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Unformatted text preview: 148 Economic Dynamics Figure 4.3. If a system has a ﬁxed point (x∗ , y∗ ) which is asymptotically stable, and if
every trajectory approaches the ﬁxed point (i.e. both points close to the ﬁxed point
and far away from the ﬁxed point), then the ﬁxed point is said to be globally
asymptotically stable. Another way to consider this is to establish the initial
set of conditions for which the given ﬁxed point is asymptotically stable, i.e.,
the largest ball from which any entering trajectory converges asymptotically to
the ﬁxed point. This set of initial conditions is called the basin of attraction. A
ﬁxed point is locally asymptotically stable if there exists a basin of attraction,
Bε (x∗ , y∗ ), within which all trajectories entering this ball eventually approach the
ﬁxed point (x∗ , y∗ ). If the basin of attraction is the whole of the (x,y)-plane, then
the system is globally asymptotically stable about the ﬁxed point (x∗ , y∗ ). ...
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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