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Economics Dynamics Problems 113

Economics Dynamics Problems 113 - Discrete dynamic systems...

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Discrete dynamic systems 97 Given such a spreadsheet, it is possible to change the initial value y 0 and see the result in the sequence and on the various graphs that can be constructed. 3 For instance, considering y t + 1 = 3 . 2 y t 0 . 8 y 2 t readily establishes that the equilibrium value is y = 2 . 75, but that this is not reached for any initial value not equal to it. For any initial value not equal to the equilibrium value, then the system will tend towards a two-cycle with values 2.05 and 3.20, as can readily be established by means of a spreadsheet. It is also easy to establish that for any value slightly above or slightly below 2.75, i.e., in the neighbourhood of the equilibrium point, then the system diverges further from the equilibrium. In other words, the equilibrium is locally unstable. What is not apparent, however, is why the system will tend towards a two-cycle result. We shall explain why in section 3.7. Nor should it be assumed that only a two-cycle result can arise from the logistic equation. For instance, the logistic equation
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