This preview shows page 1. Sign up to view the full content.
232
Economic Dynamics
Considering the canonical form
z
t
=
Vz
t
−
1
, then
z
t
=
J
t
z
0
=
±
(
α
+
β
i
)
t
0
0(
α
−
β
i
)
t
²
z
0
To investigate the stability properties of systems with complex conjugate roots,
we employ two results (see Simon and Blume 1994, appendix A3):
(i)
α
±
i
β
=
R
(cos
θ
±
i
sin
θ
)
(ii)
(
α
±
i
β
)
n
=
R
n
[cos(
n
θ
)
±
i
sin(
n
θ
)]
(De Moivre’s formula)
where
R
=
³
α
2
+
β
2
and
tan
θ
=
β
α
From the canonical form, and using these two results, we have
z
1
t
=
(
α
+
β
i
)
t
z
10
=
R
t
[cos(
t
θ
)
+
i
sin(
t
θ
)]
z
2
t
=
(
α
−
β
i
)
t
z
20
=
R
t
[cos(
t
θ
)
−
i
sin(
t
θ
)]
It follows that such a system must oscillate because as
t
increases, sin(
t
θ
) and
cos(
t
θ
) range between
+
1 and
−
1. Furthermore, the limit of
z
1
t
and
z
2
t
as
t
→∞
is governed by the term
´
´
R
t
´
´
= 
R

t
.If

R

<
1, then the system is an
asymptotically
stable focus
;if

R

>
1, then the system is an
unstable focus
; while if

R
 =
1, then
we have a
centre
.
6
We now illustrate each of these cases.
Example 5.13

R

This is the end of the preview. Sign up
to
access the rest of the document.
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

Click to edit the document details