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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "To examine the phase diagr
am close to the equilibrium point (0,100)" }}{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 0 106 "DE
plot(\{A,B\},\{x(t),y(t)\},0..100,[[x(0)=.1,y(0)=99],[x(0)=.5,y(0)=101
]],x=-.1..0.6,y=99..101,stepsize=.1); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "To examine th
e phase diagram of the linearisation at the point (0,100)" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "First find
 the linearisation as follows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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" {MPLTEXT 1 0 27 "f:=.2*x-.004*x^2-.003*x*y; " }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "g:=.1*y-.001*y^2-.003*x*y;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "v:=[f,g];" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "J:=jacobian(v,[x,y]);" }{TEXT -1 0 "" }}{PARA 
11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "
J100:=subs(x=0,y=100,evalm(J));" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The linearisation
 is given by" }}{PARA 256 "" 0 "" {OLE 1 4096 1 "[xm]Br=WfoRrB:::wk;ny
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o>JSjuMj:jt:N@>hAF:nH><<jwcyB:>l;>:DZ<B>aTXDpql`G^:f?=J<Jr?:MJ:N`DVE[T
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:^DP@:JP:US:;J<JrG:MJ:N@qlfHj:B:[KZ;_;uC;=:]C:;@N@qn`Hj:>:[;JSj^Lj:jS>
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:VZ:B:;J<jP:>X:;::::::::::::::::::::::3:" }{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 follo
ws" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "A1:=diff(x(t),t)=-0.1*x;" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "B1:=diff(y(t),t)=-0.3*x-0.1*
y;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "DEplo
t(\{A1,B1\},\{x(t),y(t)\},-50..50,[[x(0)=0.1,y(0)=-1],[x(0)=.5,y(0)=.6
]],x=-.1..0.6,y=-1..1,stepsize=.1); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "20 0 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }