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<jw?<IZ:>:sg:>Z:FZ<NZ\\VdsWhngfgcUC:UK:^:>X=j;<Z:>::::::::::::::::1:" 
}{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "b)" }}{PARA 256 "
" 0 "" {OLE 1 4122 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::
yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::fyyyyya:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=
C:Qj;>:;JZC:bKB::>\\K>d=IdJoq;V:>r>Z:j:vCSmlJ::::::::::OJ;@jyyyyyy;jys
y;Z:::::::^<>:F::]KRnC=MtFGgml>:;::::::JGN:ry:>:<::::::=J:>L<:F:AlqfG[
maNFO=;::::::::_J;vyyuy:>:<::::::AZ:^E>:nYN:<j>J<JyK=j=B:K:OJ:nYV;^;;j
@>:W:YJ:>\\:B:]:_J:nYvY:::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::F:DZ:B::::::::::^K^Z;r:Eb<=r:WJ:@J;TRUEk;U_;N
:BXN`=@i@N:?b[Qs:W:ZP?:ZsKSFi@N:Vn<>kjJ:tZ<Z;>d:Xb[og]gffk=v:;J:::::j:
<Jyyy;d:yayY;AB:>:;P:<:TNC>:UTRcETcTX[US;SK]UW=EWMuUWm>B:ETV:::^xsNpk?
:;B::::::::vYxy;J<<:=J:vYxY;J:J`h:SF;NZ=j:F;Hj:A:AB:e:;j<>:Mb:B:CZ:NZ;
F:E:=b:yyyyI:E:M:<J<JQ@JFf:F`<F:<j<j?D:<j<J@DJJLJZdJAB:Gc;YJCvYxY:<J>>
Z:>ryipye<ex:QZ:n>^;UTR:;Jj=j><Z:Ny<J;<ja^G>D_mlVH[KR<:;B:::::::JFNZ;J
:N:;B:yayA:;B::::::f:<JOH:<:?ja:[LsjsJRJ:<:::::::>=;jysy:>:<::::::wqy[
:::::::::::::vYxI:;Z::::::::::::::::::::yay=J:B::::::::>\\:B:JK^:f_;j<
B:cb:uY:;l<B:ab:uY:Md:E:c:kG;EJ:V<Fr@f:;j>DJVlP;?afOJ[x]?NbK:>::::::::
:[J:<jwEJ:uI;B:>l;Z<b:;B>aTXUeRYEUHL<jPF:MJ:N`DVg;vD=:gOY=j>JSFk=fK=B:
nP:_KjDJEIj:>:QkcG;N@QmaMj:>:gG::_k>gZ?F:nP:_KjN_BF:VNapDjw;<:[V:=J:>Z
<>:cTTPps<LaPpu<Lcxpp@PqfEC:UK;^:>x;F:MJ:N@QmD@j:>:gLoG;N@Mn_@j:JvNOM:
_Kjfm<F:;Jv:N@Qm@Qj:Je:N@Mn[Qj:jvNOJS>F=s;=:m=]:qAB:>l;>:DJ:DZJfcN?^??
inOh[K<jPF:C:[Q;<j>>:_KdDjZEj:>Z:n^:>j:F[:JSjYXj:JMjDjw?JB:>l;Z:b:^D::
>@C:UK;^:>X=j>>:_KjDjc`j:JR>r:N;<JMT:OB:yyyxy:J:><<jw?<I:;JXE:<j:NZ\\V
dsWhngfgcUC:Uk:^:>X?B:AZ:>::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::3:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "LHS:=Vx*f+Vy*g;" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "To determine if his expression is \+
always negative it is necessary to complete the square." }}{PARA 0 "" 
0 "" {TEXT -1 63 "First load the \"with(student)\" package then procee
d as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(stude
nt):" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new definition for D" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "LHS1:=completesquare(LHS,x)
;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 10 "Therefore " }{OLE 1 3594 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>
N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ:
:::::::gj<vyyuyJ<j?V:;J:>R<:T><::[w_]tj]DNwW:A:;t:B:F:YLpfF>:::::::::J
?NZ;vyyyyyY:vYxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::::::j
:>:C<;:=j[vGUMrvC?MoJ::::::::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z::C:J=j=B:K
:M:OJ:V;;J@Jyky;::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::j:b:<Z::::::::::JZ<Z:^Z;r:Eb<=b>AFM_rZKd>AvN?B\\]
c:Gvb?B\\[[:f>_;;:::::::::::::::F:wyyAb<vYxy?VZ:J:>R<:TNC>:UTRcETcTX[U
S;SK]UW=EWMuUWm>j\\@A:::cYH_WV>Z::::::::jysyA:CB:F:;jysy?:;:;h;vn;JaA:
=j>r:IA?jDJ:f:;j>D:<J<J;@j:f:FZ<vyyyy:f:F;B:C:WS:kj<jRDj:B:E:Qb:B:E:Sb
:;V:;c;WJ:n^@v;_jysy;B:K:;xy;yy;Y;;U:Q:G;Sj`@@J:>^:f:<j>J;<ja^G>D_mlVH
[KR<:;B:::::::JFNZ;J:N:;B:yayA:;B::::::f:<JuAZ:B:N:Y<>D_=a=[;;B:::::::
JF>:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::::::::::::::::vYxI:;Z::
:::::::eZ:Vy<>jx]:JBAj:J:DZJ^dcgg_WhZnc_whZNdigg[oGMK<jP@:SB:;B:_c<;T:
kl;J`\\ncfTcR:;pDjw;<:[N:<J:<Z<::::f?=J<JrA:MB:;B:_;Ir:=J:<jR<JZG;N@F^
?F:>@;PBB:qAB:>l;Z<b:<kc`a\\wgfc>C:UK:^:>X=j>>:_c<>y=F:;B:AocO;<JMJ?vy
yuy=:;JBB:qQBv:>:sg:B:=J;Dlc`qsLqlp`h_:f?=J<Jr?:AB:<J:::::::::::::::::
:::::::::::::::::::5:" }{TEXT -1 8 "for all " }{TEXT 256 1 "x" }{TEXT 
-1 5 " and " }{TEXT 257 1 "y" }{TEXT -1 40 " and is zero only at the p
oint (0,0) and" }}{PARA 260 "" 0 "" {OLE 1 4106 1 "[xm]Br=WfoRrB:::wk;
nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:jy;::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt
>^:fBWMtNHm=;:::::::n<Ejyyiy=C:Qj;>:;JZC:bKB::>pQHd=IdJoq;V:>Z?B:<:=ja
^GE=;:::::::::N;?R:yyyyyyA:yayA:<::::::JDJ:j::F@[KaFFcmnnHEM:>:::::::o
J;Zy=J:B::::::F:;J<K:j:VBYmp>HYLkNG>::::::::N<?jyyiy=J:B::::::V:<JhJ:J
yK;B:M:C:wAG:IZ:>;N;;Jyk?J@j@>:W:YJ:>\\:B:]:_J:V<nYvY:::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::j:b:<Z::::::::::
J:C:CR:Hj<dj:Dk;Uo;@Z:FZLVjrOZ<`_LVjsc:gFD?b_MK;TrXDJ>QMCHRvHjAr<]R:?B
\\_[DNZ@X_<>kpN\\=@iLVjsOZ@X?P=mu:Y::<J:v:IJ:>:::::FZ:nyyYZDjysyQj;J:>
R<:TNC>:UTRcETcTX[US;SK]UW=EWMuUWm>j\\@A:::cYH_WV>Z::::::::jysyA:CB:F:
;jysy?:;:SG>nm<JfH:=j>r:SW:N:e:;j<j>D:;B:C:?R:=j<j:Djyyyy=j<j>Z:^:n_;>
=E:]c:=Z:f:V[<Z:f:^[<>f<>^In[:JMTjAN<yayAZ:>;>ryapyAvH^\\<V;n>^;UTR:;J
jf:<j>J;<ja^G>D_mlVH[KRJ:<:::::::>=?R:>:?J:<jysy;J:<::::::EZ:>j=Z:J;vC
JbNHVH>@>Z::::::::kJ:vYxI:;Z::::::Jyyyy[:::::::::::::vYxI:;Z::::::::::
::::::::::yay=J:B::::::::jDB:qi:;fyB:>l;F:>Z<B>cTTUUSaEBWTSiEB_tUUURWM
:eX<jP@:MB:;B:_c<;W:[B:=B:;B:;:::::::::::gLoG;N@Vs;F:>IOo>JSjPHj:jvNOJ
SJKUj:>:k=OZ:N@f_DN:<Jv`QOOOM:_;SW==:m=e:qAB:>l;>:<Z<B>::JBAJ<jPF:CJ:>
x;j>>:_;Ef:=J:<JP>F[Z:VY;RyB:>l;>:<Z<B:GTSWuREuVdK<jP>:C:[q:F;;JSFq;n_
;F:;B:?MyF;N@oOnIj:jmnI[Z:VY;><:[V:b:DZJVdscRYEUdK<jPF:C:[Q:F;;JSdJJOq
>F:;B:uC:ap>JSj\\\\j:jOVTJSJbij:jOjDjw;<:[V:B:D:c\\_;rUJB_:f?;J<JrG:MJ
:N@^u?F:<jT<JZ=j>JSjg`j:JU>r::_;oH==B:f@f<VY;syB:>l;>:<Z<:VH:J:;Uk:^:>
X;j>>:_KtwfAF:<Z:>HM:d:]e==:<:OB:n>N;yyyxy:J:><<jw?<IZ:>:sg:>Z:FZ<NZ\\
VdsWhngfgcUC:UK:^:>X=j;<Z:>:::::::::::::::::::::::::::::::::::::::::::
::3:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "is a Lyapunov func
tion for the nonlinear system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 259 "" 0 "" {TEXT -1 30 "To find the equilibrium point." 
}}}{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 -1 179 "To classify the equilibrium point of
 the linearisation first prove the equilibrium point is a stationary v
alue of the Lyapunov function and then find the type of stationary poi
nt." }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 36 "To prove (0,0) is a stat
ionary point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(\{x=0,
y=0\},Vx);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(\{x=0,y=
0\},Vy);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}}{EXCHG {PARA 
262 "" 0 "" {OLE 1 3590 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyy
yy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::fyyyyya:nYf::wyyyqy;::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::g
j<vyyuyJ<j?V:;J:>R<:T><::[dAatj]DNwW:A:;B;B:F:YLpfF>:::::::::J?NZ;vyyy
yyY:vYxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::::::j:>:C=;:=
j[vGUMrvC?MoJ::::::::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z::C:J=j=B:K:M:OJ:V;
;J@j@>:W:YJ:nYvY::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::F:DZ:B::::::::::>F<J<@Z=fZDFZ=n;;R:?b[gg>AfO@J;ZrK
;TrXHJA:bO:`A?Z<>kjJb<RMt>;B::::::::::FZ:nyyYZDjysyQj;J:>R<:TNC>:UTRcE
TcTX[US;SK]UW=EWMuUWm>B:ETV:::^xsNpkK:<::::::::yayY:^Z:j:>:yayQ:>:neCJ
@M:Kf:F:MZ=^b;B:AB:e:;j<>:Mb:>Z:^:<J;@j:f:FZ<vyyyy:f:F;B:C:WS:kj<jRDj:
B:E:Qb:B:E:Sb:;C;;F<W:Gc;YJCvYxY:<J>B:;xyewy;D<ex:Q:G;Sj`@@J:>VMB:<JwE
:?B:YLpJbNHEms>@[C:>Z::::::::kJ;@:;J;>Z:vYxY:>Z::::::j<B:[g:B:N:Y<>D_=
a=[;;B:::::::JF>:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::::::::::::
::::vYxI:;Z::::::::JB<Z::?K<jP@:EZ:^\\<fy;>BEZ:V\\<fy;Fc<f:^<Vw=f:V<^x
?f:F[<Z:><;B:qi:;fy>Z:JBA:DZ<B>aTXUeRYEUHL<jPF:MJ:N`DVg;vD=:gOY=j>JSFk
=fK=B:nP:_KjDjREj:>:ukcG;N@QM;Lj:JU;N@MNVLj:JU;N@;MhTj:>:u;e:qAB:>l;F:
;J:DJ:^dcSSaEBWTSiEB_tUUURWM@<U<jPNZ:^:>x;F:MJ:N@QmD@j:>:gLoG;N@Mn_@j:
JvNOM:_ko>r>F:nD:_k>ox>F:FIOoDjw;<:[V:;Z:b::::sL<jPF:C:[Q;<j>>:_KjNkAF
:>@;H:OZ:n>N;yyyxy:J:><<jw?<I:;JXE:<j:NZ\\VdsWhngfgcUC:UK;^:>X=j;<Z:>:
:::5:" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "and (0,0
) is a stationary point of " }{TEXT 258 1 "V" }{TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 42 "To determine t
he type of stationary point." }}{PARA 0 "" 0 "" {TEXT -1 63 "First fin
d the Hessian matrix and then look at its eigenvalues." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "H:=hessian(V,[x,y]);" }}{PARA 11 "
" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " eig
envals:=eigenvalues(H);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 74 "Since the eigenvalues are both positive t
he stationary point is a minimum." }}{PARA 0 "" 0 "" {TEXT -1 60 "Thus
 the equilibrium point (0,0) is stable and an attractor." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "This can be confi
rmed by finding the Jacobian matrix of the linearisation and using the
 Linearisation theorem. " }}{PARA 0 "" 0 "" {TEXT -1 25 "Try this meth
od yourself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 68 "A look at the phase portrait should also confirm the above anal
ysis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "A:=diff(x(t),t)=-10*x-6*y;" }}{PARA 11 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B:=diff(y(t),t)=-
6*x-10*y;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 131 "DEplot(\{A,B\},[x(t),y(t)],-50..50,[[x(0)=0,y(0)=1],
[x(0)=0,y(0)=-1],[x(0)=-1,y(0)=0],[x(0)=1,y(0)=0]],x=-1..1,y=-1..1,ste
psize=0.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }