{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 50 "Math 2280 - 1 \+ FOURTH MAPLE ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 61 "Treibergs \+ Due Wednesday, November 5, 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "This lab may be downloade d from our homepage as" }}{PARA 0 "" 0 "" {HYPERLNK 17 "http://www.mat h.utah.edu/~treiberg/M2281L4.mws" 1 "M2281L4.mws" "" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 461 "We explore phase portraits of first orde r systems. The behavior of the linear systems is determined by the eig envalues of the system. Nonlinear systems locally behave like linear o nes, and the local behavior of the trajectories is determined by the l inear approximations at the critical points. We end by examining Rayle igh's equation of the clarinet reed and van der Pol's equation for a c ircuit with nonlinear resistor. These systems have stable limit cycles ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "1. L inear 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 trace have \+ been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warni ng, 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 eigenvalues, 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]]);eige nvects(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$!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 334 "Define a lin ear system using the matrix. Since the matrix has distinct real eigenv alues, its eigendirections are the principal directions of the matrix, and indicate the directions of expansion/contraction of the system. T he matrix A gives the coefficients of the linear system. IC1 is a lis t of initial data for the integral curves." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ODE1:=\{diff(x(t),t)=(A[1,1]*x(t)+A[1,2]*y(t)),diff(y (t),t)=A[2,1]*x(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "+A[2,2]*y(t) \};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ODE1G<$/-%%diffG6$-%\"yG6#% \"tGF-,&*&#\"\"\"\"\"&F1-%\"xGF,F1F1*&#\"\"%F2F1F*F1F1/-F(6$F3F-,&F3F1 *&#\"\"#F2F1F*F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "IC1:= [seq([x(0)=evalf(0.2*cos(k*Pi/12)),y(0)=evalf(0.2*sin(k*Pi/12))],k=0.. 25)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 368 "The instruction phasepo rtrait plots the direction field, and integrates the system ODE1 whose variables are [x(t),y(t)] and for which t ranges from 0.0..3.0, wit h initial data IC1, and plotting range -2 " 0 "" {MPLTEXT 1 0 81 "P1:=phaseportrait(ODE1,[x(t),y(t)], 0.0..3.0,IC1,x=-2..2,y=-2..2,linecolor=brown):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "P2:=plot(\{[P[1,1]*t,P[1,2]*t,t=-2..2],[P[2,1]*t,P[2 ,2]*t,t=-2..2]\},color=[maroon,gold],thickness=7):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "display(\{P1,P2\},scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$!\"% \"\"!$!\"#F*7$$!3ommmmFiDQ!#<$!3MLLL$Q6G\">F07$$!34LLLo!)*Qn$F0$!3amm; M!\\p$=F07$$!3nmmmwxE.NF0$!3LLLL))Qj^1DBF0$!3gmmmc4`i6F07$$!3jmm mw))yr@F0$!3KLLLQW*e3\"F07$$!3;+++S(R#**>F0$!3w++++()>'***!