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Economics Dynamics Problems 193

# Economics Dynamics Problems 193 - Systems of ﬁrst-order...

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Unformatted text preview: Systems of ﬁrst-order differential equations 177 Figure 4.27. of the cycle, which is 2π/β . The critical point is called the centre. These situations are illustrated in ﬁgure 4.27. Summary From the ﬁve cases discussed we arrive at a number of observations. 1. 2. 3. After a sufﬁcient time interval, the trajectory of the system tends towards three types of behaviour: (i) the trajectory approaches inﬁnity (ii) the trajectory approaches the critical point (iii) the trajectory traverses a closed curve surrounding the critical point. Through each point (x0 , y0 ) in the phase plane there is only one trajectory. Considering the set of all trajectories, then three possibilities arise: (i) All trajectories approach the critical point. This occurs when (a) tr(A)2 > 4det(A), r < s < 0 (b) tr(A)2 < 4det(A), r = α + β i, s = α − β i and α < 0. ...
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