{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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 " " 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }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 0 }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 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Math 2280 - 1 \+ FOURTH MAPLE ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 56 "Treibergs \+ Due Wednesday, Nov 9, 2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "This lab may be downloaded fro m our homepage as http://www.math.utah.edu/~treiberg/M2282L 4.mws" }}{PARA 0 "" 0 "" {TEXT -1 457 "We explore phase portraits of f irst order systems. The behavior of the linear systems is determined b y the eigenvalues of the system. Nonlinear systems locally behave like linear ones, and the local behavior of the trajectories is determined by the linear approximations at the critical points. We end by examin ing Rayleigh's equation of the clarinet reed and the Brusselator equat ion for an exotic chemical reaction. These systems have stable limit c ycles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "1. Linear systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(linalg):with(DEtools):with(plots):# " }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name adjoint has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "Enter a matrix. Compute its eigen values, their algebraic multiplicity, and a basis for their eigenspace . Pick off the eigenvectors and store them as rows in the matrix P." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A:=matrix([[1,2/5],[1/5,4/5]]);ei genvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$ \"\"\"#\"\"#\"\"&7$#F*F-#\"\"%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7% #\"\"$\"\"&\"\"\"<#-%'vectorG6#7$!\"\"F'7%#\"\"'F&F'<#-F*6#7$\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "P:=matrix([op(op(3,op(1,[ %]))),op(op(3,op(2,[%])))]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG -%'matrixG6#7$7$!\"\"\"\"\"7$\"\"#F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 333 "Define a linear system using the matrix. Since the matrix has \+ distinct real eigenvalues, its eigendirections are the principal direc tions of the matrix, and indicate the directions of expansion/contract ion of the system. The matrix A gives the coefficients of the linear s ystem. IC1 is a list of initial data for the integral curves." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ODE1:=\{diff(x(t),t)=(A[1,1]*x(t)+A [1,2]*y(t)),diff(y(t),t)=A[2,1]*x(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "+A[2,2]*y(t)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ODE1G<$/-% %diffG6$-%\"xG6#%\"tGF-,&F*\"\"\"*&#\"\"#\"\"&F/-%\"yGF,F/F//-F(6$F4F- ,&*&#F/F3F/F*F/F/*&#\"\"%F3F/F4F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "IC1:=[seq([x(0)=evalf(0.3*cos(k*Pi/8)),y(0)=evalf(0.3 *sin(k*Pi/8))],k=" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "0..15)]:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "The instruction phaseportrait plo ts the direction field, and integrates the system ODE1 whose variables are [x(t),y(t)] and for which