#=7$$!30+++ +@)f#=F0$!3E++++0\"*H\"*Fdo7$$!3-+++gi,f;F0$!35++++83&H)Fdo7$$!3qmmm\" G&R2:F0$!3[LLL3k(p`(Fdo7$$!3WLLLtK5F8F0$!3Anmmmj^NmFdo7$$!3eLLL$HsV<\" F0$!3(zmmmYh=(eFdo7$$!3+-++]&)4n**Fdo$!3+,++v#\\N)\\Fdo7$$!37PLLL\\[%R )Fdo$!3commmCC(>%Fdo7$$!3F)*****\\&y!pmFdo$!39*****\\FRXL$Fdo7$$!3Y*** ***\\O3E]Fdo$!3t*****\\#=/8DFdo7$$!3NKLLL3z6LFdo$!37$$!3gSnmmmr[R!#?$!3IqLLL$eV(>F`s7$$\"3yE LL$=2Vs\"Fdo$\"3)Qjmm\"f`@')F\\s7$$\"3)e*****\\`pfKFdo$\"3%z****\\nZ)H ;Fdo7$$\"36HLLLm&z\"\\Fdo$\"3bkmm;$y*eCFdo7$$\"3=(******z-6j'Fdo$\"3f) ******R^bJ$Fdo7$$\"3q\"******4#32$)Fdo$\"3&e*****\\5a`TFdo7$$\"3r$**** *\\#y'G**Fdo$\"3&o****\\7RV'\\Fdo7$$\"3G******H%=H<\"F0$\"3X'*****\\@f keFdo7$$\"35mmm1>qM8F0$\"3_ILLL&4Nn'Fdo7$$\"3%)*******HSu]\"F0$\"3A*** ****\\,s`(Fdo7$$\"3'HLL$ep'Rm\"F0$\"3%[mm;zM)>$)Fdo7$$\"3')******R>4N= F0$\"3L*******pfa<*Fdo7$$\"3#emm;@2h*>F0$\"38HLLeg`!)**Fdo7$$\"3]***** \\c9W;#F0$\"3v****\\#G2A3\"F07$$\"3Kmmmmd'*GBF0$\"3:LLL$)G[k6F07$$\"3j *****\\iN7]#F0$\"3\")****\\7yh]7F07$$\"3`LLLt>:nEF0$\"3wmmm')fdL8F07$$ \"35LLL.a#o$GF0$\"3bmmm,FT=9F07$$\"3ammm^Q40IF0$\"3FLL$e#pa-:F07$$\"3y ******z]rfJF0$\"3*)******Rv&)z:F07$$\"3gmmmc%GpL$F0$\"3HLLLGUYo;F07$$ \"3/LLL8-V&\\$F0$\"3_mmm1^rZF07$$\"\"%F*$\"\"#F*-%'COLOURG6&%$RGBG$\")viob !\")$\")!\\DP\"Fa[l$\")%yg>%Fa[l-%*THICKNESSG6#\"\"(-F$6%7S7$FizF+7$$ \"3MLLL$Q6G\">F0F17$$\"3amm;M!\\p$=F0F67$$\"3LLLL))Qj^'***FdoFbo7$$\"3E++++0\"*H\"*FdoFho7$ $\"35++++83&H)FdoF]p7$$\"3[LLL3k(p`(FdoFbp7$$\"3Anmmmj^NmFdoFgp7$$\"3( zmmmYh=(eFdoF\\q7$$\"3+,++v#\\N)\\FdoFaq7$$\"3commmCC(>%FdoFfq7$$\"39* ****\\FRXL$FdoF[r7$$\"3t*****\\#=/8DFdoF`r7$$\"3F`sFas7$$!3)Qjmm\"f`@')F\\sFfs7$$!3% z****\\nZ)H;FdoF[t7$$!3bkmm;$y*eCFdoF`t7$$!3f)******R^bJ$FdoFet7$$!3&e *****\\5a`TFdoFjt7$$!3&o****\\7RV'\\FdoF_u7$$!3X'*****\\@fkeFdoFdu7$$! 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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-%'matrixG6#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'F6F'F)F*F'#F'F6F5" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Define a li near system using the matrix. 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" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deqtnG<$/-%%diffG6$-%\"yG6#%\"tGF-,&F*!\"\"*$)-%\"xG F,\"\"#\"\"\"F//-F(6$F2F-,*F2F5F*F/F0F/*&F*F5F2F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ICs:=[[x(0)=-1,y(0)=3],[x(0)=0,y(0)=3]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "[x(0)=.9,y(0)=3]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "[x(0)=.6,y(0)=3],[x(0)=.8,y(0)=3]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "[x(0)=1,y(0)=3],[x(0)=2,y(0)=3]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "[x(0)=3,y(0)=3],[x(0)=3,y(0)=0]," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "[x(0)=3,y(0)=-1.5],[x(0)=0.05,y(0)=0]," }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 38 "[x(0)=-.05,y(0)=0],[x(0)=0,y(0)=-.05]," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "[x(0)=0,y(0)=.05]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "[x(0)=1.5,y(0)=-3]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "[x(0)=1,y(0)=-3],[x(0)=0,y(0)=-3]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "[x(0)=-1,y(0)=-3],[x(0)=-2,y(0)=-3]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "[x(0)=-1.1,y(0)=-1],[x(0)=-1,y(0)=-.9]," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "[x(0)=-.9,y(0)=-1],[x(0)=-1,y(0)=-1 .1]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "[x(0)=-1,y(0)=-.95],[x(0)=- 1,y(0)=-1.05]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "#initial conditions to make a picture like Figure 6.1.16" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "DEplot(deqtn,[x(t),y(t)],t=0..15,ICs,x=-3..3, y=-3..3,arrows=small, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "stepsize= .1,color=black,linecolor=red,title=`Figure 6.1.16`);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6]dl7%7$$!\"$\"\"!$! 3o%*y:j_5UG!#<7$F($!3J0@%ot%*y:$F-7$$!3*QZ*y:j_gHF-$!3ZWvA5V_*3$F-7%7$ $!3Z/JI\\\\JEFF-$!3f.()[&eAy%GF-7$$!3Gu%GL51@k#F-$!3S'H6XTx@:$F-7$$!3Z ++m*\\$HAEF-$!31f*R`bcd2$F-7%7$$!3-'oxLe$>oCF-$!3%zsc_vAw(GF-7$$!3]ra) =_['oAF-$!30sKuWsPAJF-7$$!3'*\\E\"Q)pD\"G#F-$!3iN.m5KWWIF-7%7$$!3s7@o* eJ#)>#F-$!3_y]-Q\\\"*QHF-7$$!3wBE@o:.2>F-$!3Z@\\(>1&3hIF-7$$!37\"oin&p ![&>F-$!3%3Ag:XM#)*HF-7%7$$!3C'*[N<]t%*=F-$!3](4h%**3C**HF-7$$!3z=9<$> \\*y:F-$!3\\-*Q05f2+$F-7$$!3kn6^a!Hrk\"F-$!3%)41Msr&4'HF-7%7$$!3!HDYZX &*Rd\"F-$!3!fR')*eJBRIF-7$$!3*3k6%3)4\"o7F-$!34/O,TowgHF-7$$!3))QHD$GW TM\"F-$!3#RC)R\"f>&RHF-7%7$$!3%*R%p!fZZ\\7F-$!3fbq&)QVHkIF-7$$!3=E.?nb ^5'*!#=$!3SWH9hcqNHF-7$$!33eJ]'fq&R5F-$!3(e8Mi-$\\FHF-7%7$$!3FV)R[,r\" [#*Fiq$!3bq\"GVy>83$F-7$$!3=s1PpEITlFiq$!3WH=n:-o=HF-7$$!3Qh,;=jlItFiq $!3m)p#)3'o0?HF-7%7$$!3?!f'e!R2F+'Fiq$!3%*4$ozxuV4$F-7$$!32:(R*>o(4Z$F iq$!30!pJ?ADc!HF-7$$!3vc**G7&\\]D%Fiq$!3JSr\\nS%[\"HF-7%7$$!3$pD$eqO@] FFiq$!3EjX51s)e5$F-7$$!3xy$)ei1\"o2%!#>$!3uOa*Qz7T*GF-7$$!3U8[lnAdz6Fi q$!3sODbT;o5HF-7%7$$\"3tyA)GF-7$$\"3W3MaL1E\")=Fiq$!3UFlq2r%p!HF-7%7$$\"3yBL5?;IoQFiq$ !3W[CHP-'=8$F-7$$\"3Z,IU!f#Q0cFiq$!3b^vqi(R\"oGF-7$$\"3cR+,NHk**[Fiq$! 3w22?3M_.HF-7%7$$\"3ykcn^1H#R(Fiq$!3(G:KU4(o\\JF-7$$\"3kq[`KI=(R)Fiq$! 37Zyw0HJ]GF-7$$\"3p76[9(*R0yFiq$!3!p'*=T;oD!HF-7%7$$\"3NReQP_aU6F-$!3r FsO'4IM:$F-7$$\"3)\\j.%y5)z1\"F-$!3HsFj.*pl%GF-7$$\"3sIWGorwX5F-$!3s/^ 3/wKAHF-7%7$$\"3Yu`r&\\\\2c\"F-$!3%=edBJ!ftIF-7$$\"36@DWndN\"G\"F-$!3; 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First, near that poin t, by the 2-dimensional Taylor's Theorem, the system is approximated b y the linear system whose matrix is " }}{PARA 0 "" 0 "" {TEXT -1 106 " [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 14 "x0:=-1;y0:=-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "A:=subs(\{x=x0,y=y0\},matrix([ [diff(F(x,y),x),d iff(F(x,y),y)]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "[diff(G(x,y),x), diff(G(x,y),y)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrix G6#7$7$\"\"#!\"#7$F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Now \+ solve the linear system as above:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Define a linear system using the matrix. 