{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 263 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 277 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Ti mes" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 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 Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 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 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 42 "The logistic family of functions \+ and chaos" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C ., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "The logistic family of functions" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 67 "For each value of the parameter a we define the quadratic funct ion " }{XPPEDIT 18 0 "proc (x) options operator, arrow; a*x*(1-x) end; " "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*(%\"aG\"\"\"F%F-,&F-F-F%!\" \"F-F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "We call thi s family of functions the " }{TEXT 259 15 "logistic family" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "It has applications in modelling population growth and can be used to illustrate the phonomenon of " } {TEXT 259 5 "chaos" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 90 "It \+ is convenient to set up the following \"double function\" to investiga te these functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "logistic := a -> (x -> a*x*(1-x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)logisticGf*6#%\"aG6\"6$%)operatorG% &arrowGF(f*6#%\"xGF(F)F(*(T$\"\"\"9$F1,&F1F1F2!\"\"F1F(F(6$F'F2F(F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "For \+ each value of the parameter " }{TEXT 266 1 "a" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "logistic(a)" "6#-%)logisticG6#%\"aG" }{TEXT -1 17 " is \+ the function " }{XPPEDIT 18 0 "proc (x) options operator, arrow; a*x*( 1-x) end;" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*(%\"aG\"\"\"F%F-,&F -F-F%!\"\"F-F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "The following picture shows some graphs o f the logistic function, where the parameter " }{TEXT 267 1 "a" } {TEXT -1 22 " is between 0.5 and 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([seq(logistic(i*0.5)(x ),i=1..8)],x=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 305 228 228 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3kh]%=\" >5m5F-7$$\"3[LL$e9ui2%F-$\"3I?nN=q0b>F-7$$\"3nmmm\"z_\"4iF-$\"3PB?)Q]3 =\"HF-7$$\"3[mmmT&phN)F-$\"3yWzyBp&*GQF-7$$\"3CLLe*=)H\\5!#=$\"3oN_e-w (fp%F-7$$\"3gmm\"z/3uC\"FB$\"35)pz0#o-faF-7$$\"3%)***\\7LRDX\"FB$\"3]+ f$=Shx?'F-7$$\"3]mm\"zR'ok;FB$\"3$)\\*yyeTy$pF-7$$\"3w***\\i5`h(=FB$\" 3rOU?#H!z?wF-7$$\"3WLLL3En$4#FB$\"3u9SAY0jw#)F-7$$\"3qmm;/RE&G#FB$\"3a &z*)\\'R5:))F-7$$\"3\")*****\\K]4]#FB$\"3o'>T4OvtP*F-7$$\"3$******\\PA vr#FB$\"3#*zroXz9&*)*F-7$$\"3)******\\nHi#HFB$\"3fpu&>$Q(\\.\"FB7$$\"3 jmm\"z*ev:JFB$\"3g3#)=37[s5FB7$$\"3?LLL347TLFB$\"3')e:'z+1C6\"FB7$$\"3 ,LLLLY.KNFB$\"3!*fL6%)QDU6FB7$$\"3w***\\7o7Tv$FB$\"3'o*)[%R#)Qs6FB7$$ \"3'GLLLQ*o]RFB$\"35Jv[ht%\\>\"FB7$$\"3A++D\"=lj;%FB$\"3r#Q[%\\ED:7FB7 $$\"31++vV&RLS#GJ9C\"FB7 $$\"3GLLeR\"3Gy%FB$\"3SP%)y%QTwC\"FB7$$\"3cmm;/T1&*\\FB$\"39;m%=y)**\\ 7FB7$$\"3&em;zRQb@&FB$\"3)e<^*frnZ7FB7$$\"3\\***\\(=>Y2aFB$\"3]G%Q#R() pT7FB7$$\"39mm;zXu9cFB$\"3()f#>^X/6B\"FB7$$\"3l******\\y))GeFB$\"3%)=h gYsk:7FB7$$\"3'*)***\\i_QQgFB$\"3QS6L-y3'>\"FB7$$\"3@***\\7y%3TiFB$\"3 9#[(GGa)H<\"FB7$$\"35****\\P![hY'FB$\"35Xng'\\?D9\"FB7$$\"3kKLL$Qx$omF B$\"35\"4R`%e#36\"FB7$$\"3!)*****\\P+V)oFB$\"31V(Q[gqC2\"FB7$$\"3?mm\" zpe*zqFB$\"3KS![2f)oL5FB7$$\"3%)*****\\#\\'QH(FB$\"3Yst#H&=4p)*F-7$$\" 3GKLe9S8&\\(FB$\"3EJYjCJ:(Q*F-7$$\"3R***\\i?=bq(FB$\"3%H*[#yh&3S))F-7$ $\"3\"HLL$3s?6zFB$\"3c@S2&HOCE)F-7$$\"3a***\\7`Wl7)FB$\"3yf`0Z'fBh(F-7 $$\"3#pmmm'*RRL)FB$\"3^J8L\\@UUpF-7$$\"3Qmm;a<.Y&)FB$\"3@P3A*RHG@'F-7$ $\"3=LLe9tOc()FB$\"3J**\\'*)H_[W&F-7$$\"3u******\\Qk\\*)FB$\"3k*ey!Hn: +ZF-7$$\"3CLL$3dg6<*FB$\"3dqE;Y(42!QF-7$$\"3ImmmmxGp$*FB$\"3s2&Gg?iY&H F-7$$\"3A++D\"oK0e*FB$\"3'=M+!y,O4?F-7$$\"3A++v=5s#y*FB$\"37![.