{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 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 -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 }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 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "vdot := (theta,v) -> -sin(theta) - R*v^2;\nthetadot := (theta,v) -> (v^2 - cos(theta))/v; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%vdotGR6$%&thetaG%\"vG6\"6$%)ope ratorG%&arrowGF),&-%$sinG6#9$!\"\"*&%\"RG\"\"\")9%\"\"#\"\"\"F2F)F)F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)thetadotGR6$%&thetaG%\"vG6\"6$% )operatorG%&arrowGF)*&,&*$)9%\"\"#\"\"\"\"\"\"-%$cosG6#9$!\"\"F3F1!\" \"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "GliderEQ := \{ diff(v(t),t) = vdot(theta(t),v(t)),\n diff(theta(t),t) = thetadot(theta(t),v(t))\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Glid erEQG<$/-%%diffG6$-%\"vG6#%\"tGF-,&-%$sinG6#-%&thetaGF,!\"\"*&%\"RG\" \"\")F*\"\"#\"\"\"F4/-F(6$F2F-*&,&*$F8F:F7-%$cosGF1F4F:F*!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "J:=jacobian( [thetadot(theta,v), vd ot(theta,v)], [theta,v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%' matrixG6#7$7$*&-%$sinG6#%&thetaG\"\"\"%\"vG!\"\",&\"\"#\"\"\"*&,&*$)F0 F3F/F4-%$cosGF-!\"\"F/*$)F0\"\"#F/F1F;7$,$F9F;,$*&%\"RGF4F0F4!\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "When vdot = 0 and thetadot=0, we h ave " }{XPPEDIT 18 0 "sin(theta) = -Rv^2;" "6#/-%$sinG6#%&thetaG,$*$) %#RvG\"\"#\"\"\"!\"\"" }{TEXT -1 0 "" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "cos(theta) = v^2;" "6#/-%$cosG6#%&thetaG*$)%\"vG\"\"#\"\"\"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "subs(\{sin(theta)=R*v^2, cos(theta)=v^2\}, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$*&%\"RG\"\"\"%\"vGF*\"\"#7$,$*$)F+F, \"\"\"!\"\",$F(!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "so at the \+ fixed point, our system acts like this matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "if R=0, system is like \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "\{diff(epsilon(t),t) = \+ 2*delta(t),\ndiff(delta(t),t) = - epsilon(t)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%%diffG6$-%(epsilonG6#%\"tGF+,$-%&deltaGF*\"\"#/-F& 6$F-F+,$F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(% ,[epsilon,delta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%&deltaG6#% \"tG,&*&-%$cosG6#*&-%%sqrtG6#\"\"#\"\"\"F(\"\"\"F4%$_C1GF4F4*(F/F3-%$s inGF-F4%$_C2GF4#!\"\"F2/-%(epsilonGF',&*(F/F3F7F3F5F3F4*&F+F3F9F3F4" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "DEplot(\{diff(epsilon(t),t) = 2*de lta(t),\ndiff(delta(t),t) = - epsilon(t)\}, [epsilon,delta], t=0..2, e psilon=-2..2, delta=-2..2);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG 6]dl7%7$$!+p(\\e!>!\"*$!+:^2Z?F*7$$!+J-:%4#F*$!+&)[#H&>F*7$$!+k2:l?F*$ !+ml%o*>F*7%7$$!+_vQ$p\"F*$!+,f)H/#F*7$$!+;)fb)=F*$!+*49q&>F*7$$!+w(*p a=F*$!+t!\\'**>F*7%7$$!+#HO5[\"F*$!+4\"\\'Q?F*7$$!+W%eon\"F*$!+\"*3Nh> F*7$$!+#)R7W;F*$!+:Uc-?F*7%7$$!+V[#)o7F*$!+(QsS.#F*7$$!+hs,o9F*$!+8w#f '>F*7$$!+M*3MV\"F*$!+U/e0?F*7%7$$!+3Fyc5F*$!+r#p#H?F*7$$!+kn+f7F*$!+H2 tq>F*7$$!+*GTDA\"F*$!+-Fo3?F*7%7$$!+j\"*Q\\%)!#5$!+>'fU-#F*7$$!+Czz\\5 F*$!+\"QSd(>F*7$$!+`&4:,\"F*$!+![_=,#F*7%7$$!+hr?LjFep$!+H(p!>?F*7$$!+ F\\j.%)Fep$!+r-$4)>F*7$$!+!\\[I+)Fep$!+`!o],#F*7%7$$!+'y@&>UFep$!+a?t8 ?F*7$$!+')Rz1jFep$!+YzE')>F*7$$!+Lk@*)eFep$!+-]I=?F*7%7$$!+c#G&3@Fep$! 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