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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 1 " " }{TEXT 256 13 "localstab.mws" }{TEXT -1 94 " A
pictorial introduction to local stability analysis based on eigenvalu
es of linear systems" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 39 "restart: with(linalg): with(DEtools):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 498 "D
on't worry about the warnings--the standard meanings of norm and trace
in Maple have to do with number theory, so this is just reminding us \+
that we are going to be using the linear algebra meanings for now. (I
f you are doing number theory and linear algebra simultaneously, then \+
you are probably smart enough to get around this little conflict.) \n
\nWe use the same matrices that appeared in eigen.mws, with one more t
hrown in for free. In each case describe what kind of eigenvalues are \+
appearing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Sink := matri
x( [[1,1],[-3,-2]] ); eigenvals(Sink);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 63 "Degen:= matrix( 2 , 2, [ -4, 5/2, -5/2, 1]); eigenva
ls(Degen);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Source:= matr
ix(2, 2, [4, 1, -2, 3]); eigenvals(Source);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 57 "NC:= matrix(2, 2, [-0.4, -4, 2, 0.4]); eigenva
ls(NC);\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Saddle:= mat
rix(2, 2, [2, 3, 4, 1]); eigenvals(Saddle); " }}}{EXCHG {PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Let's now take a lo
ok at the behavior of the linear systems corresponding to each of the \+
matrices M, A, B, C, and DD. " }{TEXT 258 144 "All have a unique equi
librium point at (0, 0), and in this case you don't even have to start
very close to the equilibrium to see the behavior. " }{TEXT -1 502 "F
irst we make equations using the matrix M to give us the coefficients.
Unfortunately the left and right hand sides end up reversed, so we ha
ve to swap them back. Once we have the rate equations we plot the phas
e plane without any initial conditions just to see the overall picture
. You can put initial conditions in if you like to see a sampling of t
he solution curves. Fill in %? with each matrix in turn, and predict \+
what the trajectories will look like given the eigenvalues that you fo
und above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 73 "eqns:= geneqns( %? , [x(t),y(t)], vector([diff(x(t
),t),diff(y(t),t)]) ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "
rate_eqns:= [seq( rhs(j) = lhs(j), j = eqns)];" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 93 "vars:= [x(t), y(t)]; domain:= 0 .. 50; # you m
ay want to extend the time domain in some cases" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 110 "DEplot(rate_eqns, vars, domain, scene=[x, y],
stepsize = 0.05, x=-8..8, y=-8..8); # arrows default to harpoons" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "init1:= [x(0)= %? , y(0)= %
? ]; init2:= [x(0)= %? , y(0)= %? ];# use the tab key to get to the f
ill-in-the-blanks " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "DEpl
ot(rate_eqns, vars, domain, [ init1, init2 ], scene=[x, y], stepsize =
0.05,x=-8..8, y=-8..8, linecolor = [black, green], arrows=medium);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }
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