Economics Dynamics Problems 241

Economics Dynamics Problems 241 - Discrete systems of...

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Discrete systems of equations 225 (the Jordan form) and the transition matrix are: J = ± 10 03 ² , V = ± 11 ² Suppose for this system ( u 10 , u 20 ) = (5 , 2), then ± u 1 t u 2 t ² = ± ²± 1 t 0 t ² 1 ± 5 2 ² i.e. u 1 t = 3 2 + 7 2 3 t u 2 t =− 3 2 + 7 2 3 t Therefore, as t increases u 1 t →+∞ and u 2 t . Turning to the canonical form, z 0 = V 1 u 0 , hence ± z 10 z 20 ² = ± 3 / 2 7 / 2 ² and z 1 t = 1 t ³ 3 2 ´ = 3 2 z 2 t = 3 t ³ 7 2 ´ Furthermore, z 2 t z 1 t = ³ 3 1 ´ t 7 / 2 3 / 2 = 3 t ³ 7 3 ´ and so for each point in the canonical phase space, the angle from the origin is increasing, and so the direction of the system is vertically upwards, as illustrated in ±gure 5.5(b). Once again, the same instability is shown in the original space, ±gure 5.5(a), and in its canonical form, ±gure 5.5(b). The fact that the trajectory in ±gure 5.5(b) is vertical arises from the fact that root r = 1. Suppose one root is greater than unity in absolute value and the other less than
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