Abstract We explain the relation between a special case of the logistic map and the renormalization group flow arising in the multiscale analysis of some interesting zero fixed point, asymptotic free quantum field theory models. Using elementary mathematics, we obtain a fine control of the asymptotic dependence of our logistic map iteration on the number of steps or time $n$. We prove that this behavior is independent of the initial state or condition, and also that it only depends on the lowest orders in a polynomial perturbation. In asymptotic free quantum field theory, this amounts to say that knowing the renormalization group $\beta$-function expansion in the coupling constant, up to higher orders, does not improve our knowledge of the asymptotics of the coupling flow. A comparison is made with a similar differential equation in continuous time.