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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
 definition for adjoint" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new d
efinition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new defini
tion for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "solve( \{
thetadot(theta,v)=0, vdot(theta,v)=0\}, \{theta,v\});" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#<$/%\"vG-%'RootOfG6#,&-F'6#,&!\"\"\"\"\"*&)%#_ZG\"
\"#\"\"\",&*$)%\"RGF2F3F.F.F.F.F.F-*$F0F3F./%&thetaG-%'arctanG6$,$*&F7
F.F*F.F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "FixedSols := \+
convert(%, radical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*FixedSolsG<
$/%\"vG*$)*&\"\"\"F+,&*$)%\"RG\"\"#F+\"\"\"F1F1!\"\"#F1\"\"%F+/%&theta
G-%'arctanG6$,$*&F/F1-%%sqrtG6#F*F+!\"\"*$F<F+" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "subs(FixedSols, [theta,v]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$-%'arctanG6$,$*&%\"RG\"\"\"-%%sqrtG6#*&\"\"\"F/,&*$
)F)\"\"#F/F*F*F*!\"\"F/!\"\"*$F+F/*$)F.#F*\"\"%F/" }}}{EXCHG {PARA 0 "
> " 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'**!#
6$\"+)3b^(**!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "jacobian
( [thetadot(theta,v), vdot(theta,v)], [theta,v]);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$
*&-%$sinG6#%&thetaG\"\"\"%\"vG!\"\",&\"\"#\"\"\"*&,&*$)F.F1F-F2-%$cosG
F+!\"\"F-*$)F.\"\"#F-F/F97$,$F7F9,$*&%\"RGF2F.F2!\"#" }}}{EXCHG {PARA 
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$,$*$-%%sqrtG6#F/
F0F8,$F+!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(J
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&,&*&%\"RG\"\"\"-%%sqrtG6#,&*
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "will have complex eigenvalues when
ever " }{XPPEDIT 18 0 "R^4-7*R^2-8;" "6#,(*$%\"RG\"\"%\"\"\"*&\"\"(F'*
$F%\"\"#F'!\"\"\"\")F," }{TEXT -1 13 " is negative." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(R^4-7*R^2-8=0,R);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&,$*$-%%sqrtG6#\"\"#\"\"\"F(,$F$!\"
#%\"IG,$F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "At this moment
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" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"+CrUGG!\"*$!+CrUGGF%,$%\"IG$
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tr:=trace(J);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#trG,$*&%\"RG\"\"\")*&\"\"\"F+,&*$)F'\"\"#F+F(F(F(!
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et(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG,&*&)%\"RG\"\"#\"\"\"-
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{XPPMATH 20 "6$,$*$-%%sqrtG6#\"\"#\"\"\"F(,$F$!\"#" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 22 "for   R between 0 and " }{XPPEDIT 18 0 "2*sqrt(2)
;" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT -1 36 ", we will have a s
piral sink.   For " }{XPPEDIT 18 0 "2*sqrt(2) < R;" "6#2*&\"\"#\"\"\"-
%%sqrtG6#\"\"#F&%\"RG" }{TEXT -1 44 " we will have a sink with 2 eigen
directions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 30 "CASE I:   neutral fixed point." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 171 "R:=0;\nDEplot( GliderEQ, [theta,v], t=0..5, \n \+
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.2*Pi, v=.01..2,\n        stepsize=0.1, linecolor=black); " }}{PARA 
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