%G)*yi5 F-7$$\"\"\"F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7SF'7$F+$\"3GB,pBQ ?K@F-7$F1$\"3dSMrOS65RF-7$F6$\"3tYSw2qhBeF-7$F;$\"3c*)edZQ\"zl(F-7$F@$ \"3Mr/<0_&>R*F-7$FF$\"3gRf6k`!=4\"FB7$FK$\"35!=n.G_:C\"FB7$FP$\"3)**yv vJovQ\"FB7$FU$\"3MZ3We!eT_\"FB7$FZ$\"3%H![C4hKb;FB7$Fin$\"35fz*Hz?Iw\" FB7$F^o$\"3MR#)=s]Zv=FB7$Fco$\"3)fVP\"*eH!z>FB7$Fho$\"3=R\\\"RmZ*p?FB7 $F]p$\"3@nAox]%G#FB7$F\\q $\"3t$z(*)ykxWBFB7$Faq$\"3Ai](Hs%*)*Q#FB7$Ffq$\"3Uln*))H00V#FB7$F[r$\" 3Ia7\"*z)G0Y#FB7$F`r$\"3#*Rm![ciG[#FB7$Fer$\"3yuodpFG&\\#FB7$Fjr$\"3GK Kpjv***\\#FB7$F_s$\"3u^B!*>VN&\\#FB7$Fds$\"3+doZyuR$[#FB7$Fis$\"3u>&Q- \"*3AY#FB7$F^t$\"3oPA@$\\%HJCFB7$Fct$\"3!3GiYgv@R#FB7$Fht$\"3Ik\\dc3(f M#FB7$F]u$\"3?!\\8K*4/&G#FB7$Fbu$\"3>#=y1p^;A#FB7$Fgu$\"39'[x'47%\\9#F B7$F\\v$\"3k!3'\\\"=xt1#FB7$Fav$\"3]uaeq$=Q(>FB7$Ffv$\"3CEp#\\iIu(=FB7 $F[w$\"3ey\\cBr,o26%H>ZA:FB7$Fjw$\"3 Imi')HW[)Q\"FB7$F_x$\"3WnT%)zecU7FB7$Fdx$\"3')**Hzf/(*)3\"FB7$Fix$\"3G zr:eMJ+%*F-7$F^y$\"39T`K#\\>9g(F-7$Fcy$\"3U:q07WK4fF-7$Fhy$\"3t$o+gN?( =SF-7$F]z$\"3Agp!olzb7#F-Faz-Fez6&FgzF(FhzF(-F$6$7SF'7$F+$\"3u%=Nbt0$) >$F-7$F1$\"3_g,2b5Lz!p$F B7$F`r$\"3gf*4s%QHCPFB7$Fer$\"3X7`OaT#Hu$FB7$Fjr$\"3G[)RbM'**\\PFB7$F_ s$\"3ZFN&)z9.VPFB7$Fds$\"3w&G:x@'4DPFB7$Fis$\"3')zxNlLJ$p$FB7$F^t$\"3! 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This means tha t the function can be applied iteratively to numbers between 0 and 1. " }}{PARA 0 "" 0 "" {TEXT -1 100 "The calculation to be performed with 15 digit precision and then the result is rounded to 10 digits." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "logistic := a -> (x -> a*x*(1-x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)logisticGf*6#%\"aG6\"6$%)operatorG%&arrowGF(f*6#%\"xGF(F)F(*( T$\"\"\"9$F1,&F1F1F2!\"\"F1F(F(6$F'F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "xx := 1/3;\nseqn \+ := evalf[14](xx):\nfor i from 1 to 11 do\n xx := evalf[14](logistic( 2.1)(xx));\n seqn := seqn,xx;\nend do:\nevalf(seqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG#\"\"\"\"\"$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6.$\"+LLLLL!#5$\"+nmmmYF%$\"+nmmE_F%$\"+n1@R_F%$\"+MM)zB&F%$\"+;k5 Q_F%$\"+?T4Q_F%$\"+]`4Q_F%$\"+F_4Q_F%$\"+R_4Q_F%$\"+Q_4Q_F%F8" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "It looks as though the sequence of successive i terates converges to the value .5238095238." }}{PARA 0 "" 0 "" {TEXT -1 48 "The Maple operation for iterating a function is " }{TEXT 0 2 "@ @" }{TEXT -1 33 ". The Maple output notation for " }{TEXT 0 4 "f@@n" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "`@@`(f,n);" "6#-%#@@G6$%\"fG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f := x -> 2*x;\n'(f@@3)(x)'=(f@@3)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$* &\"\"#\"\"\"9$F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%#@@G6$ %\"fG\"\"$6#%\"xG,$*&\"\")\"\"\"F+F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "The following command immediately g ives the result of iteratively applying the logistic function for the \+ parameter value 2.1 twelve times to the starting number " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "evalf(eval f[14]((logistic(2.1)@@11)(1/3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+Q_4Q_!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "A different starting value produces convergence to the same num ber." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf(evalf[14]((logistic(2.1)@@50)(5/7)));\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Q_4Q_!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "However if we change the \+ parameter value, the nature of the sequence changes." }}{PARA 0 "" 0 " " {TEXT -1 98 "For example, if we increase the parameter to 2.3 the co nvergence is towards the value 0.565217391." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "xx := 3/7;\nseqn \+ := evalf[14](xx):\nfor i from 1 to 19 do\n xx := evalf[14](logistic( 2.3)(xx));\n seqn := seqn,xx;\nend do:\nevalf(seqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG#\"\"$\"\"(" }}{PARA 12 "" 1 "" {XPPMATH 20 "66$\"+'G9dG%!