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Unformatted text preview: 152 Economic Dynamics
the forces are directing the system away from the ﬁxed point. We can immediately
conclude, therefore, that the ﬁxed point cannot be a stable point. Can we conclude
that for any initial value of x and y, positioning the system in quadrants I or III,
that the trajectory will tend over time to the ﬁxed point? No, we cannot make
any such deduction! For instance, if the system began in quadrant I, and began to
move towards the ﬁxed point, it could over time pass into quadrant IV, and once
in quadrant IV would move away from the ﬁxed point. In fact, this is precisely the
trajectory shown in ﬁgure 4.5. Although the trajectory shown in ﬁgure 4.5 moves
from quadrant I into quadrant IV, this need not be true of all initial points beginning
in quadrant I. Depending on the initial value for x and y, it is quite possible for the
system to move from quadrant I into quadrant II, ﬁrst moving towards the ﬁxed
point and then away from it once quadrant II has been entered. This would be the
situation, for example, if the initial point was (x0 , y0 ) = (4, 2) (see exercise 2).
This complex nature of the solution paths can be observed by considering the
direction ﬁeld for the differential equation system. The direction ﬁeld, along with
the equilibrium lines are shown in ﬁgure 4.7. Why the dynamic forces seem to
operate in this way we shall investigate later in this chapter.
Example 4.5
The following system of linear differential equations
˙
x = −3x + y
˙
y = x − 3y
with initial condition x0 = 4 and y0 = 5 has solution equations
x(t) = 9e2t − 1
2e4t and y(t) = 1 + 9e2t
2e4t which gives rise to a trajectory which approaches the ﬁxed point (x∗ , y∗ ) = (0, 0),
as shown in ﬁgure 4.8. The path of x(t) and y(t), represented by the phase line, as
t increases is shown by the direction of the arrows.
Figure 4.7. ...
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 Fall '11
 Dr.Gwartney
 Economics

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