Economics Dynamics Problems 109

Economics Dynamics - 1,y 2 is called a periodic orbit Geometrically a k-periodic point For the discrete system y t 1 = f y t is the y-coordinate oF

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Discrete dynamic systems 93 curve but with opposite sign. Then b = d and p t = a c d ± b d ² p t 1 = ± a c d ² p t 1 or p t + 1 =− p t + k where k = a c d which is identical to the situation shown in fgure 3.4, and must produce a two-cycle result. In general, a solution y n is periodic iF y n + m = y n For some fxed integer m and all n . The smallest integer For m is called the period oF the solution. ±or example, given the linear cobweb system q d t = 10 2 p t q s t = 4 + 2 p t 1 q d t = q s t it is readily established that the price cycles between p 0 and 3– p 0 , while the quantity cycles between 4 + 2 p 0 and 10 2 p 0 (see exercise 12). In other words p 0 = p 2 = p 4 = ... and p 1 = p 3 = p 5 = ... so that y n + 2 = y n For all n and hence we have a two-cycle solution. More Formally: DEFINITION If a sequence {y t } has (say) two repeating values y 1 and y 2 , then y 1 and y 2 are called period points, and the set {y
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Unformatted text preview: 1 ,y 2 } is called a periodic orbit. Geometrically, a k-periodic point For the discrete system y t + 1 = f ( y t ) is the y-coordinate oF the point where the graph oF f k ( y ) meets the diagonal line y t + 1 = y t . Thus, a three-period cycle is where f 3 ( y ) meets the line y t + 1 = y t . In establishing the stability/instability oF period points we utilise the Following theorem. THEOREM 3.2 Let b be a k-period point of f. Then b is (i) stable if it is a stable Fxed point of f k (ii) asymptotically stable (attracting) if it is an attracting Fxed point of f k (iii) repelling if it is a repelling Fxed point of f k ....
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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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