#5$\"+hIlKcF%$\"+CD%zl&F%$\"+wcV]cF%$\"+*o%p_cF%$\"+=w,_ cF%$\"+'z?Al&F%$\"+[)f@l&F%$\"+L\"y@l&F%$\"+ZE<_cF%$\"+$Hu@l&F%$\"+*zt @l&F%$\"+ZR<_cF%$\"+.R<_cF%$\"+;R<_cF%$\"+7R<_cF%$\"+8R<_cF%FDFDFD" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Bifurcation" }}{PARA 0 "" 0 "" {TEXT -1 103 "We increase the parameter value to 3.2 and for m a sequence with no particularly special starting point." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "logis tic := a -> (x -> a*x*(1-x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)lo gisticGf*6#%\"aG6\"6$%)operatorG%&arrowGF(f*6#%\"xGF(F)F(*(T$\"\"\"9$F 1,&F1F1F2!\"\"F1F(F(6$F'F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "randomize():\nxx := Float(r and(),-12);\nseqn := xx:\nfor i from 1 to 38 do\n xx := evalf[14](lo gistic(3.2)(xx));\n seqn := seqn,xx;\nend do:\nevalf(seqn);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"--b(*Hni!#7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6I$\"+b(*Hni!#5$\"+EW1'[(F%$\"+uZBAgF%$\"+T:hlwF%$\"+O [CEdF%$\"++>AJyF%$\"+;%Q\\V&F%$\"+W^YRzF%$\"+%Ha]B&F%$\"+L)>B)zF%$\"+# *e%Q:&F%$\"+hgU#*zF%$\"+1O_M^F%$\"+)34U*zF%$\"+?y5J^F%$\"+P%*\\%*zF%$ \"++9bI^F%$\"+Ega%*zF%$\"+6@YI^F%$\"+%[`X*zF%$\"+=yWI^F%$\"+xYb%*zF%$ \"+JbWI^F%$\"+o[b%*zF%$\"+l^WI^F%$\"+**[b%*zF%$\"+1^WI^F%$\"+/\\b%*zF% $\"+(4X/8&F%$\"+0\\b%*zF%$\"+'4X/8&F%Fhn$\"+&4X/8&F%FhnF\\oFhnF\\oFhnF \\o" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 " Notice that this sequence seems to be settling into a repetitive patte rn of period 2. Let's increase the value of the parameter a little to \+ 3.5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "xx := sqrt(2)/3;\nseqn := evalf[14](xx):\nfor i from 1 to 38 do\n xx := evalf[14](logistic(3.5)(xx));\n seqn := seqn,x x;\nend do:\nevalf(seqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$* &\"\"$!\"\"\"\"##\"\"\"F)F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6I$\"+3_/ 9Z!#5$\"+]/Q@()F%$\"+TY'H!RF%$\"+)pz(G$)F%$\"+.!H<([F%$\"+$HTUu)F%$\"+ ***HK%QF%$\"+3\"f;G)F%$\"+D(\\2)\\F%$\"+*Hq)\\()F%$\"+eaYGQF%$\"+'QF'p #)F%$\"+lyL3]F%$\"+oc(*\\()F%$\"+r))=GQF%$\"+Q0Sp#)F%$\"+Wq&)3]F%$\"+V D(*\\()F%$\"+tq>GQF%$\"+lsSp#)F%$\"+Z;%)3]F%$\"+RE(*\\()F%$\"+Bo>GQF%$ \"+gqSp#)F%$\"+<@%)3]F%$\"+OE(*\\()F%$\"+Io>GQF%$\"+mqSp#)F%$\"+.@%)3] F%FVFXFZFfnFVFXFZFfnFVFX" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Now we get a repeating pattern of 4 numbers or a 4-cycle." }}{PARA 0 "" 0 "" {TEXT -1 17 "If the parameter " }{TEXT 268 1 "a" }{TEXT -1 68 " i s increased again we would see period 8, and then period 16, etc. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "This proc ess is called " }{TEXT 259 15 "period doubling" }{TEXT -1 11 " is call ed " }{TEXT 259 11 "bifurcation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "It culminates at the " }{TEXT 259 16 "Feigenbaum Point" } {TEXT -1 8 " (about " }{XPPEDIT 18 0 "a = 3" "6#/%\"aG\"\"$" }{TEXT -1 18 ".5699456718...). \n" }}{PARA 0 "" 0 "" {TEXT -1 66 "Beyond the \+ Feigenbaum point other cycle lengths are possible, but " }{TEXT 259 5 "chaos" }{TEXT -1 14 " predominates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 "A procedure for displaying the bifurcation diagram of the logistic family: " }{TEXT 0 15 "logisticdi splay" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 15 "logisticdisplay" }{TEXT -1 28 " constructs sections of the " }{TEXT 259 19 "bifurcation diagram" }{TEXT -1 24 " of the logistic \+ family." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "logisticdisplay: usage" }}{PARA 256 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT 260 17 "Calling Sequence:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " logisticdisplay( rn g ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 12 " Parameters: " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 " rng - a range of real values left..righ t over which to display the bifurcation diagram" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 214 "For each of a \+ number of parameter values within a specified subinterval of the inter val from 0 to 4 a sequence of 250 successive iterates of the number 0. 