Discrete dynamic systems
97
Given such a spreadsheet, it is possible to change the initial value
y
0
and see
the result in the sequence and on the various graphs that can be constructed.
3
For
instance, considering
y
t
+
1
=
3
.
2
y
t
−
0
.
8
y
2
t
readily establishes that the equilibrium value is
y
∗
=
2
.
75, but that this is not
reached for any initial value not equal to it. For any initial value not equal to the
equilibrium value, then the system will tend towards a twocycle with values 2.05
and 3.20, as can readily be established by means of a spreadsheet. It is also easy
to establish that for any value slightly above or slightly below 2.75, i.e., in the
neighbourhood of the equilibrium point, then the system diverges further from
the equilibrium. In other words, the equilibrium is locally unstable. What is not
apparent, however, is why the system will tend towards a twocycle result. We
shall explain why in section 3.7.
Nor should it be assumed that only a twocycle result can arise from the logistic
equation. For instance, the logistic equation
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 Fall '11
 Dr.Gwartney
 Economics, initial value, Stability theory

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