Master equation for the logistic map

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Abstract
A master equation is constructed that provides a stochastic description underlying the logistic map. In an appropriate macroscopic limit, the underlying master map (equation) yields the logistic map. It also describes intrinsic fluctuations associated with the logistic map. When the logistic map parameters are chosen so that the map produces a chaotic trajectory, the variance of the associated fluctuations diverges. This means that the distribution function determined by the master map becomes very broad and that the logistic map no longer results from averaging with respect to the master map distribution function. Numerical examples of this behavior and its interpretation are discussed.