94
Economic Dynamics
In deriving the stability of a periodic point we require, then, to compute [
f
k
(
y
)] ,
and to do this we utilise the chain rule
[
f
k
(
y
)]
=
f
(
y
∗
1
)
f
(
y
∗
2
)
. . .
f
(
y
∗
n
)
where
y
∗
1
,
y
∗
2
, . . . ,
y
∗
k
are the
k
periodic points. For example, if
y
∗
1
and
y
∗
2
are two
periodic points of
f
2
(
y
), then
[
f
2
(
y
)]
=
f
(
y
∗
1
)
f
(
y
∗
2
)
and is asymptotically stable if
f
(
y
∗
1
)
f
(
y
∗
2
)
<
1
All other stability theorems hold in a similar fashion.
Although it is fairly easy to determine the stability/instability of linear dynamic
systems, this is not true for nonlinear systems. In particular, such systems can
create complex cycle phenomena. To illustrate, and no more than illustrate, the
morecomplexnatureofsystemsthatarisefromnonlinearity,considerthefollowing
quadratic equation
y
t
+
1
=
ay
t
−
by
2
t
First we need to establish any fixed points. It is readily established that two fixed
points arise since
y
∗
=
ay
∗
−
by
∗
2
=
ay
∗
1
−
by
∗
a
which gives two fixed points
y
∗
=
0
and
y
∗
=
a
−
1
b
The situation is illustrated in figure 3.5, where the quadratic is denoted by the graph
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 Fall '11
 Dr.Gwartney
 Economics, Equilibrium point, Nonlinear system, Stability theory, Economic Dynamics, y∗, two 12 1 2 k

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