{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 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 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 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 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 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 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 "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warni ng" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 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 "Time s" 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 36 "Initial conditions and it erated maps" }}{PARA 19 "" 0 "" {TEXT -1 31 "by Alex Quandt, Uni Greif swald " }}{PARA 0 "" 0 "" {TEXT -1 59 "The following worksheet will de monstrate another effect of " }{TEXT 258 17 "finite precision " } {TEXT -1 4 "for " }}{PARA 0 "" 0 "" {TEXT 256 13 "iterated maps" } {TEXT -1 15 ". Just imagine " }{TEXT 271 17 "small differences" } {TEXT -1 36 " in the initial data for two states." }}{PARA 0 "" 0 "" {TEXT -1 20 "For some maps, even " }{TEXT 275 19 "extremely accurate \+ " }{TEXT -1 35 "arithmetics cannot prevent the two " }}{PARA 0 "" 0 " " {TEXT -1 12 "states from " }{TEXT 276 9 "diverging" }{TEXT -1 49 " o ver time. The following worksheet allows for a " }{TEXT 283 7 "direct \+ " }}{PARA 0 "" 0 "" {TEXT 284 11 "observation" }{TEXT -1 20 " of such \+ a process. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 286 9 "Remark : " }{TEXT -1 4 "for " }{TEXT 273 23 "ma ny dynamical systems " }{TEXT -1 9 "like our " }{TEXT 281 16 "planetar y system" }{TEXT -1 13 ", it becomes " }}{PARA 0 "" 0 "" {TEXT 279 19 "extremely difficult" }{TEXT -1 58 " to follow and predict motion over a long period of time. " }{TEXT 280 1 " " }{TEXT -1 0 "" }{TEXT 272 0 "" }{TEXT -1 5 " " }{TEXT 257 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "General settings " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 " Let us specify a number of things : " }{MPLTEXT 1 0 1 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 "Clear everything + general settings " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "restart; \nwith(plots): \nD igits:=16; " }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name change coords has been redefined\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Digi tsG\"#;" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Specify your map " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 " Initial settings " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Specify some " }{TEXT 260 11 "parameters " }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a:=2. 50000000; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"*+++]#!\")" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Specify your " }{TEXT 261 18 "star ting points :" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x[0]:=0.70000000; \ny[0]:=0.70000001; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!$\")+++q!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"!$\"),++q!\")" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 18 "Iterate your maps " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "nit : number of " }{TEXT 262 16 "recursion steps " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nit:=50; " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$nitG\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 12 "Iterate map:" }}{PARA 0 "" 0 "" {TEXT -1 45 "x[],y [] : functional values of " }{TEXT 282 16 "iterated poin ts " }{TEXT -1 5 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "printlevel:=0; \n\nfor k from 1 to nit do\n\n \n x[k]:= evalf(a*x [k-1]*(1-x[k-1]^2)); \n y[k]:= evalf(a*y[k-1]*(1-y[k-1]^2)); \n if \+ (k mod 10 = 0) then \n printf(\"step = %d\\n\",k);\n printf(\" x = %g\\n y = %g\\n\",x[k],y[k]); \n end if; \n\nend do; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+printlevelG\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 9 "step = 10" }}{PARA 6 "" 1 "" {TEXT -1 12 " x = .948295" }} {PARA 6 "" 1 "" {TEXT -1 12 " y = .948294" }}{PARA 6 "" 1 "" {TEXT -1 9 "step = 20" }}{PARA 6 "" 1 "" {TEXT -1 12 " x = .789894" }}{PARA 6 " " 1 "" {TEXT -1 12 " y = .789736" }}{PARA 6 "" 1 "" {TEXT -1 9 "step = 30" }}{PARA 6 "" 1 "" {TEXT -1 12 " x = .881887" }}{PARA 6 "" 1 "" {TEXT -1 12 " y = .890603" }}{PARA 6 "" 1 "" {TEXT -1 9 "step = 40" }} {PARA 6 "" 1 "" {TEXT -1 12 " x = .598137" }}{PARA 6 "" 1 "" {TEXT -1 12 " y = .329388" }}{PARA 6 "" 1 "" {TEXT -1 9 "step = 50" }}{PARA 6 " " 1 "" {TEXT -1 12 " x = .72954 " }}{PARA 6 "" 1 "" {TEXT -1 12 " y = \+ .546454" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Plot that map" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "xplo t, yplot : plot structures that display an iterated sequence of \+ x- and y-values " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "xpoint s:=\{seq([n,x[n]],n=0..nit)\}: \nypoints:=\{seq([n,y[n]],n=0..nit)\}: \+ \nxplot:=pointplot(xpoints,style=line,color=red): \nyplot:= \npointplo t(ypoints,symbol=circle, color=blue): \ndisplay([xplot,yplot],axes=box ed,labels=[n,x_n]); " }}{PARA 13 "" 1 "" {GLPLOT2D 424 212 212 {PLOTDATA 2 "6&-%'CURVESG6%7U7$$\"\"!F)$\")+++q!\")7$$\"\"\"F)$\"1++++ ++D*)!#;7$$\"\"#F)$\"1+](=nQ$RXF27$$\"\"$F)$\"1&)R3eA&*4!*F27$$\"\"%F) $\"1cMo&z`$RUF27$$\"\"&F)$\"1A!Hm/IOp)F27$$\"\"'F)$\"1aLzzUj2`F27$$\" \"(F)$\"1Y&>A+a5`*F27$$\"\")F)$\"1[aAz`P#=#F27$$\"\"*F)$\"1%\\u(o]3'>& F27$$\"#5F)$\"1-Rbwh%H[*F27$$\"#6F)$\"1%y$R9k@)Q#F27$$\"#7F)$\"1:3N#f1 +j&F27$$\"#8F)$\"1Zt2M@m8'*F27$$\"#9F)$\"1V]))4#)=@=F27$$\"#:F)$\"1&Rs \\%4'>S%F27$$\"#;F)$\"1]=>,QXs))F27$$\"#F)$\"1\\N^y@4VOF27$$\"#?F)$\"1$=/cZT*)*yF27$$\" #@F)$\"1Y]A8FLEuF27$$\"#AF)$\"1(*3#)H!)pE$)F27$$\"#BF)$\"1q1pWQo$Q'F27 $$\"#CF)$\"1+VIl^gb%*F27$$\"#DF)$\"1^I%3PLP]#F27$$\"#EF)$\"1-#Q[zbp'eF 27$$\"#FF)$\"1BbQtBq='*F27$$\"#GF)$\"1jhOCJ$))z\"F27$$\"#HF)$\"1?$=z_0%F27$$\"#RF)$\"1EkV'pm3Z)F27$$\"#SF)$\"1v/AqtO\")fF27$$\"#T F)$\"17Q(*e/d.'*F27$$\"#UF)$\"1x>=bz$e'=F27$$\"#VF)$\"1Z'\\V&R?-XF27$$ \"#WF)$\"1o/c!*f.u*)F27$$\"#XF)$\"1h%=PtmtO%F27$$\"#YF)$\"1,\"=)y(\\e$ ))F27$$\"#ZF)$\"19rUV&fP%[F27$$\"#[F)$\"1D))\\44Ho#*F27$$\"#\\F)$\"1kz y+2zmKF27$$\"#]F)$\"1][[0MS&H(F2-%'COLOURG6&%$RGBG$\"*++++\"F,F(F(-%&S TYLEG6#%%LINEG-%'POINTSG6W7$F($\"),++qF,7$F.$\"1&***\\#))**\\#*)F27$F4 $\"10-4!3R$RXF27$F9$\"1K\"exk_*4!*F27$F>$\"1)*)ftR_$RUF27$FC$\"1vCjN%G Op)F27$FH$\"1rz8%QRwI&F27$FM$\"1L3\\yf0J&*F27$FR$\"1+yTboO#=#F27$FW$\" 13(3S!o1'>&F27$Ffn$\"1,8q*\\PH[*F27$F[o$\"1'y\")HC`#)Q#F27$F`o$\"1UXZ4 H3IcF27$Feo$\"1NXy(\\rOh*F27$Fjo$\"1))zz7n9@=F27$F_p$\"1hhgCv'=S%F27$F dp$\"1K0q9gNs))F27$Fip$\"1enxt9Q?ZF27$F^q$\"1G:-O]Yr\"*F27$Fcq$\"1wIx$ HT?k$F27$Fhq$\"1]N@A+O(*yF27$F]r$\"1y#*f-!y(HuF27$Fbr$\"1Q'e[&R0@$)F27 $Fgr$\"1FYg(>.*)R'F27$F\\s$\"1.\\d$zEqW*F27$Fas$\"1%\\$y,(e(RDF27$Ffs$ \"1x_L+p$)RfF27$F[t$\"1)4PZC4/h*F27$F`t$\"1XDJqjfN=F27$Fet$\"1lnZ@)oVV %F27$Fjt$\"1'4R(3x-1*)F27$F_u$\"128A.4,0YF27$Fdu$\"1S%Q))Gm62*F27$Fiu$ \"17.;uG:3B)4X$[F27$Fgw$\"1.9\">#=Th#*F27$F \\x$\"1RC'[CzQH$F27$Fax$\"1\"f&RsHETtF27$Ffx$\"1sn;)p!)=Y)F27$F[y$\"1< %Rxv5s+'F27$F`y$\"1(z;8WM&)f*F27$Fey$\"1j\"G&)\\k!))=F27$Fjy$\"1>K%e_( *=b%F27$F_z$\"1CL2Yp)=-*F27$Fdz$\"1!*\\ua0V'>%F27$Fiz$\"1!*en]ZfV')F27 $F^[l$\"1\"p<%**)QXY&F2-Fc[l6&Fe[lF(F(Ff[l-%'SYMBOLG6#%'CIRCLEG-%*AXES STYLEG6#%$BOXG-%+AXESLABELSG6$%\"nG%$x_nG" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }