Economics Dynamics Problems 199

Economics Dynamics Problems 199 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations 183 Figure 4.31. illustrate a stable limit cycle, but they also illustrate that the limit cycle shrinks as β increases. 4.11 Euler’s approximation and differential equations on a spreadsheet10 Although differential equations are for continuous time, if our main interest is the trajectory of a system over time, sometimes it is convenient to use a spreadsheet to do this. To accomplish this task we employ Euler’s approximation. For a single variable the situation is shown in figure 4.32. We have the differential equation dx = f (x, t) x(t0 ) = x0 dt Let x = φ (t) denote the unknown solution curve. At time t0 we know x0 = φ (t0 ). We also know dx/dt at t0 , which is simply f (x0 , t0 ). If we knew x = φ (t), then the value at time t1 would be φ (t1 ). But if we do not have an explicit form for x = φ (t), we can still plot φ (t) by noting that at time t0 the slope at point P is f (x0 , t0 ), which is given by the differential equation. The value of x1 at time t1 (point R) is given by x1 = x0 + f (x0 , t0 ) t (4.30) t = t1 − t0 This process can be repeated for as many steps as one wishes. If f is autonomous, so dx/dt = f (x), then xn = xn−1 + f (xn−1 ) t It is clear from figure 4.32 that point R will deviate from its ‘true’ value at point Q, the larger the step size, given by t. If the step size is reduced, then the approximation is better. 10 See Shone (2001) for a treatment of differential equations with spreadsheets. (4.31) ...
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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.

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