25 is first performed, but the actual values are not saved. " }}{PARA 0 "" 0 "" {TEXT -1 111 "The sequence of the next \"maxcycle\" values i s then formed into a set so that any duplicate numbers are removed." } }{PARA 0 "" 0 "" {TEXT -1 85 "This set of values is plotted as points \+ on a graph at the associated parameter value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "maxcycle=n" }}{PARA 0 "" 0 "" {TEXT -1 60 "The maximum cycle length to record. The default value is 16." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "numpoints=n" }}{PARA 0 "" 0 "" {TEXT -1 96 "The number of equally spaced parameter values (less one) used to cover the interval left..right." }}{PARA 0 "" 0 "" {TEXT -1 25 "The default value is 200 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" } {TEXT -1 154 "To make the procedure active open the subsection, place \+ the cursor anywhere after the prompt [ > and press [Enter].\nYou can \+ then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 " logisticdisplay: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1531 "logisticdisplay := proc(rn g::realcons..realcons)\n local x,d,seqn,pts,ss,i,j,left,right,a,n,ma xlen,Options;\n\n maxlen := 16;\n n := 200;\n if nargs>1 then\n \+ Options:=[args[2..nargs]];\n if not type(Options,list(equati on)) then\n error \"each optional argument must be an equation \"\n end if;\n if hasoption(Options,'maxcycle','i','Option s') then\n if type(i,posint) then\n maxlen := i;\n \+ else\n error \"\\\"maxcycle\\\" must be a positive \+ integer\"\n end if;\n end if;\n if hasoption(Options ,'numpoints','n','Options') then\n if not type(n,posint) then \n error \"\\\"numpoints\\\" must be a positive integer\"\n end if;\n end if;\n if nops(Options)>0 then\n \+ error \"%1 is not a valid option for %2\",op(1,Options),procname;\n end if;\n end if;\n \n left := evalf(op(1,rng));\n right := evalf(op(2,rng));\n if right>4 or left<1 then\n error \"the interval, %1, should be a subinterval of 1..4 %1\",rng;\n end if;\n x := 0.25;\n d := evalf((right-left)/n);\n pts := NULL;\n for j from 0 to n do\n a := left+j*d;\n for i from 1 to 250 do \n x := evalf(a*x*(1-x),15);\n end do;\n seqn := NUL L:\n for i from 1 to maxlen do\n x := evalf(a*x*(1-x),15) ;\n seqn := seqn,x;\n end do;\n ss := evalf(\{seqn\} ):\n pts := pts,seq([a,ss[i]],i=1..nops(ss));\n end do;\n plo ts[pointplot]([pts],color=red,symbol=POINT,\n \+ view=[left..right,0..1]);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "The bifurcation diagram of the logistic family" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 104 "The following bifurcation diagram co vers parameter values from 1 to the Feigenbaum point at around 3.57." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "logisticdisplay(1..3.57);" }}{PARA 13 "" 1 "" {GLPLOT2D 376 267 267 {PLOTDATA 2 "6&-%'POINTSG6]\\m7$$\"\"\"\"\"!$\"+Sy\"QF,7$F'$\"+J*ytz$F,7$F'$\"+X)eHy$F, 7$F'$\"+n!['oPF,7$F'$\"+f`WaPF,7$F'$\"+*\\\\.u$F,7$F'$\"+%Gfjs$F,7$F'$ \"+JNZ7PF,7$F'$\"+r5p)p$F,7$F'$\"+^9VrOF,7$F'$\"+U?&zl$F,7$F'$\"+H9dWO F,7$F'$\"+c2,&o$F,7$$\"+++&G,\"!\"*$\"+tPws6!#67$Fen$\"+.L!R<\"Fjn7$Fe n$\"++;\"Fjn7$Fen$\"+Lj=j6Fjn7$Fen$\"+#RHW;\"Fjn7 $Fen$\"+>\"fc;\"Fjn7$Fen$\"+Ec(o;\"Fjn7$Fen$\"+E!z!o6Fjn7$Fen$\"+G%p#p 6Fjn7$Fen$\"+ZpWq6Fjn7$Fen$\"+\"p6;<\"Fjn7$$\"+++qD5Fgn$\"+:\"><]#Fjn7 $Fiq$\"+^)==]#Fjn7$Fiq$\"+'*QS,DFjn7$Fiq$\"+cnh,DFjn7$Fiq$\"+2<^,DFjn7 $Fiq$\"+Kg\">]#Fjn7$Fiq$\"+()H5-DFjn7$Fiq$\"+')G>-DFjn7$Fiq$\"+#[!G-DF jn7$Fiq$\"+LeO-DFjn7$Fiq$\"+&**[C]#Fjn7$Fiq$\"+E+`-DFjn7$Fiq$\"+!)*3E] #Fjn7$Fiq$\"+4fo-DFjn7$Fiq$\"+m3w-DFjn7$Fiq$\"+A2,-DFjn7$$\"+++bQ5Fgn$ \"+Q9$=r$Fjn7$F[u$\"+8V$=r$Fjn7$F[u$\"+yq$=r$Fjn7$F[u$\"+O(R=r$Fjn7$F[ u$\"+\"HU=r$Fjn7$F[u$\"+[Z%=r$Fjn7$F[u$\"+5r%=r$Fjn7$F[u$\"+\"Q\\=r$Fj n7$F[u$\"+k:&=r$Fjn7$F[u$\"+kO&=r$Fjn7$F[u$\"+)f6=r$Fjn7$F[u$\"+Q_\"=r $Fjn7$F[u$\"+Q(==r$Fjn7$F[u$\"+-@#=r$Fjn7$F[u$\"+P`#=r$Fjn7$F[u$\"+[%G =r$Fjn7$$\"+++S^5Fgn$\"+xzr))[Fjn7$F]x$\"+yyr))[Fjn7$F]x$\"+r!=())[Fjn 7$F]x$\"+g\"=())[Fjn7$F]x$\"+W#=())[Fjn7$F]x$\"+C$=())[Fjn7$F]x$\"++%= ())[Fjn7$F]x$\"+s%=())[Fjn7$F]x$\"+T&=())[Fjn7$F]x$\"+=qr))[Fjn7$F]x$ \"+hrr))[Fjn7$F]x$\"+(H<())[Fjn7$F]x$\"+Eur))[Fjn7$F]x$\"+[vr))[Fjn7$F ]x$\"+kwr))[Fjn7$F]x$\"+uxr))[Fjn7$$\"+++Dk5Fgn$\"+w_6PgFjn7$F_[l$\"+! 