Economics Dynamics Problems 73

Economics Dynamics Problems 73 - x ∗ = a is an...

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Continuous dynamic systems 57 Figure 2.15. If any trajectory starting ‘close to’ a Fxed point 8 stays close to it for all future time, then the Fxed point is said to be stable . A Fxed point is asymptotically stable if it is stable as just deFned, and also if any trajectory that starts close to the Fxed point approaches the Fxed point as t →∞ . Considering the logistic equation as shown in Fgure 2.15, it is clear that
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Unformatted text preview: x ∗ = a is an asymptotically stable rest point . ±igure 2.15 also illustrates another feature of the characteristics of a Fxed point. The origin, x ∗ = 0, is a repellor while x ∗ = a is an attractor. In the neighbourhood of the origin, the differential equation has a positive slope. In the neighbourhood 8 We shall be more explicit about the meaning of ‘close to’ in section 4.2....
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