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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 39 "Stability of equilibria \+
investigation 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 41 "Consider the system with diagonal matrix " }}{PARA 256 "
" 0 "" {OLE 1 3592 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::
yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::fyyyyya:nYf::wyyyqy;:::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyy
uyJ<j?V:;J:>R<:T><::;FkWWO_v`wW:A:;B;B:F:YLpfF>:::::::::J?NZ;vyyyyyY:v
YxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::::::j:>:K=;:=j[vGU
MrvC?MoJ::::::::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z::C:J=j=B:KJ:F;N;;j?>:S:
UJ:n;v;;Jyky;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::j:b:<Z::::::::::JlJ<@Z=fZD^Z;FZ;N`=@i@N:?b[uc:KfF_rZkE
:=bFAVXGb::RX;Z:jysy:>:<::::::Ki>q<;::::=Jyyy;d:yayY;A:;JZC:bK_J:fc[_h
b_ds?h_?^?GhoGfnGgioGM:ETV:::^xsNpkK:<::::::::yayY:^Z:j:>:yayQ:>:nx>jQ
M:Ge:<j:F;HjS@Z:N:e:;j<>:Mb:>Z:^:<J;@j:f:FZ<vyyyy:f:F;B:C:WS:kj<jRDj:B
:E:Qb:B:E:Sb:;D;;F;WJ:n^@v;_jysy;B:K:;xyYxy;D;YB;Q:G;Sj`@@J:>VEZ:F;NZ:
vCS=[LsfFaMR>@>Z::::::::kJ;@:;J;>Z:vYxY:>Z::::::j<B:Wf:B:N:Y<>D_=a=[;;
B:::::::JF>:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::::::::::::::::v
YxI:;Z:::::::::]:qi:;fy>Z:JBAZ:b:DJ:<k`LOJPNvLQs@`p^:f_;j><J:<JSdj\\@J
rFZ:B:GC:Cn>JSfj=nH=:G;eZ:VY;><:[V:=:;b:^dcgg_WhZnc_whZNdigg[oGss`C:Uk
:^:>x;j>>:_kb@J`F:<Jv>o:F;N@>g=F:>I:_kBIJfFZ:jv>OJSJ:Ij:<:m=[B:<jw;<:[
V:;Z<b:<kc`a\\wgfs?CJ:f?;J<JrG:MJ:N`DFDsT:=J:<jX>UM:_kB?g;F:fAf<VYZ:JB
A:<Z<JdRS:::Uk:^:>X;j>>:_kbNk<F:;B:<JZG;b<]NEEj:Z:J?B:Gc;N;yyyxy:J:><<
jw?<I:;JXE:<j:DJ;Dlc`qsLqlp`h_:f?;J<JrG:AB:<J::::::::::fK<5D" }}{PARA 
0 "" 0 "" {TEXT 256 29 "To find the equilibrium point" }}{PARA 0 "" 0 
"" {TEXT -1 38 "The equilibrium point is clearly (0,0)" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 23 "To find the eigenva
lues" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "A:=matrix(2,2,[2,0,0,2]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 265 "" 0 "" {TEXT -1 3 "tr(" }{TEXT 267 1 "A" }{TEXT -1 
3 ")=4" }}{PARA 266 "" 0 "" {TEXT -1 4 "det(" }{TEXT 268 1 "A" }{TEXT 
-1 3 ")=4" }}{PARA 267 "" 0 "" {TEXT -1 9 "discrim=0" }}{PARA 0 "" 0 "
" {TEXT -1 52 "Thus there is only one repeated positive eigenvalue." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eigenvalues:=eigenvals(A);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "Thus as expected there is on
e repeated positive eigenvalue. However for a diagonal matrix there is
 an infinite number of directrices and every point lies on a directrix
." }}{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):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P:=diff(x
(t),t)=2*x;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "Q:=diff(y(t),t)=2*y;" }}{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 36 "Adding some initial conditions giv
es" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "DEplot([P,Q],[x(t),y
(t)],-2..2,[[x(0)=1,y(0)=1.7],[x(0)=1,y(0)=1],[x(0)=1,y(0)=.6],[x(0)=1
,y(0)=-1.7],[x(0)=1,y(0)=-1],[x(0)=1,y(0)=-.6],[x(0)=-1,y(0)=1.7],[x(0
)=-1,y(0)=1],[x(0)=-1,y(0)=.6],[x(0)=-1,y(0)=-1.7],[x(0)=-1,y(0)=-1],[
x(0)=-1,y(0)=-.6]],x=-5..5,y=-5..5,linecolor=black);" }}{PARA 13 "" 1 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "This equilibrium point is called a
n " }{TEXT 259 19 "unstable star node " }{TEXT -1 170 "since the eigen
value is positive and hence the trajectories are all moving away from \+
the equilibrium point. If the eigenvalue is negative the equilibrium p
oint will be a " }{TEXT 260 16 "stable star node" }{TEXT -1 65 " as al
l the trajectories will move towards the equilibrium point." }}{PARA 
0 "" 0 "" {TEXT -1 54 "You can see this for yourself by looking at the
 system" }}{PARA 258 "" 0 "" {OLE 1 4104 1 "[xm]Br=WfoRrB:::wk;nyyI;G:
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:::vYxI:;Z::::::::J\\<K;PA;B::::::::::jP@:MB:;B:_c<ET:[m:<Z:n^s?JM:_k<
IJuF:<JMJHjw?JB:>l;F:<J:DZJ^dcgg_WhZnc_whZNdigg[oGqHYUAfxAN[hJ?:Uk:^:>
x;j>>:_kb@J`F:<Jv`QOG;N@Vb>F:;Jv:N@]v:kl:<:mUXO?N@Ve>F:FI^=VYJ:<:[V:;Z
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:::::::::::::::::::::::::::::::::::::::::::::::::3:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 104 "If the equilibrium point is not at the origin \+
 every point will still be on a trajectory of the system. " }}{PARA 0 
"" 0 "" {TEXT -1 54 "You can see this for yourself by looking at the s
ystem" }}{PARA 259 "" 0 "" {OLE 1 4104 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:
j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::fyyyyya:nYf::G:jy;:::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtN
Hm=;:::::::n<Ejyyiy=C:Qj;>:;JZC:bKB::>n_opOOlUpq;V:>r>Z:j:vCSmlJ::::::
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E:G:IZ:F;nYN;V;;J@>:UJ:n;v;;JB<Z:F<N<;Jyky;:::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::j:b:B:::kTtoJeFg:^FEmq>>AL
pFG[m>VI[KRJSN@gKRJhJ<@Z=fZD^Z;FZ;N`=@i@N:?b[uc:GVE?B\\_c>AFX?b[Qc:KfF
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>=<j<jRDj:j<j?DZ:B:E:Sb:;D;;u;W:Gc;YJCvYxY:<J>JZyyayYJXjALj?JMJ@fc[;PP
:<Jjg:<j>Z:^=<jwEJ:uI;B:>l;B:DJ:DJ:<k`LOJPNvLQs@`BN[EOtvgwqWuvgwq?taww
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::::::::::::::yay=J:B:::::::::::::::::::f_;j><J:<JSdj\\@JrFZ:B:GSX;n>J
Sfj=nH=Z:n>^=VY;><:[V:=Z:>Z<B>cTTUUSaEBWTSiEB_tUUURWmgmqp;AB<OB><J?:Uk
:^:>x;j>>:_kb@J`F:<Jv`QOG;N@Vb>F:;Jv:N@]v:kl:jv`QO;_;qD;=:m=s:qAB:>l;>
:DZ<B>aTXDpql`:FZBN[FOtvgWq@[@=:;J?:UK:^:>X=j>>:_c<]Lh@j:>Z:fasWTM:_;_
f:=J:f_sWTJSJXPj:jO`qc;_kB?g;F:fA:_;wf:=:U;JSjcPj:jPJHB:qAB:>l;Z:b:^DP
@:::AZK>:O:f_;F:C:[Q:F;;JSFDEu:=B:;B:`QZG;b<Nd@F:;Zs;b<]nrHj:Zs;b<>g@F
:^`s?r:N;;B:Gc;N;yyyxy:>:;JHjw?<IZ:>:sg:B:=b:?bBaTXaEWEUUXaBnfev:OV:N;
G:IJ?;C:UK:^:>X=j;<Z:>::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::1:" }}{PARA 0 "" 0 "" {TEXT -1 65 "which has an equilibri
um point at (1,-2) and negative eigenvalue." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 261 "" 0 "" {TEXT -1 44 "Consider the system with non dia
gonal matrix" }}{PARA 260 "" 0 "" {OLE 1 4104 1 "[xm]Br=WfoRrB:::wk;ny
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::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:jy;::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^
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:::::::::::::::::::::::::::::::::::::::::::::::::::::=Z<Z:::f?[KRFB=LR
>@]K]V@]KU>Au=[ljVFQmjvFEMxJ<@Z=fZD^Z;FZ;N`=@i@N:?b[uc:KvF_rZDJ=iLCHrv
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^YJ;nE;B:;:::::US:F[:>Z:N`Dfb;>H=B:<JM`Q:G;N@Ev:gm:B:G;s:qQ:[:JBAj:B:;
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<kc`a\\wgfC:]B<OrU_HYUqgKbkBJ:<J?:UK:^:>x;FZ:F;;JSdjb^e;F:;B:uSXap>JSj
^Lj:>:QSXa@N@^i@F:;jO:N@]Nn@j:jX:N@nm<F:;jP`qc;_;WF;=:Q;JSJZUj:jPJHB:q
AB:>l;Z:b:^DP@:::YZ[J?:US:=J<Jr?:MJ:N@]L?Ej:>Z:V`s?RM:_;_v;=:a;JSFLMx;
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?n:v:O>^:f?;J<JrG:AB:<J::::::::::::::::::::::jysy:>:;B::::::::::::::::
:::jysy:>:<:::::::::3:" }}{PARA 0 "" 0 "" {TEXT 261 29 "To find the eq
uilibrium point" }}}{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 12 "E1:=3*x+y+3;" }}{PARA 11 "" 
1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E2:=-x
+y-5;" }}{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 31 "the equilibrium p
oint is (-2,3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 262 24 "To find the eigenvalues." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "A:=matrix(2,2,[3,1,-1,1]);" }}}{EXCHG {PARA 268 "" 0 
"" {TEXT -1 8 "tr(A)=4," }}{PARA 269 "" 0 "" {TEXT -1 8 "det(A)=4" }}
{PARA 270 "" 0 "" {TEXT -1 9 "discrim=0" }}{PARA 0 "" 0 "" {TEXT -1 
47 "Thus there is only one real positive eigenvalue" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "eigenvalues:=eigenvals(A);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 263 23 "To find the eigenvector" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eigensystem:=eigenvects(A);" }}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 
21 "To find the directrix" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The \+
directrix is given by" }}{PARA 263 "" 0 "" {OLE 1 4616 1 "[xm]Br=WfoRr
B:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I:wAyA::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyJ<j?V:;J:>R<:T><::;WSgWO_v`wW:A:
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FjFJG>:wAyA:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
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Q:F;;JS>r;^tBF:<J`j>JSvAcH<=:W<JSF`=^tBF:fC:_KZo[EF:vC:_kYn[EF:FD:_kRn
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y=<::vIB::jyZ:::y=<::vIB::jyZ:::y=<::vIB::jyZ:::y=<::vIB::jyfoX]:JIPJU
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Points on the directrix are taken \+
with k=1 and k=-1" }}{PARA 0 "" 0 "" {TEXT -1 27 "These are (-3,4) and
 (-1,2)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 27 "To plot the phase por
trait." }}}{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 24 "P:=diff(x(t),t)=3*x+y+3;" }}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "Q:=diff(y(t),t)=-x+y-5;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DEplot([P,Q],[x(t),y(t)],-2.
.2,x=-7..3,y=-2..8);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 73 "Putting in some initial conditions includ
ing those on the directrix gives" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 134 "DEplot([P,Q],[x(t),y(t)],-2..2,[[x(0)=-3,y(0)=4],[x(
0)=-1,y(0)=2],[x(0)=0,y(0)=6],[x(0)=-4,y(0)=-1]],x=-7..3,y=-2..8,linec
olor=black);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 160 "As the eigenvalues are positive all the trajectories \+
are moving away from the equilibrium point (-2,3). Thus the equilibriu
m point is unstable and is called an " }{TEXT 265 24 "unstable imprope
r node. " }{TEXT -1 108 "If the eigenvalues are negative the trajector
ies will move towards the equilibrium point and it is called a " }
{TEXT 266 22 "stable improper node. " }{TEXT -1 54 "You can see this f
or yourself by looking at the system" }}{PARA 264 "" 0 "" {OLE 1 4100 
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