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1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Finding a \+ parameter value for the 3-cycle" }}{PARA 0 "" 0 "" {TEXT -1 33 "We try to find a parameter value " }{TEXT 269 1 "t" }{TEXT -1 89 " such that the associated logistic function produces a 3-cycle which includes th e number " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "The problem is to find " }{TEXT 270 1 "t" }{TEXT -1 10 " so that " }{XPPEDIT 18 0 "`@@`(logistic(t),3 )(1/2)-1/2;" "6#,&--%#@@G6$-%)logisticG6#%\"tG\"\"$6#*&\"\"\"F/\"\"#! \"\"F/*&F/F/F0F1F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "logistic := a -> (x -> a*x*( 1-x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)logisticGf*6#%\"aG6\"6$%) operatorG%&arrowGF(f*6#%\"xGF(F)F(*(T$\"\"\"9$F1,&F1F1F2!\"\"F1F(F(6$F 'F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "h3 := unapply((logistic(t)@@3)(1/2)-1/2,t);\nplot(h 3(t),t=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h3Gf*6#%\"tG6\"6$% )operatorG%&arrowGF(,&*&#\"\"\"\"\"%F/*()9$\"\"$F/,&F/F/*&#F/F0F/F3F/! \"\"F/,&F/F/*&#F/F0F/*&)F3\"\"#F/F5F/F/F8F/F/F/#F/F>F8F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 308 179 179 {PLOTDATA 2 "6%-%'CURVESG6$7^o7$$\"\" !F)$!3++++++++]!#=7$$\"3Hmmmm;')=()!#>$!3s'*y&G9#Q)*\\F,7$$\"3RLLLe'40 j\"F,$!3K3NG&*4n*)\\F,7$$\"3mmmm;6m$[#F,$!3;O>B)='fk\\F,7$$\"3fmmm;yYU LF,$!3N6kg!)\\j;\\F,7$$\"3%HLL$eF>(>%F,$!3E*oMgUp5%[F,7$$\"3Qmmm\">K'* )\\F,$!3\"*fa*fV()Hu%F,7$$\"3P*****\\Kd,\"eF,$!3cH&30a66h%F,7$$\"3-mmm \"fX(emF,$!35\"R2=v6;W%F,7$$\"3.*****\\U7Y](F,$!34So@%=\"zRUF,7$$\"3'Q LLLV!pu$)F,$!3A'\\pMCt***RF,7$$\"3xmmm;c0T\"*F,$!3!\\(*zU%>DkPF,7$$\"3 #*******H,Q+5!#<$!350]\"ovS_Z$F,7$$\"3)*******\\*3q3\"Fbo$!3W,OR>JkkJF ,7$$\"3)*******p=\\q6Fbo$!39]Cs))><^GF,7$$\"3mmm;fBIY7Fbo$!3+%))*f%Q]* eDF,7$$\"3GLLLj$[kL\"Fbo$!3mhR\\c=!y?#F,7$$\"3?LLL`Q\"GT\"Fbo$!3\\S0fg [z6>F,7$$\"3!*****\\s]k,:Fbo$!3O%\\Vj_`Vd\"F,7$$\"39LLL`dF!e\"Fbo$!3UM pWZez&G\"F,7$$\"33++]sgam;Fbo$!3r)oLf&zmQ)*F07$$\"3/++] .y#G8(F07$$\"3QLLLe/TM=Fbo$!3LE#z3!*oO]%F07$$\"3JLL$eDBJ\">Fbo$!3-v@#[ f?)pAF07$$\"3immmTc-)*>Fbo$!3Pw3wUKxS\\!#@7$$\"3Mmm;f`@'3#Fbo$\"31M:&R A&)p1#F07$$\"3y****\\nZ)H;#Fbo$\"3Oxv'y&*=gx$F07$$\"3YmmmJy*eC#Fbo$\"3 %Q\\;VO$)o^&F07$$\"3')******R^bJBFbo$\"3)G%\\7V)4jC(F07$$\"3f*****\\5a `T#Fbo$\"3#4Dp_&41A*)F07$$\"3o****\\7RV'\\#Fbo$\"3#G?m*\\@#z0\"F,7$$\" 3k*****\\@fke#Fbo$\"3MR$GFbo$\"3+qp>36/ t=F,7$$\"3$*******pfa)pDJF,7$$\"31LLe**o4#H$Fb o$\"3F9oYlFU;KF,7$$\"3xmmm')fdLLFbo$\"3]eC/W='zG$F,7$$\"3mmm;WV*fP$Fbo $\"3wdOAdv*fL$F,7$$\"3bmmm,FT=MFbo$\"3iphGy\\^_LF,7$$\"37L$3xDY%RMFbo$ \"3a&yV'Q;!pM$F,7$$\"39++v8)z/Y$Fbo$\"3M(He/i94L$F,7$$\"3qm;zpL^\"[$Fb o$\"3*pHyk`tNI$F,7$$\"3FLL$e#pa-NFbo$\"3)>x#z/=%QE$F,7$$\"3emm\"HB-7a$ Fbo$\"3'*\\8t)))=[:$F,7$$\"3!*******Rv&)zNFbo$\"3wui5$4wG*HF,7$$\"3#om mT)3;COFbo$\"3S)\\*f_EvJFF,7$$\"3ILLLGUYoOFbo$\"3w4.uD)*4xBF,7$$\"3\"* ****\\n'*33PFbo$\"3mcA$Qrp&o>F,7$$\"3_mmm1^rZPFbo$\"3?Y4g/vth9F,7$$\"3 3LLe*3k**y$Fbo$\"3?QA10\")z()zF07$$\"34++]sI@KQFbo$!3O(yBd()[B3(Fas7$$ \"35++D1>V_QFbo$!3MpY(3(fm&[%F07$$\"33+++S2lsQFbo$!3Un!*=6Ob!H*F07$$\" 33++vt&pG*QFbo$!3oW*z`Ne2X\"F,7$$\"34++]2%)38RFbo$!3'G\"fQnG(f,#F,7$$ \"31+]i0j\"[$RFbo$!3C\\x\"3L1Yn#F,7$$\"3/++v.UacRFbo$!3KdK6F%G$*Q$F,7$ $\"3E+D\"G:3u'RFbo$!3i;?)G9?(oPF,7$$\"3-+](=5s#yRFbo$!3OR:1VqMjTF,7$$ \"3!)*\\P40O\"*)RFbo$!3c7L+V/jtXF,7$$\"\"%F)F*-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!Fh`l-%%VIEWG6$;F(F[`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "It looks \+ as though one parameter value which will work is 2." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "seq((logis tic(2)@@j)(1/2),j=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,#\"\"\"\" \"#F#F#F#F#F#F#F#F#F#" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "This value produces a constant sequence." }}{PARA 0 "" 0 "" {TEXT -1 29 "There is another solution of " }{XPPEDIT 18 0 "`@ @`(logistic(t),3)(1/2) = 1/2;" "6#/--%#@@G6$-%)logisticG6#%\"tG\"\"$6# *&\"\"\"F/\"\"#!