{VERSION 4 0 "IBM INTEL NT" "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 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 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 "" 1 14 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 "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 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 Out put" -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 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 "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "restart:with(DEtools ):with(linalg):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, t he name adjoint has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "W arning, the protected names norm and trace have been redefined and unp rotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoo rds has been redefined\n" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 53 "Periodic orbits of systems of differenti al equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 71 "1. A system with two periodic orbits, one attracting a nd one repelling." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 92 "system1:=diff(x(t),t)=y+x*(1-x^2-y^2)*(4-x^2 -y^2),diff(y(t),t)=-x+y*(1-x^2-y^2)*(4-x^2-y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(system1G6$/-%%diffG6$-%\"xG6#%\"tGF-,&%\"yG\"\"\"*(F +F0,(F0F0*$)F+\"\"#F0!\"\"*$)F/F5F0F6F0,(\"\"%F0F3F6F7F6F0F0/-F(6$-F/F ,F-,&F+F6*(F/F0F2F0F9F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 257 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 45 "The same system in \+ polar coordinates becomes:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(r(t) ,t) = r*(1-r^2)*(4-r^2);" "6#/-%%diffG6$-%\"rG6#%\"tGF**(F(\"\"\",&F,F ,*$)F(\"\"#F,!\"\"F,,&\"\"%F,*$F/F,F1F," }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(theta(t),t) = -1;" "6#/-%%diffG6$-%&thetaG6#%\"tGF*,$\"\"\" !\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 270 83 "The system is going to have an attracting limit cycle \+ r=1 and a repelling cycle r=2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 11 "P hase plot:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 142 "DEplot(\{system1\},[x(t),y(t)],-10..10,\{[0,0.7,0] ,[0,-0.7,0],[0,1,1],[0,-1,-1],[0,-2,-1],[0,2,1]\},x=-3..3,y=-3..3,step size=0.03,linecolor=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 354 266 266 {PLOTDATA 2 "6'-%'CURVESG6]dl7%7$$!+RM)y)G!\"*$!+`;#)))GF*7$$!+hl67JF* $!+Z$y66$F*7$$!+zSxNIF*$!+^e1\"4$F*7%7$$!+*e%HyDF*$!+]$**G)GF*7$$!+jk7 !z#F*$!+]15F*$!+1e[qGF*7$$! +,n%H9#F*$!+%>9&HJF*7$$!+d0Yr?F*$!+%o6g4$F*7%7$$!+V!*=c;F*$!+<#eU'GF*7 $$!+l^\\<=F*$!+$yTd8$F*7$$!+>fj[ V\"G&GF*7$$!+6>Ui6F*$!+\"o&=ZJF*7$$!+C](35\"F*$!++@u(4$F*7%7$$!+Ly]hu! #5$!+w\\;[GF*7$$!+8e'zK)Fer$!+C]$=:$F*7$$!+*f%ygxFer$!+b#>p4$F*7%7$$!+ nQOaWFer$!+?ElWGF*7$$!+0.K>]Fer$!+!QZ`:$F*7$$!+)>N'3XFer$!+Z>9&4$F*7%7 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Examples of Poincare -Bendixson" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "system2:=diff(x(t) ,t)=y,diff(y(t),t)=-x+y*(4-x^2-4*y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(system2G6$/-%%diffG6$-%\"xG6#%\"tGF-%\"yG/-F(6$-F.F,F-,&F+!\" \"*&F.\"\"\",(\"\"%F6*$)F+\"\"#F6F4*&F8F6)F.F;F6F4F6F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 261 92 "Note that this looks like a mechanical system wit h \"friction\" given by the nonlinear term in" }}{PARA 0 "" 0 "" {TEXT 271 19 "the second equation" }{TEXT -1 2 ". " }{TEXT 272 54 "Let us try to use the energy of the \"undamped\" system:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "L(x,y) = (x^2+y^2)/2;" "6#/-%\"LG6$%\"xG%\"yG*&,&*$) F'\"\"#\"\"\"F.*$)F(F-F.F.F.F-!\"\"" }}{PARA 0 "" 0 "" {TEXT 273 17 "W e can calculate:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(L(x(t),y(t)),t ) = y^2*(4-x^2-4*y^2);" "6#/-%%diffG6$-%\"LG6$-%\"xG6#%\"tG-%\"yG6#F-F -*&)F/\"\"#\"\"\",(\"\"%F4*$)F+F3F4!\"\"*&F6F4*$F2F4F4F9F4" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 17 "This implies that" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "0 <= diff(L(x(t),y(t)),t);" "6#1\"\"!-%%diffG6$-%\"L G6$-%\"xG6#%\"tG-%\"yG6#F.F." }{TEXT -1 1 " " }{TEXT 275 2 "if" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/,&*$)%\"xG\"\"#\"\"\" F)*$)%\"yGF(F)F)F)" }{TEXT -1 1 " " }{TEXT 276 3 "and" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "diff(L(x(t),y(t)),t) <= 0;" "6#1-%%diffG6$-%\"LG6$-% \"xG6#%\"tG-%\"yG6#F-F-\"\"!" }{TEXT -1 1 " " }{TEXT 277 2 "if" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2 = 4;" "6#/,&*$%\"xG\"\"#\"\"\"* $%\"yGF'F(\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 278 47 "There fore, the system has an invariant annulus:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "L:=(x,y)->(x^2+y^ 2)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGR6$%\"xG%\"yG6\"6$%)ope ratorG%&arrowGF),&*$)9$\"\"#\"\"\"#F2F1*&F3F2)9%F1F2F2F)F)F)" }}} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "solve(2*L(x,y)=1);solve(2*L(x,y)=4);" }}{PARA 11 "> " 1 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"yGF%/%\"xG*$ -%%sqrtG6#,&\"\"\"F-*$)F%\"\"#F-!\"\"F-<$F$/F',$F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"xG*$-%%sqrtG6#,&\"\"%\"\"\"*$)%\"yG\"\"#F,!\" \"F,/F/F/<$/F%,$F&F1F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "A :=plot(\{sqrt(1-y^2),-sqrt(1-y^2),sqrt(4-y^2),-sqrt(4-y^2)\},y=-3..3,x =-3..3,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dis play(A);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7P7$$!3S+++&[Ib***!#= $\"3cG3'*pp[*)H!#>7$$!3s******\\XF`**F*$\"3GQ7#>f,dl*F-7$$!3E+++.