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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Local Stability Worked Exa
mple 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "Consider the nonlinear system" }}{PARA 
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:::::::::::::6:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
105 "1) Find the equilibrium points of the system and classify the equ
ilibrium points of their linearisations." }}{PARA 0 "" 0 "" {TEXT -1 
70 "2) What does ths tell you about the stability of the nonlinear sys
tem." }}{PARA 0 "" 0 "" {TEXT -1 90 "3) Verify the answer to (2) by lo
oking at the phase plane diagram of the nonlinear system." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 31 "To find the equil
ibrium points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{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 "" {MPLTEXT 1 0 11 "f:=x+y-x^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "g:=-x+y;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{f
=0,g=0\},\{x,y\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The equilib
rium points are (0,0), (2,2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 257 27 "To find the Jacobian matrix" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "v:=[f,g];" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 18 "At the point (0,0)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "J0:=subs(\{x=0,y=0\},evalm(J
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::::::::::::::::4:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The eigenva
lues are imaginary with positive real parts and the linearisation is a
n unstable spiral." }}{PARA 0 "" 0 "" {TEXT -1 119 "By the linearisati
on theorem the nonlinear system mimics this behaviour and the point (0
,0) is a repeller and unstable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 258 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 18 
"At the point (2,2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "J1:=
subs(\{x=2,y=2\},evalm(J));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
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"" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The eigen
values are real and negative and the linearisation is a stable node." 
}}{PARA 0 "" 0 "" {TEXT -1 117 "By the linearisation theorem the nonli
near system mimics this behaviour and the point(2,2) is an attractor a
ndstable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 259 26 "To plot the phase portrait" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "A:=diff(x(t),t)=x+y-x^2;" }}{PARA 11 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "B:=diff(y(t),t)=-
x+y;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 183 "DEplot(\{A,B\},\{x(t),y(t)\},-50..50,[[x(0)=1,y(0)=1
],[x(0)=.1,y(0)=.1],[x(0)=1.5,y(0)=2],[x(0)=2.6,y(0)=2.1],[x(0)=1.5,y(
0)=1.5],[x(0)=2.2,y(0)=2.2]],x=-.5..2.5,y=-0.5..2.5,stepsize=.1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Th
e phase portrait supports the above analysis" }}{PARA 0 "" 0 "" {TEXT 
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