Discrete dynamic systems97Given such a spreadsheet, it is possible to change the initial valuey0and seethe result in the sequence and on the various graphs that can be constructed.3Forinstance, consideringyt+1=3.2yt−0.8y2treadily establishes that the equilibrium value isy∗=2.75, but that this is notreached for any initial value not equal to it. For any initial value not equal to theequilibrium value, then the system will tend towards a two-cycle with values 2.05and 3.20, as can readily be established by means of a spreadsheet. It is also easyto establish that for any value slightly above or slightly below 2.75, i.e., in theneighbourhood of the equilibrium point, then the system diverges further fromthe equilibrium. In other words, the equilibrium is locally unstable. What is notapparent, however, is why the system will tend towards a two-cycle result. Weshall explain why in section 3.7.Nor should it be assumed that only a two-cycle result can arise from the logisticequation. For instance, the logistic equation
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