{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 }{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 "Warning" -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 O utput" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 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 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Institutionen f\366r matem atik, " }{TEXT 256 3 "KTH" }}{PARA 0 "" 0 "" {TEXT -1 9 "Mats Boij" }} {PARA 18 "" 0 "" {TEXT -1 29 "Exempel. Styrning av flygplan" }}{PARA 0 "" 0 "" {TEXT -1 367 "Vi g\345r tillbaka till det system av differen tialekvationer som vi fick genom modellen av styrning av hastighet hos ett flygplan i f\366rel\344sning 1. Den andra modellen f\366r luftmot st\345ndet gav upphov till ett system av differentialekvationer som vi inte kunde l\366sa. Vi ska nu studera kritiska punkter och fasportr \344tt f\366r detta system. F\366rst importerar vi tre programpaket. \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "restart: with(linalg): with(plo ts): with(DEtools):" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {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 50 "Warning, the name changecoords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name adjoint has been redefined\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "De tv\345 differentialekvationerna \344r" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(u,t) = g1(u,v);" "6#/-%% diffG6$%\"uG%\"tG-%#g1G6$F'%\"vG" }{TEXT -1 5 " och " }{XPPEDIT 18 0 " diff(v,t) = g2(u,v);" "6#/-%%diffG6$%\"vG%\"tG-%#g2G6$%\"uGF'" }{TEXT -1 22 " d\344r g1 och g2 ges av " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "g1:=(u,v)->-k*u*sqrt(u^2+v^2)/m+M(u,v)/m; g2:=(u,v)->-k*v*sqrt(u^2+ v^2)/m-g+L(u,v)/m; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1G R6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF),&*&*(%\"kG\"\"\"9$F1-%%sqrtG6# ,&*$)F2\"\"#F1F1*$)9%F9F1F1F1F1%\"mG!\"\"F>*&-%\"MG6$F2FF1F)F)F )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2GR6$%\"uG%\"vG6\"6$%)operato rG%&arrowGF),(*&*(%\"kG\"\"\"9%F1-%%sqrtG6#,&*$)9$\"\"#F1F1*$)F2F:F1F1 F1F1%\"mG!\"\"F>%\"gGF>*&-%\"LG6$F9F2F1F=F>F1F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Vi beh\366ver v\344lja konstanternas v\344rden \+ f\366r att kunna f\345 numeriska resultat. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "k:=25.0: M:=(u,v)->1.0*10^6: m:=5.0*10^4: g:=9.8: a:= 2.5*10^3: L:=(u,v)->a*u*(1-0.01*v):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "F\366r att finna de kritiska punkterna ser vi p\345 sk\344rning en mellan kurvorna g1(u,v)=0 och g2(u,v)=0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "k1:=implicitplot(g1(u,v),v=-50..50,u=150..250,color= BLUE): k2:=implicitplot(g2(u,v),v=-50..50,u=150..250,color=RED): \+ Kp:=solve(\{g1(u,v),g2(u,v)\},\{u,v\}); display(k1,k2) ; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#KpG6%<$/%\"uG$\"+3v)***>!\" (/%\"vG$\"+hp(o***!#5<$/F-^$$!+HdA19F+$\"+7]vK:F+/F(^$$\"+r\"oDX&!\")$ \"+)))pw'=F+<$/F(^$F:$!+)))pw'=F+/F-^$F4$!+7]vK:F+" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6W7$7$$!#]\"\"!$\"3=E 2+[m\"*o>!#:7$$!3l9&RGON4!\\!