The logistic map comes from concatenations of the function , where and is an argument within the domain . We'll begin by defining a function as instances of concatenated together, observing the fractional form of each incrementation of :
The denominator of each iteration is the square of the previous denominator. Using the fact that is , we can rewrite
(here just denotes the remaining sections of .)
Note that the leading power of in each term cannot be taken out in this state, due to it being concentated into . However, this option may be feasible within sums of . To derive bounds for each power, we can:
The lowest order of is , as a constant times times , and the highest order is , as times produces . Given the coefficients , we can express as
More iterations of will be needed to figure out a general formula for these coefficients.