\"\"*&F/F/F0F1" }{TEXT -1 5 " for " }{TEXT 271 1 "t" } {TEXT -1 10 " near 3.8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf[14](fsolve((logistic(t)@@3)(1 /2)=1/2,t=3.8..3.9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/LGbS(=$Q! #8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Le t's check that this value works." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalf(evalf[14](seq((logisti c(3.83187405528332)@@j)(1/2),j=1..10)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+Q^oz&*!#5$\"+rz*Ga\"F%$\"+++++]F%F#F&F(F#F&F(F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The 3 associated \+ 3-cycle is: 0.5, 0.9579685138, 0.1542897971." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "If we start with a number between 0 and 1 different from " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$ \"\"#!\"\"" }{TEXT -1 113 ", the associated sequence converges to the \+ same 3-cycle. In fact the 3-cycle depends only on the parameter value. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "xx := 0.4;\nseqn := xx:\nfor i from 1 to 53 do\n xx := eval f[14](logistic(3.83187405528332)(xx));\n seqn := seqn,xx;\nend do:\n evalf(seqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"%!\"\"" }} {PARA 12 "" 1 "" {XPPMATH 20 "6X$\"\"%!\"\"$\"+Lx\\'>*!#5$\"+7w_JGF($ \"+SN$yx(F($\"+&[tGi'F($\"+QuZq&)F($\"+POp%p%F($\"+$[nRa*F($\"+\\$pxm \"F($\"+EP'[K&F($\"+.\\CR&*F($\"+Jm?%o\"F($\"+,ntm`F($\"+Q![\"G&*F($\" +3GwA7&>C#F($\"+2D%[m'F($\"+(Q/w^)F($\"+4-IQ[F($\"+0gmp&*F($\"+6U-y:F($\" +D>f#4&F($\"+r**Rw&*F($\"+oRUa:F($\"+&['[I]F($\"+%**G$z&*F($\"+,z9W:F( $\"+75J.]F($\"+P4oz&*F($\"+:F\"Ha\"F($\"+j!R++&F($\"+Q^oz&*F($\"+tz*Ga \"F($\"+0+++]F(Fgo$\"+rz*Ga\"F($\"+++++]F(FgoF]pF_pFgoF]pF_pFgoF]pF_pF goF]pF_p" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We can go straight to the 3-cycle as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "xx := evalf [14]((logistic(3.83187405528332)@@50)(0.4)):\nevalf(evalf[14](seq((log istic(3.83187405528332)@@j)(xx),j=1..3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+Q^oz&*!#5$\"+rz*Ga\"F%$\"+++++]F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Finding a parameter value for the 1st 5-cycle" }}{PARA 0 "" 0 "" {TEXT -1 33 "We try to find a paramete r value " }{TEXT 272 1 "t" }{TEXT -1 89 " such that the associated log istic function produces a 5-cycle which includes the number " } {XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 23 "The problem is to find " }{TEXT 273 1 "t " }{TEXT -1 10 " so that " }{XPPEDIT 18 0 "`@@`(logistic(t),5)(1/2)-1 /2;" "6#,&--%#@@G6$-%)logisticG6#%\"tG\"\"&6#*&\"\"\"F/\"\"#!\"\"F/*&F /F/F0F1F1" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "logistic := a -> (x -> a*x*(1-x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)logisticGf*6#%\"aG6\"6$%)operat orG%&arrowGF(f*6#%\"xGF(F)F(*(T$\"\"\"9$F1,&F1F1F2!\"\"F1F(F(6$F'F2F(F (F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "h5 := unapply((logistic(t)@@5)(1/2)-1/2,t);\nplot(h5( t),t=0..4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#h5Gf*6#%\"tG6\"6$%)o peratorG%&arrowGF(,&*,)9$\"\"&\"\"\",&F1F1*&#F1\"\"%F1F/F1!\"\"F1,&F1F 1*&#F1F5F1*&)F/\"\"#F1F2F1F1F6F1,&F1F1*&#F1F5F1*()F/\"\"$F1F2F1F7F1F1F 6F1,&F1F1*&#F1F5F1**)F/F5F1F2F1F7F1F=F1F1F6F1#F1F5#F1F$!3[%)zW.x)***\\F,7$$\"3 .+++e'40j\"F,$!37r:#osD(**\\F,7$$\"3/+++<6m$[#F,$!3Gr>#)Gd#y*\\F,7$$\" 3&******p\"yYULF,$!3cF%y![&*y!*\\F,7$$\"37+++eF>(>%F,$!3=dwZdzis\\F,7$ $\"3A+++\">K'*)\\F,$!3HN$4Q_O%Q\\F,7$$\"3[+++Dt:5eF,$!3;`koLZcw[F,7$$ \"3S+++\"fX(emF,$!3Sz)=eyZWx%F,7$$\"39+++DCh/vF,$!3T,2=xvDDYF,7$$\"3[+ ++L/pu$)F,$!3QN*pAmWjT%F,7$$\"30+++;c0T\"*F,$!3SM%)3Vqh%=%F,7$$\"3#*** ****H,Q+5!#<$!3;\")*=d\"R$R(QF,7$$\"3)*******\\*3q3\"Fbo$!3L\"QT/o)zFbo $!3C#Q75+U0F#F07$$\"3)******>kD!)*>Fbo$!3_NIyECxS\\!#@7$$\"35+++f`@'3# Fbo$\"3#H.H%e^Jm?F07$$\"3?+++nZ)H;#Fbo$\"3WCmitM\"yw$F07$$\"3#)*****>$ y*eC#Fbo$\"3E7R\\`Z&pZ&F07$$\"3')******R^bJBFbo$\"3B*ect\">BDrF07$$\"3 .+++0TN:CFbo$\"33/JkQLna')F07$$\"33+++7RV'\\#Fbo$\"3M(\\u#=b?55F,7$$\" 33+++:#fke#Fbo$\"3;rU(y7)4v6F,7$$\"39+++`4NnEFbo$\"34))G2NEYO8F,7$$\"3 #*******\\,s`FFbo$\"3a\")>zYzLM:F,7$$\"3!)******yM)>$GFbo$\"3%oo$)G2)p X\"yh]KFbo$\"3GD_H#Had7$F,7$$\"38+++()fdLLFb o$\"3$pupeee'4LF,7$$\"3\"******>q7%=MFbo$\"3'[Q7J/sg^$F,7$$\"3;+++9)z/ Y$Fbo$\"3)GK;EwU*QOF,7$$\"3&******f#pa-NFbo$\"3UF;y+X9cPF,7$$\"3>+++`2 @7NFbo$\"3?:3@^0ixPF,7$$\"3*)******yX(=_$Fbo$\"39'pdM*4H&z$F,7$$\"38++ +1%Q:`$Fbo$\"3;%[L\\R%33QF,7$$\"3#******HB-7a$Fbo$\"3h:PR*H%z9QF,7$$\" 33+++fg'3b$Fbo$\"3lo\")3\"\\yS\"QF,7$$\"3')*****f))H0c$Fbo$\"36t\\(p:h W!QF,7$$\"35+++8P>qNFbo$\"3#H4>n`QVy$F,7$$\"3!*******Rv&)zNFbo$\"3oDk_ z'))>v$F,7$$\"39+++%)3;COFbo$\"3M*o`AeITR$F,7$$\"3%******zAk%oOFbo$\"3 3$H_9\\_gb#F,7$$\"3%******z%pF)o$Fbo$\"3e([cL?sr)>F,7$$\"3&******zm*33 PFbo$\"3s]$z_=ZOH\"F,7$$\"3&******z-'*zr$Fbo$\"3JwZh>_0K!*F07$$\"3'*** ***zQ-zs$Fbo$\"3y%zz^!))Qs[F07$$\"3'******zu3yt$Fbo$\"3[!o,-OsF*\\!#?7 $$\"3))*****p5:xu$Fbo$!3k#p63&H%=.%F07$$\"33+++atFePFbo$!3ionFH;Vd*)F0 7$$\"35+++*fR)oPFbo$!3eTK1&>wwQ\"F,7$$\"3?+++X=SzPFbo$!3WE(=vm=i'=F,7$ $\"3y*******3k**y$Fbo$!3o\"*G#ow^jJ#F,7$$\"3#******4e)36QFbo$!3k]4%)y> VgIF,7$$\"30+++sI@KQFbo$!3#\\Y&*RCt)fMF,7$$\"3!*******zxEPQFbo$!3#\\N. K6][[$F,7$$\"3%)******)[AB%QFbo$!3M(p(*f[YvZ$F,7$$\"39+++(>xt%QFbo$!3r _4y#QNcV$F,7$$\"33+++1>V_QFbo$!3a\\d*\\SuJ@\"3$F,7$$\"33+++S2lsQFbo$!3,`%=j94vj#F,7$$\"3))*****p:gF)QFbo $!38(3X#H`+>?F,7$$\"35+++u&pG*QFbo$!3`z8&o/QzA\"F,7$$\"3/+++$GCz*QFbo$ !3&zDEA+*\\\\-#F,7$$ \"31+++JVQHRFbo$\"3>B-!e'QTQEF,7$$\"35+++1j\"[$RFbo$\"3pP,DI)zyA$F,7$$ \"38+++\"G[-%RFbo$\"3kW0>WRcrPF,7$$\"33+++b-oXRFbo$\"3FW3*>?1NC%F,7$$ \"37+++IA6^RFbo$\"37dH.-#))Hh%F,7$$\"31+++/UacRFbo$\"3MD:&onAR%[F,7$$ \"3#)*****4>g#fRFbo$\"3U3Cxv^_%*[F,7$$\"3-+++yh(>'RFbo$\"3Unw!H&4=%*[F ,7$$\"3y*****\\;#pkRFbo$\"3sH\\1D#el$[F,7$$\"31+++`\"3u'RFbo$\"3r!4YTy ][r%F,7$$\"3++++F,%G(RFbo$\"3=K,?4+U\\UF,7$$\"3/+++-@FyRFbo$\"3oYFl/$y GV$F,7$$\"3!)******)3))4)RFbo$\"3eiV?241qGF,7$$\"33+++xSq$)RFbo$\"3O/a z)[J2>#F,7$$\"3#)*****R1?k)RFbo$\"3E<[e&R>RQ\"F,7$$\"3-+++^g8*)RFbo$\" 39]3#zG'>zVF07$$\"3')******Q?&=*RFbo$!3s!H@+liuf'F07$$\"3/+++E!oX*RFbo $!3jm?y4-PA>F,7$$\"3!)*****H,%G(*RFbo$!3!>WH " 0 "" {MPLTEXT 1 0 56 "evalf[14](fsolve((logistic(t)@@5)(1/2)=1/2,t=3.7..3.8 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/2(H\"\\\"*QP!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Let's check tha t this works." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "xx := 0.5;\nseqn := xx:\nfor i from 1 to 10 do \n xx := evalf[14](logistic(3.7389149129707)(xx));\n seqn := seqn, xx;\nend do:\nevalf(seqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$ \"\"&!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$\"\"&!\"\"$\"+#G(GZ$*!# 5$\"+no9\"G#F($\"+t1U$e'F($\"+t'e)4%)F($\"+++++]F(F&F)F+F-F/" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The 3 ass ociated 3-cycle is: 0.5, 0.9347287282, 0.2281146867, 0.6583420673, 0. 8409858673" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "If we start with a number between 0 and 1 different from " } {XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 58 ", the asso ciated sequence still converges to this 5-cycle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "xx := 0.6; \nseqn := xx:\nfor i from 1 to 43 do\n xx := evalf[14](logistic(3.73 891491297068)(xx));\n seqn := seqn,xx;\nend do:\nevalf(seqn);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"'!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6N$\"\"'!\"\"$\"+\"z&Rt*)!#5$\"+'[NVW$F($\"+XaVU%)F($\"+Q y`;\\F($\"+!y#oW$*F($\"+?6h*G#F($\"+iEg+mF($\"+sQS*Q)F($\"+;***>0&F($ \"+$GwiM*F($\"+pIV%G#F($\"+`y4!f'F($\"+be$>S)F($\"+/%y,-&F($\"+Y]8Z$*F ($\"+` " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Finding parameter values for the \+ other two 5-cycles" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "There are two more solutions for the equation " } {XPPEDIT 18 0 "`@@`(logistic(t),5)(1/2)-1/2;" "6#,&--%#@@G6$-%)logisti cG6#%\"tG\"\"&6#*&\"\"\"F/\"\"#!\"\"F/*&F/F/F0F1F1" }{TEXT -1 55 " in \+ addition to the value obtained in the last section." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "logistic : = a -> (x -> a*x*(1-x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)logisti cGf*6#%\"aG6\"6$%)operatorG%&arrowGF(f*6#%\"xGF(F)F(*(T$\"\"\"9$F1,&F1 F1F2!