220 Economic Dynamics When considering the phase space of this canonical form it is useful to consider the following: ± z 1 t z 2 t ² = ± r t00 s t ²± z 10 z 20 ² Now take the ratio of z 2 / z 1 , then z 2 t z 1 t = s t z 20 r t z 10 = ³ s r ´ t µ z 20 z 10 ¶ and so the path of the system is dominated by the value/sign of s / r . 5.5 Graphing trajectories of discrete systems The mathematics of solving simultaneous equation systems is not very straightfor-ward and it is necessary to obtain the eigenvalues and the eigenvectors. However, it is possible to combine the qualitative nature of the phase plane discussed in the previous section and obtain trajectories using a spreadsheet or the recursive features of Mathematica and Maple . 5.5.1 Trajectories with Excel Example 5.8 Consider the following system of equations x t =− 5 +0 . 25 x t − 1 +0 . 4 y t − 1 y t = 10 − x t − 1 + y t − 1 x0 = 10 , y0 = 5 In cells B8 and C8 we place the initial values for x and y , namely x0 = 10 and y0 = 5. In cells B9 and C9 we place the formulas. These are
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