t ranges from 0.0..1.0, with initial \+ data IC1, and plotting range -2 \+ " 0 "" {MPLTEXT 1 0 80 "P1:=phaseportrait(ODE1,[x(t),y(t)],-2..3.0,IC1 ,x=-2..2,y=-2..2,linecolor=brown):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "P2:=plot(\{[P[1,1]*t,P[1,2]*t,t=-1..1],[P[2,1]*t,P[2, 2]*t,t=-3..3]\}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "color=[gold,mar oon], thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "disp lay(\{P1,P2\},scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 213 232 232 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$!\"'\"\"!$!\"$F*7$$!3z******\\ TVQd!#<$!3*)*****\\2<#pGF07$$!3l****\\-r%3^&F0$!3#)***\\7bBav#F07$$!3A +++l;!\\D&F0$!36++]K3XFEF07$$!3o*****\\lfs*\\F0$!3%)****\\F)H')\\#F07$ $!3%)****\\s@%3u%F0$!3\"****\\i3@/P#F07$$!3J++]U.6.XF0$!3:++Dr^b^AF07$ $!3&)****\\-G&pD%F0$!3#****\\7Sw%G@F07$$!3(*****\\AjP-SF0$!3*****\\7;) =,?F07$$!32++]sih[PF0$!3.++DO\"3V(=F07$$!3%)******pGf([$F0$!3#******\\ V'zViUC\"F07$$!3/++]AH4hAF0$!3-++DhkaI6F07$$!3%*******4 \\l!*>F0$!3s******\\XF`**!#=7$$!3&*******R%e:w\"F0$!3u*******>#z2))Fhp 7$$!33++]#yk]\\\"F0$!3R++]7RKvuFhp7$$!3M+++SFam%**Fhp$\"3*y******4FL( \\Fhp7$$\"3k*****\\JigC\"F0$\"3A)****\\d6.B'Fhp7$$\"3%*****\\Pw7#F07$$\"3q++]x2k2XF0$\"3N++v)Q?QD#F07$$\"3d+++?EdRZF0$\"3G +++5jypBF07$$\"3M+++&o#R0]F0$\"3<++]Ujp-DF07$$\"3++++?`9V_F0$\"3++++gE d@EF07$$\"3G++]<#Rm\\&F0$\"39++v3'>$[FF07$$\"3E++]A_ERdF0$\"37++D6EjpG F07$$\"\"'F*$\"\"$F*-%'COLOURG6&%$RGBG$\")+++!)!\")$\")AR!)\\F`[l$\")V yg>F`[l-%*THICKNESSG6#\"\"%-F$6%7S7$$\"\"\"F*$!\"\"F*7$$\"3ommm;p0k&*F hp$!3ommm;p0k&*Fhp7$$\"3vKL$3s%HaFhp$!3hmmm\">s%HaFhp7$$\"3Q+++]$*4)*\\Fhp$!3Q++ +]$*4)*\\Fhp7$$\"38+++]_&\\c%Fhp$!38+++]_&\\c%Fhp7$$\"30+++]1aZTFhp$!3 0+++]1aZTFhp7$$\"3umm;/#)[oPFhp$!3umm;/#)[oPFhp7$$\"3hLLL$=exJ$Fhp$!3h LLL$=exJ$Fhp7$$\"3)RLLLtIf$HFhp$!3)RLLLtIf$HFhp7$$\"3]++]PYx\"\\#Fhp$! 3]++]PYx\"\\#Fhp7$$\"3EMLLL7i)4#Fhp$!3EMLLL7i)4#Fhp7$$\"3c****\\P'psm \"Fhp$!3c****\\P'psm\"Fhp7$$\"3')****\\74_c7Fhp$!3')****\\74_c7Fhp7$$ \"3)3LLL3x%z#)!#>$!3)3LLL3x%z#)F]cl7$$\"3KMLL3s$QM%F]cl$!3KMLL3s$QM%F] cl7$$\"3\\^omm;zr)*!#@$!3\\^omm;zr)*Fhcl7$$!3%pJL$ezw5VF]cl$\"3%pJL$ez w5VF]cl7$$!3s*)***\\PQ#\\\")F]cl$\"3s*)***\\PQ#\\\")F]cl7$$!3GKLLe\"*[ H7Fhp$\"3GKLLe\"*[H7Fhp7$$!3H*******pvxl\"Fhp$\"3H*******pvxl\"Fhp7$$! 3#z****\\_qn2#Fhp$\"3#z****\\_qn2#Fhp7$$!3U)***\\i&p@[#Fhp$\"3U)***\\i &p@[#Fhp7$$!3A)****\\2'HKHFhp$\"3A)****\\2'HKHFhp7$$!3ElmmmZvOLFhp$\"3 ElmmmZvOLFhp7$$!3h******\\2goPFhp$\"3h******\\2goPFhp7$$!3UKL$eR<*fTFh p$\"3UKL$eR<*fTFhp7$$!3l******\\)Hxe%Fhp$\"3l******\\)Hxe%Fhp7$$!3ckm; H!o-*\\Fhp$\"3ckm;H!o-*\\Fhp7$$!3y)***\\7k.6aFhp$\"3y)***\\7k.6aFhp7$$ !3#emmmT9C#eFhp$\"3#emmmT9C#eFhp7$$!32****\\i!*3`iFhp$\"32****\\i!*3`i Fhp7$$!3%QLLL$*zym'Fhp$\"3%QLLL$*zym'Fhp7$$!3wKLL3N1#4(Fhp$\"3wKLL3N1# 4(Fhp7$$!3Mmm;HYt7vFhp$\"3Mmm;HYt7vFhp7$$!3Y*******p(G**yFhp$\"3Y***** **p(G**yFhp7$$!3]mmmT6KU$)Fhp$\"3]mmmT6KU$)Fhp7$$!3fKLLLbdQ()Fhp$\"3fK LLLbdQ()Fhp7$$!3Z++]i`1h\"*Fhp$\"3Z++]i`1h\"*Fhp7$$!3W++]P?Wl&*Fhp$\"3 W++]P?Wl&*Fhp7$F_\\lF]\\l-F[[l6&F][l$\")viobF`[l$\")!\\DP\"F`[l$\")%yg >%F`[lFe[l-F$6]dl7%7$$!3?Lh8XQM9>F0$!3/&4SA.<)Q>F07$$!3zmQ'[:cc3#F0$!3 s/*fx'H=h?F07$$!3W*es)4-FL?F0$!3IqjH/S5c?F07%7$$!3J!evSa4bq\"F0$!39`XV =\\^O>F07$$!3MsaMCyVt=F0$!3&oWl:3&[j?F07$$!3k?Ax9!47#=F0$!37\"o+*pi)p0 #F07%7$$!3;5A+GM'p\\\"F0$!3))\\pqMv(R$>F07$$!3:&*)R)38$4m\"F0$!36]IHlC -m?F07$$!3)*)4k\\gD*3;F0$!3&>j:V))Hz0#F07%7$$!3,gw%G'Gx)G\"F0$!3[E&fxx x6$>F07$$!3%z\\:CCp![9F0$!3^t/CAA#)o?F07$$!3c?i**3]P'R\"F0$!3/zQGtL$*e ?F07%7$$!3@`[thS-\"3\"F0$!31knGB()3G>F07$$!3Sd$\\>TlZB\"F0$!3$fB8nF6>2 #F07$$!3:yl/9;]$=\"F0$!3#=$RE\\/**f?F07%7$$!3J]Fs!Qu#Q()Fhp$!3d)fS^%po C>F07$$!3C))H./%44-\"F0$!3U,%f[08`2#F07$$!3:S+y0'pBq*Fhp$!3Gk?B\"z'3h? 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This time, we can't plot the eigenspaces. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A:=matrix([[-1/5,-3/5],[4/5,1/5 ]]);eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG 6#7$7$#!\"\"\"\"&#!\"$F,7$#\"\"%F,#\"\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%*&^##\"\"\"\"\"&F'\"#6#F'\"\"#F'<#-%'vectorG6#7$F',&* &^##!\"\"\"\"$F'F)F*F'#F'F6F57%*&^##F5F(F'F)F*F'<#-F.6#7$F',&*&^##F'F6 F'F)F*F'#F'F6F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Define a linea r system using the matrix. 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"Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dsolve(ODE1, \{x(t),y(t)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"xG6#%\"tG,&* &%$_C1G\"\"\"-%$sinG6#,$*(\"\"&!\"\"\"#6#F,\"\"#F(F,F,F,F,*&%$_C2GF,-% $cosGF/F,F,/-%\"yGF',**&#F,\"\"$F,*(F+F,F9F,F4F5F,F3*&#F,FAF,*(F8F,F-F ,F4F5F,F,*&#F,FAF,F*F,F3*&#F,FAF,F7F,F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Let's do the same thing with a linear system whose matri x has complex eigenvalues with negative real parts. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A:=matrix([[-2/5,-3/5],[4/5,1/5]]);eigenvects(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$#!\"#\"\"&# !\"$F,7$#\"\"%F,#\"\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,&#\"\" \"\"#5!\"\"*&^##F&F'F&\"#R#F&\"\"#F&F&<#-%'vectorG6#7$,&#\"\"$\"\")F(* &^##F&F7F&F,F-F&F&7%,&#F&F'F(*&^##F(F'F&F,F-F&F&<#-F16#7$,&#F6F7F(*&^# #F(F7F&F,F-F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "IC3:=[se q([x(0)=evalf(cos(k*Pi/8)),y(0)=evalf(sin(k*Pi/8))],k=-3..5)]:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Define a linear system using the m atrix. 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First , near that point, by the 2-dimensional Taylor's Theorem, the system i s approximated by the linear system whose matrix is [F(-1+u,-1+v),G(- 1+u,-1+v)]=[0 +diff(F(x,y),x)*u +diff(F(x,y),y)*v, 0+ diff(G(x,y),x)*u +diff(G(x,y),y)*v]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "A:=subs(\{x= x0,y=y0\},matrix([ [diff(F(x,y),x),diff(F(x,y),y)],[diff(G(x,y),x),dif f(G(x,y),y)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6# 7$7$\"\"#!\"#7$F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "e igenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,&#\"\"\"\"\"#F&*&^ #F%F&\"\"(F%F&F&<#-%'vectorG6#7$F&,&#\"\"$\"\"%F&*&^##!\"\"F3F&F*F%F&7 %,&F%F&*&^##F7F'F&F*F%F&F&<#-F-6#7$F&,&F1F&*&^##F&F3F&F*F%F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Define a linear system using the m atrix. 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