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Unformatted text preview: 194 Economic Dynamics Figure 4.38. The resulting phase line is shown in ﬁgure 4.38. This goes beyond the possibilities of a spreadsheet, and ﬁgure 4.38 should be compared with the three twodimensional plots given in ﬁgure 4.35. It is worth noting that ﬁgure 4.38 is the
default plot and the orientation can readily be changed by clicking on the ﬁgure
and revolving. Appendix 4.1 Parametric plots in the phase plane:
continuous variables
A trajectory or orbit is the path of points {x(t), y(t)} in 2dimensional space and
{x(t), y(t), z(t)} in 3dimensional space as t varies. Such plots are simply parametric
plots as far as computer programmes are concerned. There are two methods for
deriving the points (x(t), y(t)) or (x(t), y(t), z(t)):
(1)
(2) Solve for these values
Derive numerical values by numerical means:
(a) by solving numerically, or
(b) deriving by recursion. Method 2(a) is used particularly in the case of differential equations, while method
2(b) is used for difference (or recursive) equations. In each of these cases initial
conditions must be supplied. 4A.1 Twovariable case
Consider the solution values for x and y in example 4.1, which are
x(t) = 2e2t and y(t) = 3et Both x and y are expressed in terms of a common parameter, t, so that when t
varies we can establish how x and y vary. More speciﬁcally, if t denotes time, then
(x(t), y(t)) denotes a point at time t in the (x,y)plane, i.e., a Cartesian representation
of the parametric point at time t. If the differential equation system which generated
x(t) and y(t) is autonomous, then there is only one solution curve, and we can
express this in the form y = φ (x), where y0 = φ (x0 ) and (x0 , y0 ) is some initial
point, i.e., x(0) = x0 and y(0) = y0 at t = 0. In the present example this is readily ...
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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
 Dr.Gwartney
 Economics

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