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positive, so the local picture is that of a spiral source. We ca n look at the original system near the point (-1,-1)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ICs1:=[" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "[x(0)=-1.1,y(0)=-1],[x(0)=-1,y(0)=- .9]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "[x(0)=-.9,y(0)=-1],[x(0)=-1 ,y(0)=-1.1]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "[x(0)=-1,y(0)=-.95] ,[x(0)=-1,y(0)=-1.05]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#look ne ar the equilibrium point [-1,-1]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "DEplot(deqtn,[x(t),y(t )],t=0..5,ICs1,x=-1.3..-.7,y=-1.3..-.7, arrows=small, stepsize=.1,colo r=green,linecolor=blue,title=`Original system near (-1,-1)`);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "#solution to nonlinear system near \+ [-1,-1]" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CU RVESG6]dl7%7$$!3/+++++++8!#<$!3Z*y:j_5UG\"F*7$F($!3i5Uot%*y:8F*7$$!3MZ *y:j_gH\"F*$!3\\aF-JC&*38F*7%7$$!3Be[#f3))>F\"F*$!3=ejq?(=YG\"F*7$$!3b *HQ$>S&[E\"F*$!3!>k$Hz7Q:8F*7$$!3Z^*Q-F`DE\"F*$!3YB[mJ#HyI\"F*7%7$$!3u mx!))*=6X7F*$!3Z:*GSU\\lG\"F*7$$!3u[&=E P$f!zG7F*$!3`'oTt&)ebI\"F*7%7$$!3#)*47/\"=,=7F*$!31*3qrX%o!H\"F*7$$!3M ttP0X^#>\"F*$!3.6*HGa:$48F*7$$!33C>9[fq&>\"F*$!3w(>I8j%4-8F*7%7$$!3s3_ \")e%z))=\"F*$!3s-f***34dH\"F*7$$!3:AuBi*)[e6F*$!3O(4/+\"4H/8F*7$$!3m% *QDqf*R;\"F*$!3eBuRzSj)H\"F*7%7$$!33q*e#)R$*y:\"F*$!3U0.70Wz*H\"F*7$$! 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Analyze three d istinct types linear systems with constant coefficients. Pick a system , find the eigenvalues of the coefficient matrix to determine its clas sification, and have MAPLE plot the vector field and several solution \+ curves. You may choose three of any of the following types: saddle, s piral source, spiral sink, node, improper node, ellipses=center or sev eral degenerate types. see p. 387 [Section 6.2]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 338 "2. Choose a nonlinear sy stem of equations, that has at least two critical points. Plot the pha se portrait with direction field and several trajectories. Make sure y ou include enough trajectories that the global behavior is captured. T his means that you should include also those trajectories that connect the critical points. This way you" }}{PARA 0 "" 0 "" {TEXT -1 402 "kn ow the fate of every point on the plane. Find the coordinates of a cr itical points. Find a linearized system near one of the critical point s. Plot the direction field of your linearized system and several traj ectories to see that it approximates the behavior of the nonlinear sys tem near the critical point. (Choose from the list of nonlinear equati ons on p. 376 [section 6.1], or make up your own!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 309 "3. Plot the direction fi eld and several trajectories of van der Pol's equation x'' + u(x^2-1)x ' +x = 0. Choose the parameter u in such a way that the equation has an attractive limit cycle and show some of the trajectories spirallin g in and others out toward the limit cycle. (See p. 420 [application 6 .4].)" }}}}{MARK "39 3" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }