Asymptotics for Some Logistic Maps and the Renormalization Group

Abstract

We explain the relation between the $r=1$ logistic map $x_{i+1}=rx_i(1-x_i)$, $x_i\in\mathbb R$, $i=0,1,\ldots$, $r>0$ and $x_0\geq0$, and the RG flow in the multiscale analysis of zero fixed point, asymptotic free QFT models as e.g. the ultraviolet (1+1)-dimensional Gross-Neveu model and QCD, and the infrared $\phi^4_4$. We obtain the asymptotics of the mapping, showing an inverse power decay to the fixed point $x^*=0$ Gaussian fixed point, with logarithmic-like corrections. This asymptotics is independent of the initial condition $x_0\in(0,1)$ (so, there is no constraint for $x_0$ to be small, as usual in QFT models), and only depends on the lowest orders in a polynomial perturbation. In asymptotic free QFT, this means that knowing the RG $\beta$ function expansion up to higher orders in the coupling does not improve our knowledge of the flow asymptotics. A comparison with an ODE with continuous time is made by analyzing stability of the solution with higher order monomial perturbations. We also obtain the asymptotics for $0<r<1$. Also, our methods work when $r\in(1,3]$. It is known, but without detailed asymptotics, that all trajectories with initial condition $x_0\in(-1,1)$ converge to the fixed point $x^*=(r-1)/r$. For the super attractive case $r=2$, we obtain an explicit exact solution with an exploding nonconstant exponential decay rate 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 1st order linear ODEs and, sometimes, a scaling transformation. We do not use the Banach contraction mapping theorem, which only provides an upper bound on the asymptotics. We expect that our methods apply to determine the asymptotics of the logistic map for a wider range of parameters, where other fixed points are present.