{VERSION 5 0 "SUN SPARC SOLARIS" "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 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warni ng" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 39 "Complex eigenvalues/eigenvector s exampl" }{TEXT 256 1 "e" }}{PARA 258 "" 0 "" {TEXT -1 11 "Math 2270- 2" }}{PARA 257 "" 0 "" {TEXT -1 24 "Friday November 16, 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 257 22 "Glucose-insulin model " }{TEXT -1 43 " (see discussion on page 33 9 of the text.)" }}{PARA 0 "" 0 "" {TEXT -1 684 " Let G(t) be the excess glucose concentration (mg of G per 100 ml of blood, say) in so meone's blood, at time t hours. Excess means we are keeping track of \+ the difference between current and equilibrium (\"fasting\") concentra tions. Similarly, Let H(t) be the excess insulin concentration at tim e t. When blood levels of glucose rise, say as food is digested, the \+ pancreas reacts by secreting insulin in order to utilize the glucose. \+ Researchers have developed mathematical models for the glucose regula tory system. Here is a simplified (linearized) version of one such mo dels. It would be meant to apply between meals, when no additional gl ucose is being added to the system:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(linalg):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been red efined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the \+ name changecoords has been redefined\n" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"GG6#,&%\"tG\"\"\"F)F),&*&%\"aGF)-F%6#F(F)F)*&%\"bG F)-%\"HGF.F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6#,&%\"tG \"\"\"F)F),&*&%\"cGF)-%\"GG6#F(F)F)*&%\"dGF)-F%F/F)F)" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 258 58 "Explai n (understand) the signs of the matrix coefficients:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "A particular choice of constants could lead to a model for you! For \+ example" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7$7#-% \"GG6#,&%\"tG\"\"\"F.F.7#-%\"HGF+*&-F%6#7$7$$\"\"*!\"\"$!\"%F97$$F.F9F 7F.-F%6#7$7#-F*6#F-7#-F1FCF." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "Now let's solve the initial value problem , say right after a big meal, when" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7$7#-%\"GG6#\"\"!7#-%\"HGF+-F%6#7$7#\"$+\" 7#F," }}}{PARA 0 "" 0 "" {TEXT -1 27 "Of course, we know that for" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=matrix(2,2,[.9,-.4,.1,.9] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$$\"\"*!\" \"$!\"%F,7$$\"\"\"F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "The so lution at time t is just given by A^t times the initial vector. So we can make a table of values, but it won't help us understand the big p icture:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "v:=vector([100, 0]):\nfor i from 1 to 30 do\nG[i]:=evalm(A^i&*v)[1]:\nH[i]:=evalm(A^i& *v)[2]:\nprint(i,G[i],H[i]);\nod:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"$\"$+*!\"\"$\"$+\"F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#$\"%+x!\"#$\"%+=F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%\"\"$$\"&+@'!\"$$\"&+R#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%$\"'+LY!\"%$\"'+sFF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&$\"(+41$!\"&$\"(+\"eHF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% \"\"'$\")+dr:!\"'$\")+QoHF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"($ \")+hqA!\"($\"*+*pGGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\")$!*+Z7F *!\")$\"++_`oDF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*$!+5j#='=!\") $\"+5#p*=AF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#5$!+j8BjD!\")$\"+e' *)3\"=F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#6$!+!4k7.$!\")$\"+cvZt8 F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#7$!+/(GvF$!\")$\"+7R.I$*!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#8$!+*=xHK$!\")$\"+>=]>^!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#9$!+W&fa>$!\")$\"+ZWd%G\"!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#:$!+nlHFH!\")$!+TDMR?!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#;$!+3K*Hb#!\")$!+a[qiZ!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%\"#<$!+$p&=2@!\")$!+xvURo!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%\"#=$!+@+*Gi\"!\")$!+70ni#)!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#>$!+)>%4I6!\")$!+\"[$Hf!*!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#?$!+!RIra'!\"*$!+K$eMG*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#@$!+?S.z@!\"*$!+Qb#)4!*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#A$\"+'f*zU;!\"*$!+'QYnK)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#B$\"+!>=#4[!\"*$!+(y\"zHtF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#C$\"+'38-E(!\"*$!+*y!*e6'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#D$\"+%4[0)*)!\"*$!+-/GyZF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#E$\"+Xa!Q***!\"*$!+_vR-MF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#F$\"+#RQb.\"!\")$!+_sxi?!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#G$\"+Vc\\95!\")$!+S8h4#)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#H$\"+S_%)e%*!\"*$\"+?iIcF!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#I$\"+q%3FS)!\"*$\"+%3_R>\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "We get a better understa nding if we plot the points onto the G-H plane, along with the deviati on vector field:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Can we get a n ice, closed form for the solution? Let's try our eigenvalue, eigenvec tor approach:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects (A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$7%^$$\"\"*!\"\"$\"+++++?!#5\" \"\"<#-%'vectorG6#7$^$$!+++++?!\"*$\"\"!F6^$F5$F+F67%^$F%$F3F*F+<#-F.6 #7$^$F2$!\"!F6^$F5$F'F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 67 "Can we use this to find a closed for m solution to the problem?....." }}{PARA 0 "" 0 "" {TEXT -1 68 "In cla ss notes we have done so! Here is the solution we have found:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "r:=sqrt(.85);\ntheta:=arctan(.2/.9);\nGt:=t->100*r^t*cos(theta*t); \nHt:=t->50*r^t*sin(theta*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"r G$\"+dWa>#*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG$\"+f%*o'=# !#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GtGf*6#%\"tG6\"6$%)operatorG %&arrowGF(,$*&)%\"rG9$\"\"\"-%$cosG6#*&%&thetaGF1F0F1F1\"$+\"F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HtGf*6#%\"tG6\"6$%)operatorG%&arro wGF(,$*&)%\"rG9$\"\"\"-%$sinG6#*&%&thetaGF1F0F1F1\"#]F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "soltn:=plot([Gt(t),Ht(t),t= 0..100],color=black):\npict1:=fieldplot([-.1*G-.4*H,.1*G-.1*H],G=-40.. 100,H=-15..40):\nactual:=pointplot(\{seq([G[i],H[i]],i=1..30)\}):\ndis play(\{actual,pict1,soltn\},title=`numerical and \nanalytic solutions` );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG 6$7]q7$$\"$+\"\"\"!$F*F*7$$\"3pug!)H]nj(*!#;$\"31HtUe7-7H!#<7$$\"3!otF v`:!*\\*F/$\"3^vUV\\eS'o&F27$$\"3fc&zAM%H3#*F/$\"3*)*o(3)Rb\"=$)F27$$ \"3]79Gs()y$*))F/$\"3v&HH.:0.3\"F/7$$\"3ipa4t9xd&)F/$\"3\">c7='Qw88F/7 $$\"3=F))\\K;]-#)F/$\"3Wg#3eY?>`\"F/7$$\"3X&46'Qm@IyF/$\"3Z:tA^fdMF/7$$\"3qto+QOjSnF/$\"3!3\"o%HFY'4AF/7 $$\"3Td)H:Vb4,'F/$\"3(30+XSQ.X#F/7$$\"39bay>mmTGF/$\"3DiU`CBxpHF/7$$\"3-U\\eV\"=5/#F/$\"38>TX(4;G)HF/7$ $\"3VWE+n)>mF\"F/$\"3;dU#4AD1&HF/7$$\"3sy:Q.qq@bF2$\"3LUb%=->o(GF/7$$! 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