{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 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" 0 13 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Exercise T1.5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Define the map f." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "f := x -> 2*x^2-5*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\" \"\"F0*&\"\"&F1F/F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Find fixed points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solv e(f(x)=x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Define a function for the second iterate \+ of f." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f2 := x -> f(f(x)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gf*6#%\"xG6\"6$%)operatorG%& arrowGF(-%\"fG6#-F-6#9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 " Note that f^2(x) is a degree 4 polynomial." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "f2(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$),&*$) %\"xG\"\"#\"\"\"F**&\"\"&F+F)F+!\"\"F*F+F**&\"#5F+F(F+F.*&\"#DF+F)F+F+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Find the fixed points of f2. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(f2(x)=x,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!\"\"$,&\"\"\"F&*$-%%sqrtG6#\"\"#F &F&,&F&F&F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "We get four \+ fixed points for f2, but two of them are also fixed points of f. The \+ new points are 1 + sqrt(2) and 1-sqrt(2). These are the period 2 poin ts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Le t's check:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x0 := 1+sqrt( 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G,&\"\"\"F&*$-%%sqrtG6#\" \"#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x1 := f(x0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G,(*$),&\"\"\"F)*$-%%sqrtG6#\"\"# F)F)F.F)F.\"\"&!\"\"*&F/F)F+F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x1 := simplify(f(x0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G,&\"\"\"F&*$-%%sqrtG6#\"\"#F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "x2 is f(f(x0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x2 := simplify(f(x1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G,&\"\"\"F&*$-%%sqrtG6#\"\"#F&F&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 55 "We see that f(f(x0))=x0, x0 and x1 are period 2 points. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "id := x -> x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#idGf*6#%\"xG6\"6$%)operatorG%&arrowGF(9$F (F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "p2data := [[x1,x2] ,[x2,x2],[x2,x1],[x1,x1],[x1,x2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'p2dataG7'7$,&\"\"\"F(*$-%%sqrtG6#\"\"#F(!\"\",&F(F(F)F(7$F/F/7$F/F' 7$F'F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot(\{id,f,p2d ata\},-0.6..3.1,scaling=constrained,color=black,thickness=2);" }} {PARA 13 "" 1 "" {GLPLOT2D 394 431 431 {PLOTDATA 2 "6*-%'CURVESG6#7S7$ $!3w**************f!#=$\"3v************>P!#<7$$!3uKL$eH0N>&F*$\"3\"Rkt If-i8$F-7$$!3Ym;/m&y<\\%F*$\"3Igyuw2T\\EF-7$$!3GLL3nMh-PF*$\"33/UFjO\\ D@F-7$$!3\\KLeps@3HF*$\"3)4sL&)=jKi\"F-7$$!3mm;a)p'f<@F*$\"3[CV#[m#[[6 F-7$$!3WL$3FA!f%Q\"F*$\"3\"*ob]I\"pjI(F*7$$!3)****\\PuWgD'!#>$\"3./maj &)H1KF*7$$\"3Ytm\"Hs'R$f\"FO$!3#[L^AN0i\"zFO7$$\"3q-+DJ\\m<%*FO$!3C4/6 SwWJXF*7$$\"3#RLL3l)eYF-7$$\"3>m;HAoHGbF*$!3GI'Qg42H:#F-7$$\" 3WLL$3OZ@O'F*$!3'e\\(f*HN:P#F-7$$\"3eLLLV\"G&oqF*$!3CfZSpA)\\`#F-7$$\" 3c+]i?p@!*yF*$!3)*e@A**z***p#F-7$$\"3>KLL=2b<')F*$!3X>d]^<`BGF-7$$\"3p ,]iq6b:%*F*$!3#RMz$3NsMHF-7$$\"3N+v=JOa<5F-$!3G9q>t!Gp,$F-7$$\"3kL$eRn Ho4\"F-$!3))z0W-x2yIF-7$$\"3sLek6!R'p6F-$!3#H[,JA%37JF-7$$\"3gm;a=P<[7 F-$!3)*p?RHL*\\7$F-7$$\"3zm\"Hs?\\(H8F-$!3')QE*y7!G7JF-7$$\"3-+v$*44w+ 9F-$!3qL'pgHU&zIF-7$$\"3om;H%\\buZ\"F-$!3cmIB'*z_@IF-7$$\"3%)****\\/&) oc:F-$!3*z5:CK%)o$HF-7$$\"38+]7ZD?M;F-$!3jB`c0oxHGF-7$$\"3')*\\i!p8?4< F-$!3?^AI0#oKq#F-7$$\"3)***\\(QxuCz\"F-$!3'[`=%RAWODF-7$$\"3CLL$=j*Hn= F-$!3eio2HB)GO#F-7$$\"31++vQ6>Z>F-$!3+cp4K!\\G:#F-7$$\"3Tm\"H#=Ze>?F-$ !3mpi$Hsy/%>F-7$$\"3E++DA+t)4#F-$!3SGIm)p9Vo\"F-7$$\"3-LeR&e*>t@F-$!3) R#*\\500/U\"F-7$$\"3;+DJOY6F-7$$\"3cL$37THLz#F-$\"3GB8+V 8tQ;F-7$$\"3'pmmOZOm'GF-$\"3H$zyf(p--@F-7$$\"3<+D1#4(zWHF-$\"3D6q_mVn> EF-7$$\"38+v$px1'>IF-$\"3/ai'**G;!QJF-7$$\"33+++++++JF-$\"3T-+++++?PF- -F$6#7S7$F(F(7$F/F/7$F4F47$F9F97$F>F>7$FCFC7$FHFH7$FMFM7$FSFS7$FXFX7$F gnFgn7$F\\oF\\o7$FaoFao7$FfoFfo7$F[pF[p7$F`pF`p7$FepFep7$FjpFjp7$F_qF_ q7$FdqFdq7$FiqFiq7$F^rF^r7$FcrFcr7$FhrFhr7$F]sF]s7$FbsFbs7$FgsFgs7$F\\ tF\\t7$FatFat7$FftFft7$F[uF[u7$F`uF`u7$FeuFeu7$FjuFju7$F_vF_v7$FdvFdv7 $FivFiv7$F^wF^w7$FcwFcw7$FhwFhw7$F]xF]x7$FbxFbx7$FgxFgx7$F\\yF\\y7$Fay Fay7$FfyFfy7$F[zF[z7$F`zF`z7$FezFez-F$6#7'7$$!3Y^4tBc8UTF*$\"3#\\4tBc8 UT#F-7$Fc^lFc^l7$Fc^lFa^l7$Fa^lFa^lF`^l-%*THICKNESSG6#\"\"#-%'COLOURG6 &%$RGBG\"\"!F`_lF`_l-%+AXESLABELSG6$Q!6\"Fd_l-%(SCALINGG6#%,CONSTRAINE DG-%%VIEWG6$;$!\"'!\"\"$\"#JF``l%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The above plot includes the graph \+ of f. The box shows the period-2 orbit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }