Discrete systems of equations
229
showed that the canonical form of
u
t
=
Au
t
−
1
is
z
t
=
J
t
z
0
In the case of a repeated root this is
z
t
=
±
λ
t
t
λ
t
−
1
0
λ
t
²
z
0
Hence
z
1
t
=
λ
t
z
10
+
t
λ
t
−
1
z
20
z
2
t
=
λ
t
z
20
Therefore if

λ

<
1, then
³
³
λ
t
³
³
→
0as
t
→∞
, consequently
z
1
t
→
0 and
z
2
t
→
0
as
t
. The system is asymptotically stable. If, on the other hand,

λ

>
1, then
³
³
λ
t
³
³
as
t
, and
z
1
t
→±∞
and
z
2
t
as
t
. The system is
asymptotically unstable.
We can conclude for repeated roots, therefore, that
(a)
if

λ

<
1 the system is asymptotically stable
(b)
if

λ

>
1 the system is asymptotically unstable.
Example 5.11
Consider the following system
x
t
+
1
=
4
+
x
t
−
y
t
y
t
+
1
=−
20
+
x
t
+
3
y
t
Then
x
∗
=
12 and
y
∗
=
4. Representing the system as deviations from equilibrium,
we have
x
t
+
1
−
x
∗
=
(
x
t
−
x
∗
)
−
(
y
t
−
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.
 Fall '11
 Dr.Gwartney
 Economics

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