{VERSION 6 0 "SGI MIPS UNIX" "6.0" } {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 } {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 "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 "" {MPLTEXT 1 0 62 "restart;\nf := (x,y) -> a*x-b*x^2-x*y;\ng := (x,y) -> m*x*y-n*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*&%\"aG \"\"\"9$F0F0*&%\"bGF0)F1\"\"#F0!\"\"*&F1F09%F0F6F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*( %\"mG\"\"\"9$F09%F0F0*&%\"nGF0F2F0!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "MLV := diff(x(t),t) = f(x(t),y(t)), diff(y(t),t) = g(x(t),y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MLVG6$/-%%diffG 6$-%\"xG6#%\"tGF-,(*&%\"aG\"\"\"F*F1F1*&%\"bGF1)F*\"\"#F1!\"\"*&F*F1-% \"yGF,F1F6/-F(6$F8F-,&*(%\"mGF1F*F1F8F1F1*&%\"nGF1F8F1F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The fixed points of the system are" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "F := solve( \{f(x,y)=0,g(x,y )=0\}, \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG6%<$/%\"yG \"\"!/%\"xGF)<$F'/F+*&%\"aG\"\"\"%\"bG!\"\"<$/F+*&%\"nGF0%\"mGF2/F(*&, &*&F/F0F7F0F0*&F1F0F6F0F2F0F7F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Note that for the third fixed point to be positive " }{XPPEDIT 18 0 "a-b*n/m;" "6#,&%\"aG\"\"\"*(%\"bGF%%\"nGF%%\"mG!\"\"F*" }{TEXT -1 11 " > 0 i.e. " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 3 " > " } {XPPEDIT 18 0 "b*n/m;" "6#*(%\"bG\"\"\"%\"nGF%%\"mG!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The Jacobian matrix of part ial derivatives is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "J := \+ Matrix([[diff(f(x,y),x),diff(f(x,y),y)],\n [diff(g(x,y),x) ,diff(g(x,y),y)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'RTABLE G6%\"*3Q+q#-%'MATRIXG6#7$7$,(%\"aG\"\"\"*(\"\"#F0%\"bGF0%\"xGF0!\"\"% \"yGF5,$F4F57$*&F6F0%\"mGF0,&*&F:F0F4F0F0%\"nGF5%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "J1 := eval(J,F[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%'RTABLEG6%\"*+n/q#-%'MATRIXG6#7$7$%\"aG\" \"!7$F/,$%\"nG!\"\"%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "T he fixed point (0,0) is a saddle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "J3 := eval(J,F[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#J3G-%'RTABLEG6%\"*w90q#-%'MATRIXG6#7$7$,(%\"aG\"\"\"**\"\"#F0%\"b GF0%\"nGF0%\"mG!\"\"F6*&,&*&F/F0F5F0F0*&F3F0F4F0F6F0F5F6F6,$*&F4F0F5F6 F67$F8\"\"!%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "lam bda := LinearAlgebra:-Eigenvalues(J3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG-%'RTABLEG6%\"*/cgq#-%'MATRIXG6#7$7#,$*(\"\"#!\"\",&*& %\"bG\"\"\"%\"nGF5F5*$,(*&)F4F0F5)F6F0F5F5**\"\"%F5F6F5%\"aGF5)%\"mGF0 F5F1**F=F5F@F5F4F5F;F5F5#F5F0F1F5F@F1F17#,$*(F0F1,&F3F5F7F5F5F@F1F1&%' VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Delta := b^2*n^2-4*n*a*m^2+4*m*b*n^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &DeltaG,(*&)%\"bG\"\"#\"\"\")%\"nGF)F*F***\"\"%F*F,F*%\"aGF*)%\"mGF)F* !\"\"**F.F*F1F*F(F*F+F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "n,m := 1,1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"nG%\"mG6$\"\" \"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Delta;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(*$)%\"bG\"\"#\"\"\"F(*&\"\"%F(%\"aGF(!\"\"*&F* F(F&F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plots[implicit plot]( [Delta=0,a-b*n/m=0], a=0..3, b=0..3, grid=[50,50], color=[blue, red], thickness=2 );" }}{PARA 13 "" 1 "" {GLPLOT2D 234 223 223 {PLOTDATA 2 "6%-%'CURVESG6\\u7$7$$\"\"!F)F(7$$\"3#H`ZgHuW3$!#>$\"3o.Va $ouz.$F-7$7$$\"3gO=fz*[C7'F-$\"3mW)oPv],.'F-F*7$F17$$\"3$e1&4#f[*ohF-$ \"3P2')3n$\\f2'F-7$7$$\"35*4hL$*fh@'F-F2F77$F=7$$\"3mgWk8J=)H*F-$\"38 \\58DQ;p!*F-7$7$$\"3Kn$=fz*[C7!#=$\"3DrT=q5l)=\"FJFA7$FG7$$\"35S%***Q8 \"GC\"FJ$\"3S%HPGDoh?\"FJ7$7$$\"3fsgUxT(>E\"FJFHFN7$7$$\"3KsgUxT(>E\"F JFH7$$\"3!z([/Nx=FJ$\"3[\"3w*Q*=hz\"FJ7$7$$\"3W())pAbu5#>FJ F\\oFao7$Fgo7$$\"3+q&=mQ[%)>#FJ$\"3Y:d4**eE(3#FJ7$7$$\"3kMn$=fz*[CFJ$ \"38(=q*[@b8BFJF[p7$Fap7$$\"3Hh8&QQO,_#FJ$\"3)z5A)*zAyP#FJ7$7$$\"3=bv' y6<*)f#FJFbpFgp7$F]q7$$\"3Wp1v[h'\\%GFJ$\"32$)>)G$zBlEFJ7$7$$\"3V=fz*[ C71$FJ$\"3S$)))fGDJbGFJFaq7$Fgq7$$\"3=0cKl[\"3<$FJ$\"3oJiE9Tj^HFJ7$7$$ \"3ev!>U(=]&H$FJFhqF]r7$Fcr7$$\"3WYb'Qmf\"*\\$FJ$\"3Auao8U`NKFJ7$7$$\" 3B-^v(QpMn$FJ$\"3QGp1p]s%Q$FJFgr7$F]s7$$\"3Vf#[1i^!HQFJ$\"3.X>'[:()y^$ FJ7$7$$\"32[WK@)G3,%FJF^sFcs7$Fis7$$\"3#=VtoO32;%FJ$\"3Wcff1`Z)z$FJ7$7 $$\"3-'G9dG9dG%FJ$\"3;I+Jibt-RFJF]t7$Fct7$$\"31820\"*4`%\\%FJ$\"3afyP! 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