Asymptotics for some logistic maps and the renormalization group

Abstract We explain the relation between the r = 1 case of the logistic map x i+1 = r x i (1 − x i ), x i ∈ R , i = 0, 1, 2, …, r > 0 and x 0 ≥ 0, and the renormalization group flow arising in the multiscale analysis of interesting zero fixed point, asymptotic free quantum field theory models such as the ultraviolet (1 + 1)-dimensional Gross-Neveu model and QCD, and the infrared ϕ 4 4 model . We obtain the asymptotics of the mapping, which shows an inverse power decay approach to the fixed point x * = 0, Gaussian fixed point, with additional logarithmic-like corrections. This asymptotic behavior is independent of the initial condition x 0 ∈ (0, 1) (hence, there is no constraint for x 0 to be small, as usual in quantum field models), and 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 β-function expansion in the coupling constant, up to higher orders, does not improve our knowledge of the asymptotics of the coupling flow. A comparison with a similar differential equation with continuous time is made by analyzing stability of this kind of solution and higher order monomial perturbations. We also obtain the detailed asymptotics for 0 < r < 1. As well, our methods can be applied when r ∈ (1, 3]. It is known, but without detailed asymptotics, that all trajectories with initial condition x 0 ∈ ( − 1, 1) converge to the fixed point x * = (r − 1)/r. For r = 2, the super attractive case, we obtain an explicit exact solution which exhibits an exploding, non-constant exponential decay rate approach to the x * = (1/2) fixed point. Our methods include the use of iterations, a discrete version of the Fundamental theorem of Calculus, a discrete version of the integrating factor method for first order linear ODEs and, sometimes, a scaling transformation. To obtain these results, we do not use the traditional Banach contraction mapping theorem, which only provides an upper bound on the asymptotics. We expect that our methods can be employed to determine the asymptotics of the logistic map for a wider range of parameters, where other fixed points are present.