>o\") )*F*$\"3:sl)zUTP`\"F*7$$!3y*****pD*35)*F*$\"3)=Y2b$fiR>F*7$$!3L+++5m\\ Q(*F*$\"3c64%)GR$>F#F*7$$!3w*****H'R!pm*F*$\"3!evr4%e[fDF*7$$!3$)***** *p'=P_*F*$\"3=ejzW0R\\IF*7$$!3s*****\\PL0Q*F*$\"3EVZw())3\\Y$F*7$$!3y* ****zyiT4*F*$\"3O()R!=Uq)eTF*7$$!3u*******>#z2))F*$\"3zd8VS:FNZF*7$$!3 ))*****z7vYZ)F*$\"3?deB7EZ3`F*7$$!3'******\\0e:9)F*$\"3AF'R2rkk!eF*7$$ !36+++$)4W3yF*$\"3Q(\\3BNfsC'F*7$$!3=+++5RKvuF*$\"3?AsTmNDUmF*7$$!3%)* ****\\!Qf&)oF*$\"3;w%Qaj*z^sF*7$$!3]*******pjeH'F*$\"3kRJm0]IpxF*7$$!3 7+++0j$)[cF*$\"3S:([sy0J%Q[#F*$\"3')zy>\"o:mo*F*7$$!34+++g6:.8F*$\"3m[vFsis9**F*7$$ !35++++!Q:'H!#?$\"3'f@nj9c*****F*7$$\"3++++!RIKH\"F*$\"3.[&4$p^-;**F*7 $$\"3/+++5:xWCF*$\"3+,CGM0b'p*F*7$$\"3-+++![n%)o$F*$\"3)Q-$))RA!\\H*F* 7$$\"3z*******G(*3L%F*$\"3B'>&*=73N,*F*7$$\"37++++rKt\\F*$\"3kOr#R(*)f v')F*7$$\"3[+++S$>=g&F*$\"3G#yF*7$$\"3R+++N,TQoF*$\"3wRi/%\\5jH(F*7$$\"3%********o3lW(F*$\"3M'fF n!=aumF*7$$\"3m*****Hd.Ty(F*$\"3M(\\E2xdvF'F*7$$\"3V+++b%)p@\")F*$\"3C 5U\\v1@MeF*7$$\"3;+++QLHf%)F*$\"3Vvbn[+&HL&F*7$$\"3!)******>#))oz)F*$ \"3WS$fKi(\\bZF*7$$\"3>+++SAB+\"*F*$\"3+L::?FdXTF*7$$\"3Y******fid.%*F *$\"3=X0*4+v=S$F*7$$\"3m******p#[_b*F*$\"39)3P*Gn5\\HF*7$$\"3&)******z -#pq*F*$\"3#\\GVCLoKS#F*7$$\"3]+++&GcFy*F*$\"3UlR%*HX3t?F*7$$\"3/+++!H #fe)*F*$\"3%o=(emfvv;F*7$$\"3S+++$H5l*)*F*$\"3;9zRx3&\\V\"F*7$$\"3e*** **\\HGW$**F*$\"3cJ?B3)*HV6F*7$$\"3%******zHYB(**F*$\"3SAq\"=c^!#<$\"3Mgod9*[N0\"F*7$$!3%******z:eK*>Ffz$\"34Ffz$\"3QqP3qeqm?F*7$$!35+++d\"G`)>Ffz$\"3=68w!o+\"=CF* 7$$!3-+++c\")Rx>Ffz$\"3s>-A[]F)*HF*7$$!33+++a\"o%p>Ffz$\"3!osb+'*48[$F *7$$!33+++^\"3O&>Ffz$\"37Hj`$***p#G%F*7$$!31+++[\"[x$>Ffz$\"3gLDp]=3^ \\F*7$$!3/+++U\"Gg!>Ffz$\"3'=3;si?&egF*7$$!3-+++O\"3V(=Ffz$\"3on%3q\\; $ypF*7$$!3!******4@!oT=Ffz$\"39)p+Qg2))z(F*7$$!3,+++'G_!4=Ffz$\"3KtYl@ MOG&)F*7$$!3/+++gVUwjo\"Ffz$\"3Q![]>'pKv5Ffz7$$!3.+++e;%)G;Ffz$\"3 aa'QY7Ffz7$$!3!******\\!)H%*\\ \"Ffz$\"3#*Q#[^\">_B8Ffz7$$!3'********=eWV\"Ffz$\"3q?Dw!*[n$R\"Ffz7$$! 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Another example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "system3:=diff(x(t),t)=y-8*x^3,diff( y(t),t)=2*y-4*x-2*y^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(system3G6 $/-%%diffG6$-%\"xG6#%\"tGF-,&%\"yG\"\"\"*&\"\")F0)F+\"\"$F0!\"\"/-F(6$ -F/F,F-,(F/\"\"#*&\"\"%F0F+F0F5*&F;F0)F/F4F0F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 102 "One can check that the rectangle with corners (-1,2), (1,2), (1,-2), (-1,-2) is positive ly invariant. " }}{PARA 0 "" 0 "" {TEXT 279 21 "For example, on y=2 \+ " }{XPPEDIT 18 0 "diff(y(t),t) = -4*x-12;" "6#/-%%diffG6$-%\"yG6#%\"tG F*,&*&\"\"%\"\"\"%\"xGF.!\"\"\"#7F0" }{TEXT -1 1 " " }{TEXT 280 38 "wh ich is negative inside the rectangle" }}{PARA 0 "" 0 "" {TEXT 281 106 "The fixed point (0,0) is unstable. 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