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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 32 "Case Study - Economic I
nvestment" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 14 "with(D
Etools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "Define the investment function" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v:=x->arctan(x):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
17 "Define the system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=r*v(x)-2*(x+y);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g:=x;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Find the equilibrium poin
ts" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "ep:=solve(\{f,g\},\{x,y\});" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Find the Jacobian matri
x" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "u:=[f,g];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "J:=jacobian(u,[x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "J0:=subs(ep,evalm(J));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 47 "Find the trace and determinant and discri
minant" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "trJ0:=trace(J0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "detJ0:=det(J0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "discJ0:=simplify(trJ0^2-4*detJ0);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 18 "solve(discJ0>0,r);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "From these results we \+
have an attractor if " }{XPPEDIT 18 0 "r < 2;" "6#2%\"rG\"\"#" }{TEXT 
-1 19 " and a repeller if " }{XPPEDIT 18 0 "2 < r;" "6#2\"\"#%\"rG" }
{TEXT -1 21 " which is a focus if " }{XPPEDIT 18 0 "r < 2+2*sqrt(2);" 
"6#2%\"rG,&\"\"#\"\"\"*&\"\"#F'-%%sqrtG6#\"\"#F'F'" }{TEXT -1 22 " and
 a node otherwise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "Find the eigenvalues of the Jacobian matrix" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evs:
=eigenvals(J0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ev1:=evs
[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ev2:=evs[2];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(r=2,ev1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(r=2,ev2);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Thus eigenvalues a
re purely imaginary at " }{XPPEDIT 18 0 "r = 2;" "6#/%\"rG\"\"#" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 36 "Also derivative of real part =1/2>0." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "To investigate A.S. of th
e origin at " }{XPPEDIT 18 0 "r = 2;" "6#/%\"rG\"\"#" }{TEXT -1 26 " u
se the Lyapunov function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "V:=m*x^2+n*y^2;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Find dV/dt" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "dV:=collect(diff(V,x)*f+diff(V,y)*g,[x,y],distributed);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Cho
ose " }{XPPEDIT 18 0 "m = 1;" "6#/%\"mG\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "n = 2;" "6#/%\"nG\"\"#" }{TEXT -1 33 " to eliminate cro
ss term and let " }{XPPEDIT 18 0 "r = 2;" "6#/%\"rG\"\"#" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "dV1:=subs(m=1,n=2,r=2,dV);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "This expression is neg
ative except on the line " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }
{TEXT -1 27 " which is not a trajectory." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(dV1,x=-2..2);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 
"Thus all the conditions of the Hopf bifurcation theorem are satisfied
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Now \+
obtain the phase portrait to show the bifurcation" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=r*v(x(t)
)-2*(x(t)+y(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "g:=x(t):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=3:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 40 "de1:=diff(x(t),t)=f:de2:=diff(y(t),t)=g:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ics:=[[0,2,0],[0,0.5,0]
]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "DEplot([de1,de2],[x(t
),y(t)],t=-10..10,ics,x=-2.5..2.5,y=-2..2,stepsize=0.1,linecolour=blue
);" }}{PARA 257 "" 0 "" {TEXT -1 35 "Unstable Focus + Stable Limit Cyc
le" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "DEplot([de1,de2],[x(t),y(t)],t=20..30,ics,x=-2.5..2.5
,y=-2..2,stepsize=0.1,linecolour=blue);" }}{PARA 258 "" 0 "" {TEXT -1 
11 "Limit Cycle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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