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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "To examine the phase diagr
am close to the equilibrium point (0,50)" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{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 14 "with(DEtools):" }}{PARA 7 "
" 1 "" {TEXT -1 35 "Warning, new definition for adjoint" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=diff(x(t),t)=.2*x-.004*x^2-.003*
x*y; " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "B:=
diff(y(t),t)=.1*y-.001*y^2-.003*x*y;" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "DE
plot(\{A,B\},\{x(t),y(t)\},0..100,[[0,49,-1],[0,49,.5],[0,51,1],[0,51,
-.5]],x=49..51,y=0..1,stepsize=.1); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "To examine th
e phase diagram of the linearisation at the point(50,0)" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "First find t
he linearisation as follows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "f:=.2*x-.004*x^2-.003*x*y; " }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g:=.1*y-.001*y^2-.003*x*y;" 
}{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "v:=[f,g];" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "J:=jacobian(v,[x,y]);" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "J50:=subs(x=50,y=0,evalm(J))
;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The linearisa
tion is given by" }}{PARA 256 "" 0 "" {OLE 1 4104 1 "[xm]Br=WfoRrB:::w
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::JcvGY
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:f_;>:C:[q:F;;JSVEmt:=Z:>@;@N@fd>F:B:;@b<^`AF:>@N;<JSJF\\J;jRPMZ?RM:_k
Ggc=F:<JRJ?JSj^LJ;JRPMZ?RJMT:OjyyiyIZ:B:;JBB:qQBv:>:sg:B:=J;Dlc`qsLqlp
`h_:f?=J<Jr?:AB:;B:;:::::::::::::::::::::::5:" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 65 "To draw the phase diagram of the linearisation proceed as
 follows" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A1:=diff(x(t),t)=-0.2*x-0.15*y;" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "B1:=diff(y(
t),t)=-0.05*y;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 103 "DEplot(\{A1,B1\},\{x(t),y(t)\},-50..50,[[x(0)=-0.6,y(0)=.4],[x(
0)=0,y(0)=.6]],x=-1..1,y=0..1,stepsize=.1); " }}}{EXCHG {PARA 0 "> " 
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1803 }