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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 40 "Stability of equilibria \+
worked example 1" }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the system \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {OLE 1 
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::2:" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 29 "To find the equilibrium
 point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 
1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "E1:=-5*x-3*y-1;" }}{PARA 11 "" 1 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "E2:=6*x+4*y+2;" }}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "solve(\{E1=0,E2=0\},\{x,y\});" }}{PARA 11 "" 1 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "the equilibrium pont is (1,-2)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 24 "To f
ind the eigenvalues." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "A:=matrix(2,2,[-5,-3,6,4]);" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eigenvalues
:=eigenvals(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "there are two \+
real eigenvalues so there will be two real eigenvectors." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 24 "To find the eigen
vectors" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "e
igensystem:=eigenvectors(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 23 "To find the directrice
s" }}{PARA 0 "" 0 "" {TEXT -1 28 "The directrices are given by" }}
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0 "" 0 "" {TEXT -1 76 "Taking k=1 and -1 gives the initial conditions \+
(0,-1), (2,-3), (2,-4), (0,0)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 258 26 "To plot the phase portrait" }}}{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 27 "P:=diff(x(t),t)=-5*x-3*y-1;" }}{PARA 11 "" 1 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Q:=diff(
y(t),t)=6*x+4*y+2;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 48 "DEplot([P,Q],[x(t),y(t)],-2..2,x=-5..5,y=
-5..5);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 29 "Adding the initial conditions" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "DEplot([P,Q],[x(
t),y(t)],-2..2,[[x(0)=-2,y(0)=3],[x(0)=0,y(0)=1],[x(0)=0,y(0)=0],[x(0)
=-2,y(0)=4],[x(0)=2,y(0)=-2],[x(0)=0,y(0)=-2],[x(0)=-3,y(0)=2],[x(0)=3
,y(0)=-4.5]],x=-4..6,y=-7..3,linecolor=black);" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "DEplot([P,Q
],[x(t),y(t)],-2..6,[[x(0)=-3,y(0)=2]],x=-4..6,y=-7..3,scene=[t,x],lin
ecolor=black);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 95 "DEplot([P,Q],[x(t),y(t)],-2..6,[[x(0)=-3,y(0)=
2]],x=-4..6,y=-7..3,scene=[t,y],linecolor=black);" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK 
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