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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 32 "Global Stability Investig
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "for that system. " }}
{PARA 0 "" 0 "" {TEXT -1 90 "You are going to examine the two diagrams
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warni
ng, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, n
ew definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
with(DEtools):" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definition
 for adjoint" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 56 "To plot the pha
se plane diagram for the nonlinear system" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "A:=diff(x(t),t)=-x^3;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "B:=diff(y(t),t)=-y^3;" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "p1:=DEplot(\{A,B\},\{x(t),y(t)\},t
=0..50,[[x(0)=1,y(0)=1],[x(0)=-.25,y(0)=1],[x(0)=-1,y(0)=-.4],[x(0)=.3
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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 " Verify that " }}{PARA 256 "
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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "is a Lyapunov function for th
e system." }}{PARA 0 "" 0 "" {TEXT -1 35 "If it  is a Lyapunov functio
n then " }}{PARA 257 "" 0 "" {OLE 1 3612 1 "[xm]Br=WfoRrB:::wk;nyyI;G:
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:E
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0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "V
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14 "Vx:=diff(V,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Vy:=d
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0 "" {TEXT -1 5 "Thus " }}{PARA 258 "" 0 "" {OLE 1 3612 1 "[xm]Br=WfoR
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:" }}{PARA 0 "" 0 "" {TEXT -1 69 "for all x and y so V is a Lyapunov f
unction for the nonlinear system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 257 29 "To plot the level curves of V" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "p2:=implicitplot(\{x^2+y^2=1
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"" {MPLTEXT 1 0 3 "p2;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "The two diagrams are obviously not the s
ame. to examine them more closely look at the two graphs plotted on th
e same axes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{p
1,p2\});" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 109 "Notice that the solution curves for the nonlinear system
 all cross the level curves of the Lyapunov function." }}}}{MARK "24 0
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