{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 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 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 "" -1 259 "" 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 "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {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 "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 -1 100 "Today we're doing an in-c lass exercise which should help you figure out what to do on the proje ct. " }}{PARA 0 "" 0 "" {TEXT -1 51 "We'll examine the system of diff erential equations:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "ode: = [ diff(x(t),t)= -y(t)+y(t)^2 +k, diff(y(t),t)= x(t)-y(t)+x(t)*y(t) ] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG7$/-%%diffG6$-%\"xG6#%\"tG F-,(-%\"yGF,!\"\"*$)F/\"\"#\"\"\"F5%\"kGF5/-F(6$F/F-,(F*F5F/F1*&F*F5F/ F5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Our goal is to describe t he behavior of solutions for various values of " }{TEXT 265 3 "k>0" } {TEXT -1 29 ". Here's what you should do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "1. Draw DEplots for " }{TEXT 263 5 "k=0.1" }{TEXT -1 5 " \+ and " }{TEXT 264 3 "k=3" }{TEXT -1 74 ". Describe in words (on your m aple sheet) what is happening to solutions." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 54 "2. Solve for the fixed points of the ode in terms of \+ " }{TEXT 262 1 "k" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "3. Do an analysis of the fixed points for " }{TEXT 256 5 "k=0. 1" }{TEXT -1 5 " and " }{TEXT 257 3 "k=3" }{TEXT -1 164 " to describe \+ the behavior of solutions.\n ( for each fixed point, compute trace and det of Jacobian at the fixed point, \n from that conclud e behavior )." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "4. Solve for th e " }{TEXT 258 1 "k" }{TEXT -1 15 "-value between " }{TEXT 259 3 "0.1 " }{TEXT -1 5 " and " }{TEXT 260 1 "3" }{TEXT -1 31 " at which the beh avior changes." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "5. If the fix ed point analysis doesn't explain everything you see in the pictures, \+ then come up with\n a way to explain the behavior of solutions f or ALL " }{TEXT 261 1 "k" }{TEXT -1 23 " (in words at least). 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Where's the fixed point?" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "2. Solve for the fixed point (in terms of k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:='k';" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "FP:=convert(solve( \{-y+y^2+k=0, x-y+x*y=0\},\{x,y\}) ,radical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FPG<$/%\"yG,&#\"\"\" \"\"#F**&F)F*-%%sqrtG6#,&F*F**&\"\"%F*%\"kGF*!\"\"F*F*/%\"xG*&,(F3F*F) F**&F)F*F-F*F*F*,&F3F*F+F*F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "3 . Analyze fixed points, start with k=0.1 and k=3." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "fp1:=subs(k=.1, FP); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fp1G<$/%\"yG$\"+YL)H())!#5/%\"xG$\"+T1U,ZF*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name adjoint has been redefined\n" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "J:= jacobian( [-y+y^2+k, x-y+x*y], [x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrixG6#7$7$\"\"!,&!\"\"\"\"\"*&\" \"#F-%\"yGF-F-7$,&F-F-F0F-,&F,F-%\"xGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "trfix:=subs(FP, trace(J));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&trfixG,&!\"\"\"\"\"*&,(%\"kGF'#F'\"\"#F'*&F+F'-%%sqr tG6#,&F'F'*&\"\"%F'F*F'F&F'F'F',&F*F'F,F'F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "detfix:= subs(FP, det(J));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'detfixG,$*&-%%sqrtG6#,&\"\"\"F+*&\"\"%F+%\"kGF+!\"\" F+,&#\"\"$\"\"#F+*&#F+F3F+F'F+F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs( k=.1, trfix);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$!+f$z&)H&!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs( k= .1, detfix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+/]*=Y\"!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "Confirms for k=.1, there is a fix ed point at (.5, .9) which is a saddle.\nFor k=3, we didn't see a fixe d point, if you look at formula for f.p. in terms of k, you see if k i s large, the f.p. is\ncomplex:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(subs( k=3, FP));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/% \"yG^$$\"+++++]!#5$\"+&R7$e;!\"*/%\"xG^$$\"+++++qF)$\"+!zCmJ$F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "If you evaluate trfix, detfix at \+ k=3, also get complex...\n4. Solve for at which k the behavior change s, i.e. the f.p. becomes complex:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve ( 1-4*k=0, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 431 "Conclude: for 0<= k <= .25, there's a fixed point which is a saddle, for k> .25, we guess the solutions just tend toward positive infinity inthe x-direct ion, and there's no real fixed point.\n\nIf k>.25, we can't use fixed \+ point analysis to describe the behavior of solutions.... what can we d o?\n\nExamine equations, at least when x and y are small, the +k for d x/dt dominates, and solutions just shift to the right (in forward time )." }}}}{MARK "31 0 0" 431 }{VIEWOPTS 1 1 0 2 1 1805 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }