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178
Economic Dynamics
(ii) All trajectories remain bounded but do not approach the critical
point as
t
→∞
. This occurs when tr(
A
)
2
<
4det(
A
) and
r
=
β
i
and
s
=−
β
i
(
α
=
0).
(iii) At least one of the trajectories tends to inFnity as
t
. This occurs
when
(a) tr(
A
)
2
>
4det(
A
),
r
>
0 and
s
>
0or
r
<
0 and
s
>
0
(b) tr(
A
)
2
<
4det(
A
),
r
=
α
+
β
i
,
s
=
α
−
β
i
and
α>
0.
4.9 Stability/instability and its matrix specifcation
Having outlined the methods of solution for linear systems of homogeneous au
tonomous equations, it is quite clear that the characteristic roots play an important
part in these. Here we shall continue to pursue just the twovariable cases.
±or the system
˙
x
=
ax
+
by
˙
y
=
cx
+
dy
where
A
=
±
ab
cd
²
and
A
−
λ
I
=
±
a
−
λ
b
−
λ
²
we have already shown that a unique critical point exists if
A
is nonsingular, i.e.,
det(
A
)
±=
0 and that
r
,
s
=
tr(
A
)
²
³
tr(
A
)
2
−
4det(
A
)
2
(4.26)
±urthermore, if:
(i)
tr(
A
)
2
>
4det(
A
) the roots are real and distinct
(ii)
tr(
A
)
2
=
4det(
A
) the roots are real and equal
(iii)
tr(
A
)
2
<
4det(
A
) the roots are complex conjugate.
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 Fall '11
 Dr.Gwartney
 Economics

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