{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Define the function that w ill display cobweb plots." }}{PARA 0 "" 0 "" {TEXT -1 125 "f will be t he function, n will iterate the given point (start) n times, domain an d range will specify the region of the plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 463 "cobweb := proc(f,n,start,domain,range)\n local a , i, s, gra, gpl, fpl, ipl;\n a := evalf(start);\n gra := [[a,f( a)],[f(a)-a,0]]; \n for i to n do a := f(a); \n gra := gra,[[a, a],[0,f(a)-a]],[[a,f(a)],[f(a)-a,0]];\n od:\n gpl := arrow([gra] ,shape=arrow,color=red,head_length=0.02,head_width=0.02);\n fpl := \+ plot(f,domain,color=black);\n ipl := plot(x->x,domain,color=blue); \n print(plots[display]([gpl,fpl,ipl],view=[domain,range]));\n e nd:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Define the standard Quadra tic Function and display the required cobweb plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "T:=x-> piecewise(x>=0 and x<1/2,2*x,x>=1/2 \+ and x<=1,2-2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TGf*6#%\"xG6\" 6$%)operatorG%&arrowGF(-%*piecewiseG6&31\"\"!9$2F2#\"\"\"\"\"#,$F2F631 F4F21F2F5,&F6F5*&F6F5F2F5!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "cobweb(T,15,0.757,0..1,0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6E-%'CURVESG6&7$7$$\"$d(!\"$$\"$'[F *7$F+F+7%7$$\"++++g]!#5$\"++++gZF2F-7$F0$\"++++g\\F2-%&STYLEG6#%,PATCH NOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!FDFC-F$6&7$F-7$F+$\"$s*F* 7%7$F3$\"++++?&*F2FH7$F6FMF8F<-F$6&7$FH7$FIFI7%7$FM$\"++++?)*F2FS7$FM$ \"++++?'*F2F8F<-F$6&7$FS7$FI$\"#cF*7%7$FV$\"*+++g(F2Fhn7$FYF]oF8F<-F$6 &7$Fhn7$FinFin7%7$F]o$\"+++++Y!#6Fco7$F]o$\"+++++mFhoF8F<-F$6&7$Fco7$F in$\"$7\"F*7%7$Ffo$\"+++++#*FhoF_p7$FjoFdpF8F<-F$6&7$F_p7$F`pF`p7%7$Fd p$\"++++?7F2Fjp7$Fdp$\"++++?5F2F8F<-F$6&7$Fjp7$F`p$\"$C#F*7%7$F`q$\"++ ++S?F2Feq7$F]qFjqF8F<-F$6&7$Feq7$FfqFfq7%7$Fjq$\"++++SBF2F`r7$Fjq$\"++ ++S@F2F8F<-F$6&7$F`r7$Ffq$\"$[%F*7%7$Ffr$\"++++!G%F2F[s7$FcrF`sF8F<-F$ 6&7$F[s7$F\\sF\\s7%7$F`s$\"++++!e%F2Ffs7$F`s$\"++++!Q%F2F8F<-F$6&7$Ffs 7$F\\s$\"$'*)F*7%7$F\\t$\"++++g()F2Fat7$FisFftF8F<-F$6&7$Fat7$FbtFbt7% 7$Fft$\"++++g!*F2F\\u7$Fft$\"++++g))F2F8F<-F$6&7$F\\u7$Fbt$\"$3#F*7%7$ F_u$\"++++!G#F2Fgu7$FbuF\\vF8F<-F$6&7$Fgu7$FhuFhu7%7$F\\v$\"++++!)>F2F bv7$F\\v$\"++++!=#F2F8F<-F$6&7$Fbv7$Fhu$\"$;%F*7%7$Fev$\"++++gRF2F]w7$ FhvFbwF8F<-F$6&7$F]w7$F^wF^w7%7$Fbw$\"++++gUF2Fhw7$Fbw$\"++++gSF2F8F<- F$6&7$Fhw7$F^w$\"$K)F*7%7$F^x$\"++++?\")F2Fcx7$F[xFhxF8F<-F$6&7$Fcx7$F dxFdx7%7$Fhx$\"++++?%)F2F^y7$Fhx$\"++++?#)F2F8F<-F$6&7$F^y7$Fdx$\"$O$F *7%7$Fay$\"++++gNF2Fiy7$FdyF^zF8F<-F$6&7$Fiy7$FjyFjy7%7$F^z$\"++++gKF2 Fdz7$F^z$\"++++gMF2F8F<-F$6&7$Fdz7$Fjy$\"$s'F*7%7$Fgz$\"++++?lF2F_[l7$ FjzFd[lF8F<-F$6&7$F_[l7$F`[lF`[l7%7$Fd[l$\"++++?oF2Fj[l7$Fd[l$\"++++?m F2F8F<-F$6&7$Fj[l7$F`[l$\"$c'F*7%7$F]\\l$\"++++gnF2Fe\\l7$F`\\lFj\\lF8 F<-F$6&7$Fe\\l7$Ff\\lFf\\l7%7$Fj\\l$\"++++gkF2F`]l7$Fj\\l$\"++++gmF2F8 F<-F$6&7$F`]l7$Ff\\l$\"$)oF*7%7$Fc]l$\"++++!o'F2F[^l7$Ff]lF`^lF8F<-F$6 &7$F[^l7$F\\^lF\\^l7%7$F`^l$\"++++!)pF2Ff^l7$F`^l$\"++++!y'F2F8F<-F$6& 7$Ff^l7$F\\^l$\"$C'F*7%7$Fi^l$\"++++SkF2Fa_l7$F\\_lFf_lF8F<-F$6&7$Fa_l 7$Fb_lFb_l7%7$Ff_l$\"++++ShF2F\\`l7$Ff_l$\"++++SjF2F8F<-F$6&7$F\\`l7$F b_l$\"$_(F*7%7$F_`l$\"++++?tF2Fg`l7$Fb`lF\\alF8F<-F$6&7$Fg`l7$Fh`lFh`l 7%7$F\\al$\"++++?wF2Fbal7$F\\al$\"++++?uF2F8F<-F$6$7U7$FCFC7$$\"3emmm; arz@!#>$\"39LLLL3VfVFabl7$$\"3[LL$e9ui2%Fabl$\"3'pmm;H[D:)Fabl7$$\"3nm mm\"z_\"4iFabl$\"3LLLLe0$=C\"!#=7$$\"3[mmmT&phN)Fabl$\"3ILLL3RBr;F^cl7 $$\"3CLLe*=)H\\5F^cl$\"3Ymm;zjf)4#F^cl7$$\"3gmm\"z/3uC\"F^cl$\"3=LL$e4 ;[\\#F^cl7$$\"3%)***\\7LRDX\"F^cl$\"3p****\\i'y]!HF^cl7$$\"3]mm\"zR'ok ;F^cl$\"3,LL$ezs$HLF^cl7$$\"3w***\\i5`h(=F^cl$\"3_****\\7iI_PF^cl7$$\" 3WLLL3En$4#F^cl$\"3#pmmm@Xt=%F^cl7$$\"3qmm;/RE&G#F^cl$\"3QLLL3y_qXF^cl 7$$\"3\")*****\\K]4]#F^cl$\"3i******\\1!>+&F^cl7$$\"3$******\\PAvr#F^c l$\"3()******\\Z/NaF^cl7$$\"3)******\\nHi#HF^cl$\"3'*******\\$fC&eF^cl 7$$\"3jmm\"z*ev:JF^cl$\"3ELL$ez6:B'F^cl7$$\"3?LLL347TLF^cl$\"3Smmm;=C# o'F^cl7$$\"3,LLLLY.KNF^cl$\"3-mmmm#pS1(F^cl7$$\"3w***\\7o7Tv$F^cl$\"3] ****\\i`A3vF^cl7$$\"3'GLLLQ*o]RF^cl$\"3slmmm(y8!zF^cl7$$\"3A++D\"=lj;% F^cl$\"3V++]i.tK$)F^cl7$$\"31++vV&R
Y2aF^cl$\"3-,+]ih2&=*F^cl7$$\"39m m;zXu9cF^cl$\"3snmmT3^q()F^cl7$$\"3l******\\y))GeF^cl$\"3q++++VAU$)F^c l7$$\"3'*)***\\i_QQgF^cl$\"33-++v%HK#zF^cl7$$\"3@***\\7y%3TiF^cl$\"3d, +]P/$y^(F^cl7$$\"35****\\P![hY'F^cl$\"3y,++DRqnqF^cl7$$\"3kKLL$Qx$omF^ cl$\"3uMLLL_CjmF^cl7$$\"3!)*****\\P+V)oF^cl$\"3R+++]#*RJiF^cl7$$\"3?mm \"zpe*zqF^cl$\"3enm;/E3SeF^cl7$$\"3%)*****\\#\\'QH(F^cl$\"3M+++],F7aF^ cl7$$\"3GKLe9S8&\\(F^cl$\"3VNL$3(>t4]F^cl7$$\"3R***\\i?=bq(F^cl$\"3?,+ ](ej*)e%F^cl7$$\"3\"HLL$3s?6zF^cl$\"3=MLL$e&exTF^cl7$$\"3a***\\7`Wl7)F ^cl$\"3#4++v$4\"pu$F^cl7$$\"3#pmmm'*RRL)F^cl$\"3;mmmm+7KLF^cl7$$\"3Qmm ;a<.Y&)F^cl$\"3Bnmm\"\\Oz!HF^cl7$$\"3=LLe9tOc()F^cl$\"3mLL$3Pls[#F^cl7 $$\"3u******\\Qk\\*)F^cl$\"3`++++Br+@F^cl7$$\"3CLL$3dg6<*F^cl$\"3]LLLe )ywl\"F^cl7$$\"3ImmmmxGp$*F^cl$\"3UnmmmWUh7F^cl7$$\"3A++D\"oK0e*F^cl$ \"3E&****\\PY$*Q)Fabl7$$\"3A++v=5s#y*F^cl$\"3i&****\\izbM%Fabl7$$\"\" \"FDFC-F=6&F?FDFDFD-F$6$7SF]bl7$F_blF_bl7$FeblFebl7$FjblFjbl7$F`clF`cl 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Map." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G:=x -> 4*x*(1-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&9$\" \"\",&F/F/F.!\"\"F/\"\"%F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "We know that the Quadratic Map is conjugate to the tent map via the c onjugacy C(x)= " }{XPPEDIT 18 0 "(1-cos(Pi*x))/2;" "6#*&,&\"\"\"F%-%$c osG6#*&%#PiGF%%\"xGF%!\"\"F%\"\"#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "C:=x -> (1-cos(Pi*x))/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,\"\"\"\"\"# F.*F.F/F.-%$cosG6#*&%#PiGF.9$F.F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "y:=C(0.757);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"yG,\"\"\"\"\"#F'*F'F(F'-%$cosG6#,$%#PiG$\"$d(!\"$F'!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "cobweb(G,15,y,0..1,0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6E-%'CURVESG6&7$ 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