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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "vdot := (theta,v) -
> -sin(theta) - R*v^2;\nthetadot := (theta,v) -> (v^2 - cos(theta))/v;
\n\nGliderEQ := \{ diff(theta(t),t) = thetadot(theta(t),v(t)), \n     \+
         diff(v(t),t) = vdot(theta(t),v(t)) \};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%vdotGR6$%&thetaG%\"vG6\"6$%)operatorG%&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)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%)GliderEQG<$/-%%diffG6$-%&thetaG6#%\"tGF-*&,&*
$)-%\"vGF,\"\"#\"\"\"\"\"\"-%$cosG6#F*!\"\"F5F2!\"\"/-F(6$F2F-,&-%$sin
GF9F:*&%\"RGF6F1F5F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "wit
h(DEtools): with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new
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efinition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new defini
tion for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "solve( \{
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1 "" {XPPMATH 20 "6#<$/%\"vG-%'RootOfG6#,&-F'6#,&!\"\"\"\"\"*&)%#_ZG\"
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" {XPPMATH 20 "6#7$-%'arctanG6$,$*&%\"RG\"\"\"-%%sqrtG6#*&\"\"\"F/,&*$
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> " 0 "" {MPLTEXT 1 0 51 "FixedPt := unapply (subs(FixedSols, [theta,v
]), R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(FixedPtGR6#%\"RG6\"6$%)o
peratorG%&arrowGF(7$-%'arctanG6$,$*&9$\"\"\"-%%sqrtG6#*&\"\"\"F8,&*$)F
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "FixedPt(0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "FixedPt(0.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+\\_'o'**!#
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" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$
*&-%$sinG6#%&thetaG\"\"\"%\"vG!\"\",&\"\"#\"\"\"*&,&*$)F.F1F-F2-%$cosG
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0 "> " 0 "" {MPLTEXT 1 0 104 "J:=simplify(eval(jacobian( [thetadot(the
ta,v), vdot(theta,v)], [theta,v]),\n                 FixedSols));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrixG6#7$7$,$*&%\"RG\"\"\")
*&\"\"\"F0,&*$)F,\"\"#F0F-F-F-!\"\"#F-\"\"%F0!\"\"F47$,$*$-%%sqrtG6#F/
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(J);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,$*&,&*&%\"RG\"\"\"-%%sqrtG6#,&*$)F'\"\"#\"\"\"F(F(F(F0
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{TEXT -1 39 "will have complex eigenvalues whenever " }{XPPEDIT 18 0 "
R^4-7*R^2-8;" "6#,(*$)%\"RG\"\"%\"\"\"F(*&\"\"(F(*$)F&\"\"#F(F(!\"\"\"
\")F." }{TEXT -1 13 " is negative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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