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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 34 "Phase Portraits for Linear Syst
ems" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Co
nsider the linear system of two first order equations" }}{PARA 257 "" 
0 "" {XPPEDIT 18 0 "dx[1]/dt = x[1]+x[2];" "6#/*&&%#dxG6#\"\"\"\"\"\"%
#dtG!\"\",&&%\"xG6#\"\"\"F)&F.6#\"\"#F)" }{TEXT -1 0 "" }}{PARA 258 "
" 0 "" {XPPEDIT 18 0 "dx[2]/dt = x[1]-x[2];" "6#/*&&%#dxG6#\"\"#\"\"\"
%#dtG!\"\",&&%\"xG6#\"\"\"F)&F.6#\"\"#F+" }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 55 "we consider this system with several initial conditi
ons" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "x[1](0) = -1.1;" "6#/-&%\"xG6#
\"\"\"6#\"\"!,$$\"#6!\"\"!\"\"" }{TEXT -1 6 "      " }{XPPEDIT 18 0 "x
[2](0) = 3;" "6#/-&%\"xG6#\"\"#6#\"\"!\"\"$" }{TEXT -1 1 "," }}{PARA 
260 "" 0 "" {XPPEDIT 18 0 "x[1](0) = -1.2;" "6#/-&%\"xG6#\"\"\"6#\"\"!
,$$\"#7!\"\"!\"\"" }{TEXT -1 6 "      " }{XPPEDIT 18 0 "x[2](0) = 3;" 
"6#/-&%\"xG6#\"\"#6#\"\"!\"\"$" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 3 "and" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "x[1](0) = -1.3;" "
6#/-&%\"xG6#\"\"\"6#\"\"!,$$\"#8!\"\"!\"\"" }{TEXT -1 6 "      " }
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-1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "de1:=diff(x1(t),t)=x1(t)+x
2(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/-%%diffG6$-%#x1G6#%\"
tGF,,&F)\"\"\"-%#x2GF+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 
"de2:=diff(x2(t),t)=x1(t)-x2(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$de2G/-%%diffG6$-%#x2G6#%\"tGF,,&-%#x1GF+\"\"\"F)!\"\"" }}}{PARA 0 "" 
0 "" {TEXT -1 53 "We now construct the phase portrait for the equation
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t),x2(t)], t=0..5, [[0,-1.1,3],[0,-1.2,3],[0,-1.3,3]],x1=-3..3,x2=-3..
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l" 2 307 289 289 2 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 17744 177 0 0 0 0 0 0 }}}{PARA 
0 "" 0 "" {TEXT -1 227 "Note that at the origin there seem to be slope
 lines pointing in differen directions. We therefore suspect that the \+
origin may be a fixed point. If we look at the equation it is obvious \+
that indeed the origing is a fixed point." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "Also note that since some slope l
ines point away from the origin it is not a sink, also since some slop
e lines point towards the origin it is not a source. This suggests tha
t the origin is a saddle." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 67 "Looking at the phase portrait and considering an \+
initial condition " }{TEXT 256 1 "x" }{TEXT -1 11 "(0) = (a,3)" }
{TEXT 257 1 " " }{TEXT -1 49 " it seems plausible that there should be
 a point " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 80 " between -1.2 a
nd -1.3 such that the solution curve tends towards the origin as " }
{XPPEDIT 18 0 "proc (t) options operator, arrow; infinity end;" "6#R6#
%\"tG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 36 ". In \+
fact this can be made rigorous." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 45 "We now write the system using matrix nota
tion" }}{PARA 262 "" 0 "" {TEXT 258 0 "" }{TEXT -1 0 "" }{TEXT 259 1 "
x" }{TEXT -1 5 "' = A" }{TEXT 260 1 "x" }{TEXT -1 3 " = " }{XPPEDIT 
18 0 "matrix([[1, 1], [1, -1]]);" "6#-%'matrixG6#7$7$\"\"\"\"\"\"7$\"
\"\",$\"\"\"!\"\"" }{TEXT -1 0 "" }{TEXT 261 1 "x" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "We now f
ind the eigenvalues and eigenvectors of A. (We begin by loading the li
near algebra program.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wi
th(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definition fo
r adjoint" }}{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 26 "A:=matrix([[1,1],[1,-1]
]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"\"F*7
$F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%*$-%%sqrtG6#\"\"#\"\"\"\"\"\"<#-
%'vectorG6#7$F*,&F$F*!\"\"F*7%,$F$F1F*<#-F-6#7$F*,&F$F1F1F*" }}}{PARA 
0 "" 0 "" {TEXT -1 25 "The eigenvalues are  and " }{XPPEDIT 18 0 "-sqr
t(2);" "6#,$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 65 " both have multiplicit
y 1 and the corresponding eigenvectors are " }{XPPEDIT 18 0 "vector([1
, sqrt(2)-1]);" "6#-%'vectorG6#7$\"\"\",&-%%sqrtG6#\"\"#\"\"\"\"\"\"!
\"\"" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "vector([1, -sqrt(2)-1]);" "
6#-%'vectorG6#7$\"\"\",&-%%sqrtG6#\"\"#!\"\"\"\"\"F-" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Corresp
onding to the eigenvalue  " }{XPPEDIT 18 0 "-sqrt(2);" "6#,$-%%sqrtG6#
\"\"#!\"\"" }{TEXT -1 18 " the eigenvector  " }{XPPEDIT 18 0 "vector([
1, -sqrt(2)-1]);" "6#-%'vectorG6#7$\"\"\",&-%%sqrtG6#\"\"#!\"\"\"\"\"F
-" }{TEXT -1 27 " generates a line of slope " }{XPPEDIT 18 0 "-sqrt(2)
-1;" "6#,&-%%sqrtG6#\"\"#!\"\"\"\"\"F(" }{TEXT -1 81 " through the ori
gin. This line contains trajectories that approach the origin as " }
{XPPEDIT 18 0 "proc (t) options operator, arrow; infinity end;" "6#R6#
%\"tG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 18 "The line of slope " }{XPPEDIT 18 0 "sqrt(
2)-1;" "6#,&-%%sqrtG6#\"\"#\"\"\"\"\"\"!\"\"" }{TEXT -1 33 " correspon
ding to the eigenvalue " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#
" }{TEXT -1 61 " contains the trajectories that move away from the ori
gin as " }{XPPEDIT 18 0 "proc (t) options operator, arrow; infinity en
d;" "6#R6#%\"tG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
40 "The eigendirection is aline of the form " }{XPPEDIT 18 0 "x[2] = m
*x[1];" "6#/&%\"xG6#\"\"#*&%\"mG\"\"\"&F%6#\"\"\"F*" }{TEXT -1 8 ", wh
ere " }{XPPEDIT 18 0 "m = -sqrt(2)-1;" "6#/%\"mG,&-%%sqrtG6#\"\"#!\"\"
\"\"\"F*" }{TEXT -1 8 ". So if " }{XPPEDIT 18 0 "x[2] = 3;" "6#/&%\"xG
6#\"\"#\"\"$" }{TEXT -1 6 " then " }{XPPEDIT 18 0 "a = 3/(-sqrt(2)-1);
" "6#/%\"aG*&\"\"$\"\"\",&-%%sqrtG6#\"\"#!\"\"\"\"\"F-F-" }{TEXT -1 
56 ". So we draw the trajectory with this initial condition." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "DEplot(\{de1,de2\},[x1(t),x2
(t)], t=0..5, [[0,3/(-sqrt(2)-1),3]],x1=-3..3,x2=-3..3);" }}{PARA 13 "
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