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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Local Stability Investigat
ion 3." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "Consider the nonlinear system " }}{PARA 256 "" 0 "" {OLE 
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{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "First plot the ph
ase portrait of the nonlinear system close to the equilibrium point." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 
"" 1 "" {TEXT -1 35 "Warning, new definition for adjoint" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}{PARA 7 "" 1 "" 
{TEXT -1 35 "Warning, new definition for adjoint" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 34 "The equilibrium point is given by " }{OLE 1 3584 
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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf:
:wyyyqy;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyJ<j?V:;J:>R<:T><:
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=y+x*(1-x^2-y^2);" }}{PARA 11 "" 
1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g:=-x+
y*(1-x^2-y^2);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "solve(\{f=0,g=0\},\{x,y\});" }}{PARA 11 "" 1 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The equilibrium
 point is (0,0)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 31 "Plot the phase plane as follows" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "A:=diff(x(t),t)=y+x*(1-x^2-y^2);" }}{PARA 11 "" 
1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B:=dif
f(y(t),t)=-x+y*(1-x^2-y^2);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 65 "Here is the phase plane plot of \+
the nonlinear system in the range" }}{PARA 259 "" 0 "" {OLE 1 4096 1 "
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yyqy;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
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;JSFDYR:=B:;B:[KZ=j>JSjsAj:jR<JZ;_;id;=J:>@:_;qs;=:];JSFLYR:=B:>@:_;aY
:=:];JSjkTj:JR:N@vbAF:F@:_kbfq;F:n_:>FM:_;Er;=:W;JSFLuW:=:W;JSjBXj:JQj
Djw;<:[V:=:;b::::Mf:C:UK:^:>X=j>>:_kbf^>F:<JvNOM:_kBG`>FZ:jvNOOZ:n>N;y
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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::1:" }
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "DEplot(\{
A,B\},\{x(t),y(t)\},-50..50,[[0,1,1],[0,-1,1],[0,0.1,-0.1],[0,-0.1,-0.
1]],x=-.1..0.1,y=-.1..0.1,stepsize=.1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The equilibri
um point resembles that of an unstable linear focus." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "H
ere is the phase plot of the nonlinear system in the range" }}{PARA 
260 "" 0 "" {OLE 1 4096 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyy
yy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::fyyyyya:nYf::G:jy;::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ej
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:::::::::::::::::::::::::::::::=B:DZ:B::::::::::>K^Z;r:Eb<CR:=R:?b[Uc:
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;fy>Z:JBA:DZ<B>aTXUeRYEUHL<jP@:MB:;B:_c<]T:sk:B:UC:ap>JSJlEj:>:Aoc;_;W
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F:f@:_kBw[;FZ:jR:N@vv;F:f@:_;Qd;=:];JSjKXj:jT:N@]lLAj:JQ<JjF;N@ft@F:n?
:_kBgn;F:n?:_;=i;=:W;eJ:VYZ:JBAj:J:D:;J:>`gxJ[Se>:;::::::::Jd`@a=:FMC:
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::::::::::::::::::::::::3:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 116 "DEplot(\{A,B\},\{x(t),y(t)\},-50..50,[[0,1,1],[0,-
1,1],[0,0.1,-0.1],[0,-0.1,-0.1]],x=-1.5..1.5,y=-1.5..1.5,stepsize=.1);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 153 "The global phase portrait does not resemble that of
 a linear unstable focus but shows the existence of a closed trajector
y. This is called a limit cycle." }}}}{MARK "19 0 0" 0 }{VIEWOPTS 1 1 
0 1 1 1803 }