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Unformatted text preview: Systems of ﬁrst-order differential equations 169
Figure 4.17. Figure 4.18. through vr then c2 = 0. The solution will therefore remain on vr . Since r is positive,
then over time the solution moves away from the origin, away from the ﬁxed point.
On the other hand, if the system starts on the line through vs , then c1 = 0, and
since s < 0, then as t → ∞ the system tends towards the ﬁxed point.
For initial points off the lines through the eigenvectors, then the positive root
will dominate the system. Hence for points above vr and vs , the solution path will
veer towards the line through vr . The same is true for any initial point below vr
and above vs . On the other hand, an initial point below the line through vs will
be dominated by the larger root and the system will veer towards minus inﬁnity.
In this case the node is called a saddle point. The line through vr is called the
unstable arm, while the line through vs is called the stable arm.
Saddle path equilibria are common in economics and one should look out for
them in terms of real distinct roots of opposite sign and the fact that det(A) is
negative. It will also be important to establish the stable and unstable arms of ...
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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