(-8.+Q))) ); z_type e=-exp(z*q2L)*cos(M_PI*z); //return U0+e; //return U0+e*(1.+e*(U2)); //return U0+e*(U1+e*(U2+e*U3)); return U0+e*(U1+e*(U2+e*(U3+e*U4))); }

z_type U(z_type z){    if(Re(z)+3*fabs(Im(z))>-6.) return H(U(z-1.));   return u(z);}

// Let me know if any problem with use of the routine "U".

C++ generator of curves
// Routine logiu.cin above and files ado.cin and conto.cin should be loaded to the working directory in order to compile the code below

using namespace std; typedef complex z_type; main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=408,M1=M+1; int N=602,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("logi2c4a.eps","w");ado(o,126,124); fprintf(o,"62 62 translate\n 20 20 scale\n"); DO(m,M1) X[m]=-3.01+.015*(m-.5); DO(n,300) Y[n]=-3.+.01*n; Y[300]=-0.002; Y[301]= 0.002; for(n=302;n-49 && p<49)   {g[m*N1+n]=p;} if(q>-49 && q<49)   {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=-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,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logi2c4a.eps"); // for linux system(   "open logi2c4a.pdf"); // for mac getchar; system("killall Preview"); }
 * 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"

// Copileft 2009, 2012 by Dmitrii Kouznetsov

Latex generator of labels
% File logi2c4T.pdf should be generated with the code above in order to compile the Latex document below % % \documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphics} % \usepackage{rotating} % \paperwidth 128pt % \paperheight 127pt % \topmargin -104pt % \oddsidemargin -90pt % \newcommand \sx {\scalebox} % \newcommand \ing \includegraphics % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \begin{document} % \newcommand \axes { % \put( 2,124){\sx{.5}{$y$}} % \put( 2,105){\sx{.5}{$2$}} % \put( 2,85){\sx{.5}{$1$}} % \put( 2, 65){\sx{.5}{$0$}} % \put(-.8, 45){\sx{.5}{$-\!1$}} % \put(-.8, 25){\sx{.5}{$-\!2$}} % \put( 22, 1){\sx{.5}{$-2$}} % \put( 42, 1){\sx{.5}{$-\!1$}} % \put( 65, 1){\sx{.5}{$0$}} % \put( 85, 1){\sx{.5}{$1$}} % \put(105, 1){\sx{.5}{$2$}} % \put(124.6, 1){\sx{.5}{$x$}} % } % \begin{picture}(122,122) \put( 4, 4){\ing{logi2c4a}} % \put(7,110){\rot{ 0.}\sx{.6}{$v\!=\!0$}\ero} % \put(7, 87.4){\rot{ 0.}\sx{.6}{$u\!=\!0$}\ero} % \put(7, 64.5){\rot{ 0.}\sx{.6}{$v\!=\!0$}\ero} % \put(7, 41.6){\rot{ 0.}\sx{.6}{$u\!=\!0$}\ero} % \put(7, 19){\rot{ 0.}\sx{.6}{$v\!=\!0$}\ero} % \put(34,70){\rot{50}\sx{.5}{$u\!=\!0.1$}\ero} % \put(44,68){\rot{58}\sx{.5}{$u\!=\!0.2$}\ero} % \put(56,68){\sx{.5}{$v\!=\!0.1$}} % \put(53,61){\sx{.5}{$v\!=\!-0.1$}} % \axes % \end{picture} % \end{document} %