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Economics Dynamics Problems 110

# Economics Dynamics Problems 110 - 94 Economic Dynamics In...

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94 Economic Dynamics In deriving the stability of a periodic point we require, then, to compute [ f k ( y )] , and to do this we utilise the chain rule [ f k ( y )] = f ( y 1 ) f ( y 2 ) . . . f ( y n ) where y 1 , y 2 , . . . , y k are the k -periodic points. For example, if y 1 and y 2 are two periodic points of f 2 ( y ), then [ f 2 ( y )] = f ( y 1 ) f ( y 2 ) and is asymptotically stable if f ( y 1 ) f ( y 2 ) < 1 All other stability theorems hold in a similar fashion. Although it is fairly easy to determine the stability/instability of linear dynamic systems, this is not true for nonlinear systems. In particular, such systems can create complex cycle phenomena. To illustrate, and no more than illustrate, the morecomplexnatureofsystemsthatarisefromnonlinearity,considerthefollowing quadratic equation y t + 1 = ay t by 2 t First we need to establish any fixed points. It is readily established that two fixed points arise since y = ay by 2 = ay 1 by a which gives two fixed points y = 0 and y = a 1 b The situation is illustrated in figure 3.5, where the quadratic is denoted by the graph
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