\"\"F1F(F(6$F'F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "h5 := unapply((logistic(t)@@ 5)(1/2)-1/2,t):\nplot(h5(t),t=3.7..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 308 179 179 {PLOTDATA 2 "6%-%'CURVESG6$7co7$$\"3;+++++++P!#<$\"3qN%*\\ TdX\"f\"!#=7$$\"3O++DY\"Rlq$F*$\"3yPsuk)RAN\"F-7$$\"3'**\\PC#)GAr$F*$ \"3Ls:Ol,$Q8\"F-7$$\"3I+]Peui=PF*$\"3U!z]&*)z:u()!#>7$$\"3;+]i3&o]s$F* $\"3UMI(3V+i3'F=7$$\"3%**\\(oX*y9t$F*$\"3:G.?g7c:LF=7$$\"3C+vVTAUPPF*$ \"3!4mgx7nEt'!#?7$$\"3.+v$*zhdVPF*$!3h2JJ<@mB@F=7$$\"3'**\\P>fS*\\PF*$ !3+xAI1Kjk]F=7$$\"3'**\\(=$f%GcPF*$!3sq9\\B\"pYP\"F-7$$\"3E++v4&G]x$F*$ !3?Eb+dGhq;F-7$$\"3G++Drc_\"y$F*$!3gW+n7[`f>F-7$$\"3/++D!*oy(y$F*$!3Ha b@79qEAF-7$$\"3A+v$pnsMz$F*$!3u!)Rc&H*ocCF-7$$\"3\"****\\siL-!QF*$!3N` 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3K4N⪙p*[F-7$$\"3o7GQWZKgRF*$\"3'zqMP4,3!\\F-7$$\"37vVt-N6hRF*$\"3hD +\\ap=+\\F-7$$\"3ePf3hA!>'RF*$\"3cNvR/)=\\*[F-7$$\"3-+vV>5piRF*$\"3=)y bn*y$[)[F-7$$\"3'[7`MfST'RF*$\"3ys!*))z\"=K&[F-7$$\"3s\\(ou;!flRF*$\"3 ]CKW?qv.[F-7$$\"3+vV[T(Rq'RF*$\"3F!*)Q$fvPNZF-7$$\"3%)****\\:$*[oRF*$ \"3#[h\"H![hpk%F-7$$\"3%)*\\7j17=(RF*$\"3YO^YlPMiVF-7$$\"3%)**\\7<[8vR F*$\"3y\"[gc/$R^RF-7$$\"3=+D1Dn5yRF*$\"3')Q@*el;OY$F-7$$\"35+++L'y5)RF *$\"3E>R'G8]$\\GF-7$$\"3'\\Pf)pHm#)RF*$\"3[BrGz'*elCF-7$$\"3%)\\(=nIZU )RF*$\"3=uR\\>B#*R?F-7$$\"39D\"yNkJe)RF*$\"3sZ!4(zB6q:F-7$$\"3-+vV!)fT ()RF*$\"3Wsc**f$QQ0\"F-7$$\"39](oHRK*))RF*$\"3!\\_iLHa&R^F=7$$\"3!)*** *\\0)[/*RF*$!35,R+!f]\"*G(FM7$$\"3#*\\7.=_'>*RF*$!3%)3Yv*fE(*3(F=7$$\" 3/+DcI;[$*RF*$!3wvOw3%RmR\"F-7$$\"3Ev=#HA6^*RF*$!3')3[TI%>f>#F-7$$\"3- ]7G:3u'*RF*$!3U6!phh=21$F-7$$\"3=P4Y6cb(*RF*$!37Q4.!3*p=NF-7$$\"3yC1k2 /P)*RF*$!3s+5FfyG%*RF-7$$\"3S7.#Q?&=**RF*$!3[iab)R5z[%F-7$$\"\"%\"\"!$ !3++++++++]F--%'COLOURG6&%$RGBG$\"#5!\"\"$FfalFfalF`bl-%+AXESLABELSG6$ Q\"t6\"Q!Febl-%%VIEWG6$;$\"#PF_blFdal%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "There are solutions for the equation " }{XPPEDIT 18 0 "`@@`(log istic(t),5)(1/2) = 1/2" "6#/--%#@@G6$-%)logisticG6#%\"tG\"\"&6#*&\"\" \"F/\"\"#!\"\"*&F/F/F0F1" }{TEXT -1 8 " , with " }{TEXT 275 1 "t" } {TEXT -1 19 " near 3.9 and with " }{TEXT 276 1 "t" }{TEXT -1 11 " near 3.99." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "evalf[20](fsolve((logistic(t)@@5)(1/2)=1/2,t=3.9..3. 91));\nevalf[20](fsolve((logistic(t)@@5)(1/2)=1/2,t=3.98..4));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5[N!HJ)pkq0R!#>" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"5$=:qtp/n-*R!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Let's check that these parameter values work." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "xx := 0.5;\nseqn := xx:\nfor i from 1 to 10 do\n x x := evalf[20](logistic(3.9057064698312903548)(xx));\n seqn := seqn, xx;\nend do:\nevalf(seqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$ \"\"&!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$\"\"&!\"\"$\"+vhEk(*!#5 $\"+e%H+**)!#6$\"+m*zb>$F($\"+R[f#\\)F($\"+++++]F(F&F)F,F.F0" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The assoc iated 5-cycle is: 0.5000000000, 0.9764266175, 0.08990029458, 0.319557 9966, 0.8492594839" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 142 "xx := 0.5;\nseqn := xx:\nfor i from 1 to 10 do\n xx := evalf[20](logistic(3.9902670469737015183)(xx));\n seqn := seqn,xx;\nend do:\nevalf(seqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#xxG$\"\"&!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$\"\"&!\"\"$\"+< wmv**!#5$\"+kak&o*!#7$\"+3yRFQ!#6$\"+<1yo9F($\"+++++]F(F&F)F,F/F1" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The assoc iated 5-cycle is: 0.5000000000, 0.9975667617, 0.009685645464, 0.038273 97808, 0.1468780617 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 99 "Find all parameter values which produce 7-cycle s for the logistic family, and check that they work." }}{PARA 0 "" 0 " " {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 107 "Find all the parameter values which prod uce 8-cycles, ruling out those which produce 2-cycles and 4-cycles." } }{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }