The iterated Logistic map is widely used to demonstrate and study chaotic oscillations. As an ecological model the iterated Logistic map has some unfortunate properties. If x>1, the population is above its carrying capacity, then at the next iteration x<0.  Stephen Wolfram has proposed closed form solutions for the map for a=2 and a=4. I will demonstrate that by completing the square that the iterated Logistic map can be written such that x only occurs once in the expression.

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First the map will be algebraically rewritten so the square can be completed.

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By completing the square the logistic map can be rewritten so x will occur only a single time in the expression .

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Because x occurs once in the expression when the Logistic map is iterated the expression will have only one occurrence of x.

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Compare this to the original Logistic map iterated 4 time steps.

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One can argue which of the two expressions is simpler. Aesthetics is subjective. The advantage of an expression with a single occurrence is that with interval arithmetic the exact range of the function can be computed. See, for example, R.E. Moore (1979) Methods and Applications of Interval Analysis, SIAM.

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If interval analysis is applied to the original iterated Logistic map then the computed bounds will be overly conservative.  

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Computing bounds on the iterated Logistic map allows us to compute how long it takes for rounding of the floating point numbers to effect the results. Starting with exact knowledge x=0.5 and a=3.9 it takes 67 iterations before the bounds on the model output are non-informative.

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Created by Mathematica  (July 5, 2005)