#;$\"3K^RGON4q>F-7$7$$!3I*************f%F 1$\"3qGv?zKnt>F-F.7$7$$!#YF*F87$$!3/!z\"4EF(Ha%F1$\"3+z\"4EF(Hu>F-7$7$ $!#UF*$\"37G#*3tT0y>F-F>7$FD7$$!32Z]e$>4B=%F1$\"3q/&e$>4By>F-7$7$$!3_! 4p#G%)R0SF1$\"$)>F*FJ7$FP7$$!3/.1Jk6L=QF1$\"33g5V;J$=)>F-7$7$$!#QF*$\" 33&>1+!H,#)>F-FV7$Ffn7$$!3PjpO-R-^MF1$\"37'pO-R-^)>F-7$7$$!#MF*$\"38zf @:Wb&)>F-F\\o7$Fbo7$$!3wv&fB)Hv!3$F1$\"3ddfB)Hv!))>F-7$7$$!#IF*$\"3Ax: 7'H8())>F-Fho7$F^p7$$!3oqkHD-T2FF1$\"31Z'HD-T2*>F-7$7$$!#EF*$\"3M`E67d [\"*>F-Fdp7$Fjp7$$!3caUF=`)3L#F1$\"3_Du#=`)3$*>F-7$7$$!#AF*$\"3#e#oj2$ oQ*>F-F`q7$7$Fgq$\"3_Doj2$oQ*>F-7$$!3Ut<*>#f1^>F1$\"3ax\"*>#f1^*>F-7$7 $$!#=F*$\"3mV:L/#ee*>F-F_r7$Fer7$$!3_:m\"o\"y$yc\"F1$\"3[h;o\"y$y'*>F- 7$7$$!#9F*$\"3W(*4U-IX(*>F-F[s7$Fas7$$!3*4i?;4&3\"=\"F1$\"3?i?;4&3\")* >F-7$7$$!#5F*$\"3Oe+c\"y]')*>F-Fgs7$F]t7$$!3/F!oIz.p!z!#<$\"35!oIz.p!* *>F-7$7$$!\"'F*$\"3%fV!H,,X**>F-Fct7$7$F[u$\"3mN/H,,X**>F-7$$!3\\p0CD9 NlRFft$\"3a0CD9Nl**>F-7$7$$!\"#F*$\"3?+0D,+&)**>F-Fcu7$Fiu7$$\"3SK,+& \\()**\\\"!#>F\\v7$7$$\"\"#F*F\\vF_v7$Fdv7$$\"3\\?;Kn\"[`.%Fft$\"3%QyE $=lk**>F-7$7$$\"\"'F*F]uFhv7$7$F_wFau7$$\"31q`)RP\"*o4)Fft$\"3UY,E'3J! **>F-7$7$$\"#5F*F`tFcw7$Fiw7$$\"3[4eo>A3?7F1$\"3+>9.y<*z*>F-7$7$$\"#9F *FdsF]x7$Fcx7$$\"3JzQ\"oCK[j\"F1$\"3C7'=`x;l*>F-7$7$$\"#=F*FhrFgx7$F]y 7$$\"3k1x0&pbS0#F1$\"3IHU\\IWf%*>F-7$7$$\"#AF*FiqFay7$7$FhyF]r7$$\"3X% ==;mmyZ#F1$\"3[\"=QQL8A*>F-7$7$$\"#EF*F]qF\\z7$Fbz7$$\"3R$=;W9wj!HF1$ \"3W\"QebQi$*)>F-7$7$$\"#IF*FapFfz7$F\\[l7$$\"3q`k5z7pRLF1$\"3xa$*3s3. 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Vi v\344ljer fyra oli ka begynnelsevillkor. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sys:=\{d iff(u(t),t)=g1(u(t),v(t)),diff(v(t),t)=g2(u(t),v(t))\}; " }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%$sysG<$/-%%diffG6$-%\"vG6#%\"tGF-,(*&F*\"\"\"- %%sqrtG6#,&*$)-%\"uGF,\"\"#F0F0*$)F*F9F0F0F0$!+++++]!#8$\"#)*!\"\"FA*( $\"+++++]!#6F0F7F0,&F0F0*&$F0!\"#F0F*F0FAF0F0/-F(6$F7F-,&*&F7F0F1F0F<$ \"+++++?!\")F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "sol:=DEp lot(sys,\{u(t),v(t)\},t=0..100,u=180..220,v=-20..20, [[u(0)=220,v(0)=- 10],[u(0)=180,v(0)=10],[u(0)=220,v(0)=10],[u(0)=180,v(0)=-10]],scene=[ u,v],stepsize=0.2,linecolor=BLUE,scaling=constrained): display([sol]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6 ]dl7%7$$\"+%3=;z\"!\"($!+ujnj?!\")7$$\"+;>Q3=F*$!+EOKO>F-7$$\"+I0;.=F* $!+::%H%>F-7%7$$\"+[X>8=F*$!+U%Q+2#F-7$$\"+;2\"*G=F*$!+e:'*H>F-7$$\"+0 rvB=F*$!+#)QkS>F-7%7$$\"+$*p([$=F*$!+8:_w?F-7$$\"+NNL\\=F*$!+([yM#>F-7 $$\"+k0HW=F*$!+'fU&Q>F-7%7$$\"+$)Qnc=F*$!+!>AH3#F-7$$\"+4>kp=F*$!+5y2< >F-7$$\"+%>hZ'=F*$!+RTxO>F-7%7$$\"+/ley=F*$!+*ez*)3#F-7$$\"+_X$)*)=F*$ !+6/-6>F-7$$\"+)yu^)=F*$!+U'*[N>F-7%7$$\"+(y01!>F*$!+f#*R%4#F-7$$\"+L0 #*4>F*$!+T2g0>F-7$$\"+]Qa0>F*$!+'QL[$>F-7%7$$\"+K1rA>F*$!+*)p*))4#F-7$ $\"+_4#*H>F*$!+6I5,>F-7$$\"+Zu)e#>F*$!+jP\"\\$>F-7%7$$\"+!4p[%>F*$!+\\ HD-@F-7$$\"+ex')\\>F*$!+^qu(*=F-7$$\"+\">Hi%>F*$!+UbxN>F-7%7$$\"+%yUq' >F*$!+JpN/@F-7$$\"+G$*zp>F*$!+pIk&*=F-7$$\"+&f$fm>F*$!+m_QP>F-7%7$$\"+ (G$>*)>F*$!+/eA0@F-7$$\"+*3a(*)>F*$!+'>uZ*=F-7$$\"+F?+()>F*$!+-tjR>F-7 %7$$\"+X\")G6?F*$!+F-7%7$$\"+aXIL?F*$!+<\"RQ5#F-7$$\"+]L&)H?F*$!+$)3;'*=F-7$$\"+! 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